Chapter 7 Fractions (Additional Questions)

The correct option is (A) Whole.

A fraction is fundamentally a way of representing a portion or a part of something complete, which we call a whole. It is typically written in the form $\frac{a}{b}$, where $a$ is the numerator and $b$ is the denominator.

The denominator ($b$) tells us into how many equal parts the whole has been divided.

The numerator ($a$) tells us how many of those equal parts we are considering or taking.

For example, if you have a whole pizza divided into 8 equal slices, and you eat 3 of those slices, the fraction representing the part you ate is $\frac{3}{8}$. Here, the whole is the entire pizza, it's divided into 8 parts, and you took 3 parts.

Similarly, if you have a collection of 10 marbles, and 4 of them are red, the fraction of red marbles is $\frac{4}{10}$. Here, the whole is the group of 10 marbles, and we are considering 4 of them.

Let's consider the other options:

• (B) Point: A point is a location in space and has no dimension or parts, so a fraction cannot represent a part of a point.
• (C) Line: A line is a one-dimensional figure that extends infinitely in both directions. While a line segment can be considered a "whole" length and divided into parts, the term "whole" is a more general concept that applies broadly to objects, quantities, or collections, whereas a line is a specific geometric concept.
• (D) Number line: A number line is a graphical representation used to order numbers, including fractions. Fractions can be located or plotted on a number line to show their value and position relative to whole numbers and other fractions. However, the number line is a tool for visualizing fractions, not the fundamental concept that a fraction represents a part of.

Therefore, the most accurate and fundamental description of what a fraction represents is a part of a whole.

The correct option is (B) 5.

In a fraction of the form $\frac{a}{b}$, $a$ is called the numerator and $b$ is called the denominator.

In the given fraction $\frac{5}{9}$, the number on the top is 5.

Therefore, the numerator is 5.

The correct option is (B) $\frac{3}{4}$.

To find the fraction representing the shaded part, we need to determine two values:

1. The total number of equal parts the whole is divided into.
2. The number of those parts that are shaded.

From the image description, we are told:

• The circle is divided into 4 equal parts.
• 3 parts are shaded.

The total number of equal parts is the denominator, which is 4.

The number of shaded parts is the numerator, which is 3.

Thus, the fraction representing the shaded part is $\frac{\text{Number of shaded parts}}{\text{Total number of equal parts}} = \frac{3}{4}$.

The correct option is (B) A point between 0 and 1, dividing the unit length into 5 equal parts and taking the 2nd part.

To represent a fraction $\frac{a}{b}$ on a number line, where $a < b$ (a proper fraction), we focus on the unit length from 0 to 1.

The denominator ($b$) tells us into how many equal parts the unit length (from 0 to 1) must be divided.

The numerator ($a$) tells us how many of these equal parts we need to move from 0 towards 1.

In the fraction $\frac{2}{5}$:

• The denominator is 5, which means the unit length between 0 and 1 is divided into 5 equal parts.
• The numerator is 2, which means we count 2 parts from 0.

So, the point representing $\frac{2}{5}$ is located at the end of the second equal part when the segment from 0 to 1 is divided into five equal parts.

Let's visualize this:

We start at 0. We divide the distance between 0 and 1 into 5 equal segments. The endpoints of these segments represent the fractions $\frac{1}{5}, \frac{2}{5}, \frac{3}{5}, \frac{4}{5},$ and finally $\frac{5}{5} = 1$. The fraction $\frac{2}{5}$ is the second mark from 0.

Comparing this with the options, option (B) accurately describes this process.

The correct option is (C) $\frac{3}{8}$.

A proper fraction is a fraction where the numerator is less than the denominator.

Let's look at each option:

• (A) $\frac{7}{5}$: The numerator (7) is greater than the denominator (5). This is an improper fraction.
• (B) $\frac{9}{9}$: The numerator (9) is equal to the denominator (9). This is an improper fraction (or equal to 1).
• (C) $\frac{3}{8}$: The numerator (3) is less than the denominator (8). This fits the definition of a proper fraction.
• (D) $1\frac{1}{2}$: This is a mixed number. To check if it's a proper or improper fraction, we convert it to an improper fraction: $1\frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2}$. The numerator (3) is greater than the denominator (2). This is an improper fraction.

Therefore, only $\frac{3}{8}$ is a proper fraction.

The correct option is (B) $3\frac{2}{3}$.

To convert an improper fraction (where the numerator is greater than or equal to the denominator) into a mixed fraction, we divide the numerator by the denominator.

The improper fraction given is $\frac{11}{3}$.

We divide 11 by 3:

$11 \div 3$

Using division, we find:

3 goes into 11 three times ($3 \times 3 = 9$).

The remainder is $11 - 9 = 2$.

The result of the division can be written as:

$11 = 3 \times 3 + 2$

The mixed fraction is formed using the quotient as the whole number, the remainder as the new numerator, and the original denominator.

Mixed fraction = $\text{Quotient}\frac{\text{Remainder}}{\text{Denominator}}$

Here:

• Quotient = 3
• Remainder = 2
• Denominator = 3

So, the mixed fraction is $3\frac{2}{3}$.

The correct option is (A) $\frac{11}{4}$.

To convert a mixed fraction into an improper fraction, we follow these steps:

1. Multiply the whole number part by the denominator of the fractional part.
2. Add the numerator of the fractional part to the result from step 1.
3. The sum from step 2 becomes the numerator of the improper fraction.
4. The denominator remains the same as the denominator of the fractional part in the mixed fraction.

The given mixed fraction is $2\frac{3}{4}$.

Here:

• Whole number = 2
• Numerator of fractional part = 3
• Denominator of fractional part = 4

Step 1: Multiply the whole number by the denominator: $2 \times 4 = 8$.

Step 2: Add the numerator to the result: $8 + 3 = 11$.

Step 3: This sum (11) is the new numerator of the improper fraction.

Step 4: The denominator remains the same (4).

So, the improper fraction is $\frac{11}{4}$.

The correct option is (C) $\frac{8}{12}$.

Equivalent fractions represent the same value, even though they have different numerators and denominators. To find an equivalent fraction, you multiply or divide both the numerator and the denominator by the same non-zero number.

We need to check which of the given options is equal to $\frac{2}{3}$. We can do this by simplifying the options or by seeing if we can multiply the numerator and denominator of $\frac{2}{3}$ by some number to get one of the options.

Let's check each option:

• (A) $\frac{4}{5}$: To get 4 from 2, we multiply by 2. If we multiply the denominator 3 by 2, we get 6. Since the denominator is 5, $\frac{4}{5}$ is not equivalent to $\frac{2}{3}$.
• (B) $\frac{6}{8}$: To get 6 from 2, we multiply by 3. If we multiply the denominator 3 by 3, we get 9. Since the denominator is 8, $\frac{6}{8}$ is not equivalent to $\frac{2}{3}$. Alternatively, simplifying $\frac{6}{8}$ by dividing both by 2 gives $\frac{3}{4}$, which is not equal to $\frac{2}{3}$.
• (C) $\frac{8}{12}$: To get 8 from 2, we multiply by 4. If we multiply the denominator 3 by 4, we get 12. Since the denominator is 12, $\frac{8}{12}$ is equivalent to $\frac{2}{3}$. $\frac{2 \times 4}{3 \times 4} = \frac{8}{12}$. Alternatively, simplifying $\frac{8}{12}$ by dividing both by their greatest common divisor, 4, gives $\frac{\cancel{8}^{2}}{\cancel{12}_{3}} = \frac{2}{3}$.
• (D) $\frac{10}{18}$: To get 10 from 2, we multiply by 5. If we multiply the denominator 3 by 5, we get 15. Since the denominator is 18, $\frac{10}{18}$ is not equivalent to $\frac{2}{3}$. Alternatively, simplifying $\frac{10}{18}$ by dividing both by 2 gives $\frac{5}{9}$, which is not equal to $\frac{2}{3}$.

Therefore, $\frac{8}{12}$ is the only fraction equivalent to $\frac{2}{3}$.

The correct option is (A) $\frac{3}{4}$.

To reduce a fraction to its lowest terms (also called simplifying a fraction), we need to divide both the numerator and the denominator by their greatest common divisor (GCD).

The given fraction is $\frac{18}{24}$.

We need to find the GCD of 18 and 24.

We can find the factors of each number:

• Factors of 18: 1, 2, 3, 6, 9, 18
• Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

The common factors are 1, 2, 3, and 6. The greatest common divisor is 6.

Now, we divide both the numerator and the denominator by the GCD (6):

Numerator: $18 \div 6 = 3$

Denominator: $24 \div 6 = 4$

The simplified fraction is $\frac{3}{4}$.

Alternatively, we can divide by common factors repeatedly until the fraction cannot be simplified further:

$\frac{18}{24}$ (Divide both by 2) = $\frac{9}{12}$

$\frac{9}{12}$ (Divide both by 3) = $\frac{3}{4}$

Since the only common factor of 3 and 4 is 1, the fraction $\frac{3}{4}$ is in its lowest terms.

The correct option is (B) $\frac{3}{7} < \frac{5}{7}$.

To compare fractions with the same denominator (like fractions), we only need to compare their numerators.

The fractions are $\frac{3}{7}$ and $\frac{5}{7}$.

Both fractions have the same denominator, which is 7.

We compare the numerators: 3 and 5.

Since $3$ is less than $5$ ($3 < 5$), the fraction with the numerator 3 is less than the fraction with the numerator 5.

Therefore, $\frac{3}{7} < \frac{5}{7}$.

The correct option is (A) $\frac{2}{5} > \frac{2}{7}$.

To compare fractions with the same numerator (like fractions), we compare their denominators. When the numerators are the same, the fraction with the smaller denominator is greater.

The fractions are $\frac{2}{5}$ and $\frac{2}{7}$.

Both fractions have the same numerator, which is 2.

We compare the denominators: 5 and 7.

Since $5$ is less than $7$ ($5 < 7$), the fraction with the denominator 5 is greater than the fraction with the denominator 7.

Therefore, $\frac{2}{5} > \frac{2}{7}$.

Think of it this way: if you divide a whole into 5 equal parts, each part is larger than if you divide the same whole into 7 equal parts. Since you are taking 2 parts in both cases, taking 2 larger parts ($\frac{2}{5}$) results in a larger total amount than taking 2 smaller parts ($\frac{2}{7}$).

The correct option is (B) $\frac{3}{4}$.

To add fractions with the same denominator (like fractions), we add the numerators and keep the denominator the same.

The given expression is $\frac{1}{4} + \frac{2}{4}$.

The denominators are both 4.

We add the numerators: $1 + 2 = 3$.

We keep the denominator as 4.

So, the sum is $\frac{3}{4}$.

Mathematically:

$\frac{1}{4} + \frac{2}{4} = \frac{1+2}{4} = \frac{3}{4}$

The correct option is (C) $\frac{5}{6}$.

To add fractions with different denominators (unlike fractions), we first need to find a common denominator. The least common multiple (LCM) of the denominators is usually the easiest to use.

The given expression is $\frac{1}{2} + \frac{1}{3}$.

The denominators are 2 and 3.

The LCM of 2 and 3 is 6.

Now, we convert each fraction into an equivalent fraction with a denominator of 6:

• For $\frac{1}{2}$: We need to multiply the denominator 2 by 3 to get 6. We must also multiply the numerator by the same number: $\frac{1 \times 3}{2 \times 3} = \frac{3}{6}$.
• For $\frac{1}{3}$: We need to multiply the denominator 3 by 2 to get 6. We must also multiply the numerator by the same number: $\frac{1 \times 2}{3 \times 2} = \frac{2}{6}$.

Now we can add the equivalent fractions, which have the same denominator:

$\frac{3}{6} + \frac{2}{6} = \frac{3+2}{6} = \frac{5}{6}$

The resulting fraction $\frac{5}{6}$ is in its lowest terms because the only common factor of 5 and 6 is 1.

Therefore, $\frac{1}{2} + \frac{1}{3} = \frac{5}{6}$.

The correct option is (D) $\frac{3}{6}$.

To subtract fractions with different denominators (unlike fractions), we first need to find a common denominator. The least common multiple (LCM) of the denominators is the most efficient common denominator.

The given expression is $\frac{5}{6} - \frac{1}{3}$.

The denominators are 6 and 3.

We find the LCM of 6 and 3:

The multiples of 3 are 3, 6, 9, ...

The multiples of 6 are 6, 12, 18, ...

The least common multiple of 6 and 3 is 6.

Now, we convert each fraction into an equivalent fraction with a denominator of 6:

• The first fraction, $\frac{5}{6}$, already has a denominator of 6, so it remains as $\frac{5}{6}$.
• For the second fraction, $\frac{1}{3}$: We need to multiply the denominator 3 by 2 to get 6 ($3 \times 2 = 6$). We must also multiply the numerator by the same number: $\frac{1 \times 2}{3 \times 2} = \frac{2}{6}$.

Now we can subtract the equivalent fractions, which have the same denominator:

$\frac{5}{6} - \frac{2}{6}$

To subtract fractions with the same denominator, we subtract the numerators and keep the denominator the same:

$\frac{5 - 2}{6} = \frac{3}{6}$

The resulting fraction is $\frac{3}{6}$. This fraction can be simplified by dividing both numerator and denominator by their greatest common divisor, which is 3, giving $\frac{1}{2}$. However, option (D) lists the result as $\frac{3}{6}$, which is mathematically correct before simplification.

Therefore, $\frac{5}{6} - \frac{1}{3} = \frac{3}{6}$.

The correct option is (A) $3\frac{3}{4}$.

To simplify the addition of mixed fractions, we can add the whole number parts and the fractional parts separately.

The given expression is $1\frac{1}{4} + 2\frac{1}{2}$.

We can rewrite this as: $(1 + \frac{1}{4}) + (2 + \frac{1}{2})$

Group the whole numbers and the fractions:

$(1 + 2) + (\frac{1}{4} + \frac{1}{2})$

First, add the whole numbers:

$1 + 2 = 3$

Next, add the fractional parts: $\frac{1}{4} + \frac{1}{2}$. These fractions have different denominators (4 and 2). We need to find a common denominator. The least common multiple (LCM) of 4 and 2 is 4.

Convert $\frac{1}{2}$ to an equivalent fraction with a denominator of 4:

Multiply the numerator and the denominator by 2: $\frac{1 \times 2}{2 \times 2} = \frac{2}{4}$.

Now add the equivalent fractions:

$\frac{1}{4} + \frac{2}{4} = \frac{1+2}{4} = \frac{3}{4}$

Finally, combine the sum of the whole numbers and the sum of the fractions:

$3 + \frac{3}{4} = 3\frac{3}{4}$

Thus, $1\frac{1}{4} + 2\frac{1}{2} = 3\frac{3}{4}$.

Alternate Method: Convert to Improper Fractions

Alternatively, we can convert the mixed fractions to improper fractions first and then add them.

$1\frac{1}{4} = \frac{(1 \times 4) + 1}{4} = \frac{5}{4}$

$2\frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{5}{2}$

Now, add the improper fractions: $\frac{5}{4} + \frac{5}{2}$. The common denominator is 4.

Convert $\frac{5}{2}$ to an equivalent fraction with a denominator of 4: $\frac{5 \times 2}{2 \times 2} = \frac{10}{4}$.

Add the improper fractions:

$\frac{5}{4} + \frac{10}{4} = \frac{5+10}{4} = \frac{15}{4}$

Finally, convert the improper fraction $\frac{15}{4}$ back to a mixed fraction by dividing 15 by 4.

$15 \div 4 = 3$ with a remainder of 3.

So, $\frac{15}{4} = 3\frac{3}{4}$.

Both methods yield the same result, $3\frac{3}{4}$.

The correct option is (B) $\frac{3}{10}$.

To multiply fractions, we multiply the numerators together and multiply the denominators together.

The given expression is $\frac{3}{4} \times \frac{2}{5}$.

Multiply the numerators: $3 \times 2 = 6$.

Multiply the denominators: $4 \times 5 = 20$.

The product is $\frac{6}{20}$.

We can reduce this fraction to its lowest terms by dividing both the numerator and denominator by their greatest common divisor, which is 2.

$\frac{\cancel{6}^{3}}{\cancel{20}_{10}} = \frac{3}{10}$

Alternatively, we can cancel common factors before multiplying:

$\frac{3}{\cancel{4}_{2}} \times \frac{\cancel{2}^{1}}{5} = \frac{3 \times 1}{2 \times 5} = \frac{3}{10}$

Both methods give the simplified answer $\frac{3}{10}$. Option (A) shows the product before simplification, which is also mathematically correct, but option (B) shows the simplified form.

The correct option is (B) $\frac{8}{7}$.

The reciprocal of a fraction is found by flipping the fraction, i.e., swapping the numerator and the denominator.

The given fraction is $\frac{7}{8}$.

The numerator is 7 and the denominator is 8.

To find the reciprocal, we make the numerator the new denominator and the denominator the new numerator.

New numerator = Original denominator = 8

New denominator = Original numerator = 7

The reciprocal of $\frac{7}{8}$ is $\frac{8}{7}$.

A property of reciprocals is that when a fraction is multiplied by its reciprocal, the result is 1:

$\frac{7}{8} \times \frac{8}{7} = \frac{7 \times 8}{8 \times 7} = \frac{56}{56} = 1$

The correct option is (C) $\frac{6}{5}$.

To divide one fraction by another, we multiply the first fraction by the reciprocal of the second fraction.

The given expression is $\frac{4}{5} \div \frac{2}{3}$.

The first fraction is $\frac{4}{5}$.

The second fraction is $\frac{2}{3}$.

The reciprocal of the second fraction ($\frac{2}{3}$) is $\frac{3}{2}$.

Now, we multiply the first fraction by the reciprocal of the second fraction:

$\frac{4}{5} \div \frac{2}{3} = \frac{4}{5} \times \frac{3}{2}$

Multiply the numerators and the denominators:

Numerator: $4 \times 3 = 12$

Denominator: $5 \times 2 = 10$

The product is $\frac{12}{10}$.

We can reduce this fraction to its lowest terms by dividing both the numerator and denominator by their greatest common divisor, which is 2.

$\frac{\cancel{12}^{6}}{\cancel{10}_{5}} = \frac{6}{5}$

Alternatively, we can cancel common factors before multiplying:

$\frac{\cancel{4}^{2}}{5} \times \frac{3}{\cancel{2}_{1}} = \frac{2 \times 3}{5 \times 1} = \frac{6}{5}$

The result of the division is $\frac{6}{5}$.

The correct option is (B) $\frac{3}{5}$.

To find the total fraction of the cake they ate together, we need to add the fraction Raghav ate and the fraction Simran ate.

Fraction eaten by Raghav = $\frac{2}{5}$

Fraction eaten by Simran = $\frac{1}{5}$

Total fraction eaten = Fraction by Raghav + Fraction by Simran

Total fraction eaten = $\frac{2}{5} + \frac{1}{5}$

Since the fractions have the same denominator (5), we can add the numerators directly and keep the denominator the same.

Total fraction eaten = $\frac{2 + 1}{5} = \frac{3}{5}$

Therefore, Raghav and Simran ate $\frac{3}{5}$ of the cake together.

The correct option is (A) (i)-(c), (ii)-(a), (iii)-(b), (iv)-(d).

Let's match each type of fraction with its correct description:

• (i) Proper Fraction: A proper fraction is one where the numerator is less than the denominator. This matches description (c) Numerator is less than the denominator.
• (ii) Improper Fraction: An improper fraction is one where the numerator is greater than or equal to the denominator. This matches description (a) Numerator is greater than or equal to the denominator.
• (iii) Mixed Fraction: A mixed fraction is a combination of a whole number and a proper fraction. This matches description (b) Contains a whole number and a proper fraction.
• (iv) Unit Fraction: A unit fraction is a fraction where the numerator is 1 and the denominator is a positive integer. This matches description (d) Numerator is 1.

Based on these matches, the correct pairing is:

• (i) $\rightarrow$ (c)
• (ii) $\rightarrow$ (a)
• (iii) $\rightarrow$ (b)
• (iv) $\rightarrow$ (d)

This corresponds to option (A).

The correct option is (D) A is false but R is true.

Let's analyze the Assertion (A) and the Reason (R).

Assertion (A): The fraction $\frac{4}{6}$ is in its lowest terms.

A fraction is in its lowest terms when the only common factor between the numerator and the denominator is 1. Let's find the factors of the numerator (4) and the denominator (6):

• Factors of 4: 1, 2, 4
• Factors of 6: 1, 2, 3, 6

The common factors of 4 and 6 are 1 and 2. Since the common factor 2 is greater than 1, the fraction $\frac{4}{6}$ is not in its lowest terms. It can be simplified by dividing both the numerator and the denominator by their greatest common factor.

So, Assertion (A) is false.

Reason (R): The HCF of 4 and 6 is 2.

As determined when evaluating Assertion (A), the common factors of 4 and 6 are 1 and 2. The Highest Common Factor (HCF) is the largest of these common factors, which is 2.

So, Reason (R) is true.

Since Assertion (A) is false and Reason (R) is true, option (D) is the correct answer.

The correct options are (A) $\frac{2}{4}$, (B) $\frac{3}{6}$, and (C) $\frac{5}{10}$.

To find fractions equivalent to $\frac{1}{2}$, we can check if each given fraction can be simplified to $\frac{1}{2}$ or if it can be obtained by multiplying the numerator and denominator of $\frac{1}{2}$ by the same non-zero integer.

A fraction is equivalent to $\frac{1}{2}$ if its numerator is half of its denominator.

Let's check each option:

• (A) $\frac{2}{4}$: The numerator is 2, the denominator is 4. Is 2 half of 4? Yes, $2 = \frac{1}{2} \times 4$. Simplifying $\frac{2}{4}$ by dividing both numerator and denominator by their GCD, which is 2: $\frac{\cancel{2}^{1}}{\cancel{4}_{2}} = \frac{1}{2}$. This fraction is equivalent to $\frac{1}{2}$.
• (B) $\frac{3}{6}$: The numerator is 3, the denominator is 6. Is 3 half of 6? Yes, $3 = \frac{1}{2} \times 6$. Simplifying $\frac{3}{6}$ by dividing both numerator and denominator by their GCD, which is 3: $\frac{\cancel{3}^{1}}{\cancel{6}_{2}} = \frac{1}{2}$. This fraction is equivalent to $\frac{1}{2}$.
• (C) $\frac{5}{10}$: The numerator is 5, the denominator is 10. Is 5 half of 10? Yes, $5 = \frac{1}{2} \times 10$. Simplifying $\frac{5}{10}$ by dividing both numerator and denominator by their GCD, which is 5: $\frac{\cancel{5}^{1}}{\cancel{10}_{2}} = \frac{1}{2}$. This fraction is equivalent to $\frac{1}{2}$.
• (D) $\frac{6}{10}$: The numerator is 6, the denominator is 10. Is 6 half of 10? No, $6 \neq \frac{1}{2} \times 10$. Simplifying $\frac{6}{10}$ by dividing both numerator and denominator by their GCD, which is 2: $\frac{\cancel{6}^{3}}{\cancel{10}_{5}} = \frac{3}{5}$. This fraction is not equivalent to $\frac{1}{2}$.

Therefore, the fractions equivalent to $\frac{1}{2}$ are $\frac{2}{4}$, $\frac{3}{6}$, and $\frac{5}{10}$.

The correct option is (C) 8 metres.

Given:

• Total length of the ribbon = 10 metres
• Fraction of the ribbon cut off = $\frac{1}{5}$

To Find:

The length of the remaining ribbon.

Solution:

First, we need to calculate the length of the ribbon that was cut off. This is $\frac{1}{5}$ of the total length.

Length of ribbon cut off = $\frac{1}{5}$ of 10 metres

Length of ribbon cut off = $\frac{1}{5} \times 10$ metres

Length of ribbon cut off = $\frac{1 \times 10}{5}$ metres

Length of ribbon cut off = $\frac{10}{5}$ metres

Length of ribbon cut off = $2$ metres

Now, we need to find the length of the remaining ribbon. We subtract the length that was cut off from the total length.

Remaining length = Total length - Length cut off

Remaining length = $10$ metres - $2$ metres

Remaining length = $8$ metres

So, the length of the remaining ribbon is 8 metres.

The correct option is (D) Both (B) and (C).

Given:

• The product of two fractions is 1.
• One fraction is $\frac{5}{8}$.

To Find:

The other fraction.

Solution:

If the product of two numbers (including fractions) is 1, then the two numbers are called reciprocals of each other.

To find the reciprocal of a fraction, we swap its numerator and denominator.

The given fraction is $\frac{5}{8}$.

The reciprocal of $\frac{5}{8}$ is $\frac{8}{5}$.

Let the other fraction be $x$. According to the problem:

$\frac{5}{8} \times x = 1$

To find $x$, we can multiply both sides of the equation by the reciprocal of $\frac{5}{8}$:

$x = 1 \div \frac{5}{8}$

Dividing by a fraction is the same as multiplying by its reciprocal:

$x = 1 \times \frac{8}{5}$

$x = \frac{8}{5}$

So, the other fraction is $\frac{8}{5}$.

Now let's look at the options:

• (A) $\frac{5}{8}$: This is the same as the given fraction. The product of $\frac{5}{8} \times \frac{5}{8} = \frac{25}{64}$, which is not 1.
• (B) $\frac{8}{5}$: This is the reciprocal we found. The product of $\frac{5}{8} \times \frac{8}{5} = \frac{40}{40} = 1$.
• (C) $1\frac{3}{5}$: This is a mixed fraction. Let's convert it to an improper fraction. $1\frac{3}{5} = \frac{(1 \times 5) + 3}{5} = \frac{5+3}{5} = \frac{8}{5}$. This is the same fraction as option (B). The product of $\frac{5}{8} \times 1\frac{3}{5} = \frac{5}{8} \times \frac{8}{5} = \frac{40}{40} = 1$.
• (D) Both (B) and (C): Since both option (B) and option (C) represent the fraction $\frac{8}{5}$, which is the correct other fraction, this option is correct.

Therefore, the other fraction is $\frac{8}{5}$, which is represented by both options (B) and (C).

The correct option is (D) 4.

The fraction $\frac{1}{4}$ represents one out of four equal parts of a whole.

To make up a complete whole (which can be represented by the number 1), you need all the equal parts that the whole was divided into.

If the whole is divided into 4 equal parts, and each part is $\frac{1}{4}$, then you need 4 of these $\frac{1}{4}$ parts to form the whole.

Mathematically, this question asks how many times $\frac{1}{4}$ fits into 1. This can be solved by division:

$1 \div \frac{1}{4}$

To divide by a fraction, we multiply by its reciprocal. The reciprocal of $\frac{1}{4}$ is $\frac{4}{1}$ or 4.

$1 \div \frac{1}{4} = 1 \times 4 = 4$

So, there are 4 quarters ($\frac{1}{4}$ parts) in a whole.

The correct option is (B) 2.

To multiply a mixed fraction by a proper fraction, we first convert the mixed fraction into an improper fraction.

The given expression is $2\frac{1}{3} \times \frac{6}{7}$.

Convert the mixed fraction $2\frac{1}{3}$ to an improper fraction:

$2\frac{1}{3} = \frac{(2 \times 3) + 1}{3} = \frac{6 + 1}{3} = \frac{7}{3}$

Now, the multiplication becomes:

$\frac{7}{3} \times \frac{6}{7}$

To multiply fractions, we multiply the numerators and the denominators. We can also cancel common factors before multiplying to simplify the calculation.

Notice that there is a 7 in the numerator and a 7 in the denominator. Also, there is a 3 in the denominator and a 6 in the numerator (which is a multiple of 3).

$\frac{\cancel{7}^{1}}{\cancel{3}_{1}} \times \frac{\cancel{6}^{2}}{\cancel{7}_{1}}$

Now multiply the remaining terms:

$\frac{1}{1} \times \frac{2}{1} = \frac{1 \times 2}{1 \times 1} = \frac{2}{1} = 2$

So, $2\frac{1}{3} \times \frac{6}{7} = 2$.

Alternate Method: Multiply first, then simplify

Convert $2\frac{1}{3}$ to $\frac{7}{3}$.

Multiply the improper fraction by the proper fraction:

$\frac{7}{3} \times \frac{6}{7} = \frac{7 \times 6}{3 \times 7} = \frac{42}{21}$

Now simplify the resulting fraction $\frac{42}{21}$ by dividing both the numerator and denominator by their greatest common divisor, which is 21.

$\frac{\cancel{42}^{2}}{\cancel{21}_{1}} = \frac{2}{1} = 2$

Both methods confirm that the simplified result is 2.

The correct option is (C) Reciprocal.

Dividing a fraction by another fraction is equivalent to multiplying the first fraction by the multiplicative inverse of the second fraction.

For fractions, the multiplicative inverse is also called the reciprocal.

To find the reciprocal of a fraction, you swap its numerator and its denominator.

For example, to calculate $\frac{a}{b} \div \frac{c}{d}$, the rule is:

$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \text{Reciprocal of } \frac{c}{d}$

The reciprocal of $\frac{c}{d}$ is $\frac{d}{c}$.

So, $\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$.

Therefore, to divide a fraction by another fraction, we multiply the first fraction by the Reciprocal of the second fraction.

The correct option is (A) $\frac{1}{2}$.

To compare fractions with the same numerator (like fractions), we look at their denominators. When the numerators are the same, the fraction with the smaller denominator is the largest.

The given fractions are $\frac{1}{2}$, $\frac{1}{3}$, $\frac{1}{4}$, and $\frac{1}{5}$.

All these fractions have the same numerator, which is 1. These are unit fractions.

We compare the denominators: 2, 3, 4, and 5.

The smallest denominator among these is 2.

According to the rule for comparing fractions with the same numerator, the fraction with the smallest denominator is the largest.

Therefore, $\frac{1}{2}$ is the largest fraction among the given options.

Think about dividing a whole: dividing a whole into 2 parts means each part is bigger than dividing it into 3, 4, or 5 parts. $\frac{1}{2}$ is half of the whole, $\frac{1}{3}$ is one-third, $\frac{1}{4}$ is one-fourth, and $\frac{1}{5}$ is one-fifth. Half is the largest share among these.

The correct option is (B) $\textsf{₹}$ 300.

Given:

• Total amount to be distributed = $\textsf{₹}$ 1000
• Fraction of the total amount Anand gets = $\frac{3}{10}$

To Find:

The amount of money Anand gets.

Solution:

To find the amount Anand gets, we need to calculate $\frac{3}{10}$ of $\textsf{₹}$ 1000.

Amount Anand gets = $\frac{3}{10}$ of $\textsf{₹}$ 1000

In mathematics, "of" means multiplication.

Amount Anand gets = $\frac{3}{10} \times 1000$

We can write 1000 as $\frac{1000}{1}$.

Amount Anand gets = $\frac{3}{10} \times \frac{1000}{1}$

We can cancel common factors before multiplying. 10 is a common factor of 10 and 1000.

$\frac{3}{\cancel{10}_{1}} \times \frac{\cancel{1000}^{100}}{1}$

Now multiply the remaining terms:

Amount Anand gets = $\frac{3 \times 100}{1 \times 1}$

Amount Anand gets = $\frac{300}{1}$

Amount Anand gets = $300$

So, Anand gets $\textsf{₹}$ 300.

Alternatively, $\frac{3}{10}$ means 3 parts out of 10 equal parts.

First, find the value of one part: $\textsf{₹} 1000 \div 10 = \textsf{₹} 100$.

Since Anand gets 3 of these parts: $3 \times \textsf{₹} 100 = \textsf{₹} 300$.

Both methods give the result $\textsf{₹}$ 300.

The correct option is (D) $\frac{5}{25}$.

We need to find which of the given fractions is NOT equivalent to $\frac{1}{4}$. A fraction is equivalent to $\frac{1}{4}$ if, when simplified to its lowest terms, it equals $\frac{1}{4}$. We can check each option by simplifying it or by checking if the denominator is four times the numerator.

Let's check each option:

• (A) $\frac{2}{8}$: Simplify by dividing numerator and denominator by their GCD, which is 2. $\frac{\cancel{2}^{1}}{\cancel{8}_{4}} = \frac{1}{4}$. This fraction IS equal to $\frac{1}{4}$.
• (B) $\frac{3}{12}$: Simplify by dividing numerator and denominator by their GCD, which is 3. $\frac{\cancel{3}^{1}}{\cancel{12}_{4}} = \frac{1}{4}$. This fraction IS equal to $\frac{1}{4}$.
• (C) $\frac{4}{16}$: Simplify by dividing numerator and denominator by their GCD, which is 4. $\frac{\cancel{4}^{1}}{\cancel{16}_{4}} = \frac{1}{4}$. This fraction IS equal to $\frac{1}{4}$.
• (D) $\frac{5}{25}$: Simplify by dividing numerator and denominator by their GCD, which is 5. $\frac{\cancel{5}^{1}}{\cancel{25}_{5}} = \frac{1}{5}$. This fraction is equal to $\frac{1}{5}$, which is NOT equal to $\frac{1}{4}$.

Alternatively, we can check if the denominator is 4 times the numerator for each fraction:

• (A) $\frac{2}{8}$: Is $8 = 4 \times 2$? Yes, $8 = 8$.
• (B) $\frac{3}{12}$: Is $12 = 4 \times 3$? Yes, $12 = 12$.
• (C) $\frac{4}{16}$: Is $16 = 4 \times 4$? Yes, $16 = 16$.
• (D) $\frac{5}{25}$: Is $25 = 4 \times 5$? No, $25 \neq 20$.

Ther

The correct option is (B) $\frac{3}{4}$.

To simplify the subtraction involving a mixed fraction and a proper fraction, we can convert the mixed fraction into an improper fraction first.

The given expression is $1\frac{1}{2} - \frac{3}{4}$.

Convert the mixed fraction $1\frac{1}{2}$ to an improper fraction:

$1\frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{2 + 1}{2} = \frac{3}{2}$

Now, the subtraction becomes:

$\frac{3}{2} - \frac{3}{4}$

These are unlike fractions (different denominators). We need to find a common denominator. The least common multiple (LCM) of 2 and 4 is 4.

Convert $\frac{3}{2}$ to an equivalent fraction with a denominator of 4:

Multiply the numerator and the denominator by 2: $\frac{3 \times 2}{2 \times 2} = \frac{6}{4}$.

Now subtract the equivalent fractions, which have the same denominator:

$\frac{6}{4} - \frac{3}{4}$

Subtract the numerators and keep the denominator the same:

$\frac{6 - 3}{4} = \frac{3}{4}$

The resulting fraction $\frac{3}{4}$ is in its lowest terms as the only common factor of 3 and 4 is 1.

So, $1\frac{1}{2} - \frac{3}{4} = \frac{3}{4}$.

The correct option is (B) 2 kg.

Given:

• Total amount of dough = 5 kg
• Fraction of dough used for bread = $\frac{2}{5}$

To Find:

The amount of dough (in kg) used for making bread.

Solution:

To find the amount of dough used for bread, we need to calculate $\frac{2}{5}$ of the total amount of dough.

Amount of dough for bread = $\frac{2}{5}$ of 5 kg

Amount of dough for bread = $\frac{2}{5} \times 5$ kg

We can write 5 as $\frac{5}{1}$.

Amount of dough for bread = $\frac{2}{5} \times \frac{5}{1}$ kg

We can cancel the common factor 5 in the numerator and the denominator.

Amount of dough for bread = $\frac{2}{\cancel{5}_{1}} \times \frac{\cancel{5}^{1}}{1}$ kg

Amount of dough for bread = $\frac{2 \times 1}{1 \times 1}$ kg

Amount of dough for bread = $\frac{2}{1}$ kg

Amount of dough for bread = $2$ kg

So, the baker used 2 kg of dough for making bread.

The correct option is (B) $\frac{3}{5}$.

When a fraction is expressed in words like "three-fifths", the first part of the phrase ("three") represents the numerator, and the second part ("fifths") represents the denominator.

"Three-fifths" means 3 out of 5 equal parts.

Numerator = 3

Denominator = 5

So, the fraction is $\frac{3}{5}$.

The correct options are (A) $\frac{5}{4}$ and (B) $1\frac{1}{3}$.

A fraction is greater than 1 if the numerator is greater than the denominator (improper fraction) or if it is a mixed number with a whole number part greater than or equal to 1.

Let's examine each option:

• (A) $\frac{5}{4}$: The numerator (5) is greater than the denominator (4). This is an improper fraction. Improper fractions are greater than or equal to 1. Since $5 > 4$, $\frac{5}{4} > 1$. This fraction is greater than 1.
• (B) $1\frac{1}{3}$: This is a mixed number. A mixed number consists of a whole number part and a proper fraction part. If the whole number part is 1 or more, the mixed number is greater than or equal to 1. In this case, the whole number part is 1. $1\frac{1}{3}$ means $1 + \frac{1}{3}$. Since $1 > 0$ and $\frac{1}{3} > 0$, their sum is greater than 1. This fraction is greater than 1. (Converting to improper fraction: $1\frac{1}{3} = \frac{(1 \times 3) + 1}{3} = \frac{4}{3}$. Since $4 > 3$, $\frac{4}{3} > 1$).
• (C) $\frac{7}{7}$: The numerator (7) is equal to the denominator (7). A fraction where the numerator equals the denominator is equal to 1 ($\frac{7}{7} = 1$). This fraction is NOT greater than 1.
• (D) $\frac{9}{10}$: The numerator (9) is less than the denominator (10). This is a proper fraction. Proper fractions are always less than 1. This fraction is NOT greater than 1.

Therefore, the fractions that represent a value greater than 1 are $\frac{5}{4}$ and $1\frac{1}{3}$.

The correct option is (C) Equivalent like.

To compare fractions that have different denominators (unlike fractions), we cannot directly compare their numerators or denominators. We need to rewrite them so they share a common denominator.

The standard method involves finding the Least Common Multiple (LCM) of the denominators. This LCM becomes the common denominator for the new fractions.

Then, we convert each original fraction into an equivalent fraction that has this common denominator. An equivalent fraction has the same value as the original fraction but a different numerator and denominator.

Since the converted fractions now share the same denominator, they are called like fractions.

So, the process converts the original fractions into equivalent like fractions, which can then be easily compared by looking at their numerators.

For example, to compare $\frac{2}{3}$ and $\frac{3}{4}$:

• The denominators are 3 and 4.
• The LCM of 3 and 4 is 12.
• Convert $\frac{2}{3}$ to an equivalent fraction with a denominator of 12: $\frac{2 \times 4}{3 \times 4} = \frac{8}{12}$.
• Convert $\frac{3}{4}$ to an equivalent fraction with a denominator of 12: $\frac{3 \times 3}{4 \times 3} = \frac{9}{12}$.

Now we compare the equivalent like fractions $\frac{8}{12}$ and $\frac{9}{12}$. Since they have the same denominator, we compare the numerators: $8 < 9$.

Therefore, $\frac{8}{12} < \frac{9}{12}$, which means $\frac{2}{3} < \frac{3}{4}$.

The fractions $\frac{8}{12}$ and $\frac{9}{12}$ are equivalent like fractions derived from the original fractions $\frac{2}{3}$ and $\frac{3}{4}$.

Definition of a Fraction:

A fraction is a numerical representation of a part of a whole or a part of a collection. It is written as a ratio of two numbers, typically with a horizontal bar or a slash separating them. The number on top is called the numerator, and the number on the bottom is called the denominator.

In essence, a fraction $\frac{a}{b}$ means the whole is divided into $b$ equal parts, and we are considering $a$ of those parts.

In the given fraction $\frac{5}{8}$:

The fraction is $\frac{5}{8}$.

The number on the top is 5. This is the numerator.

The number on the bottom is 8. This is the denominator.

So, in the fraction $\frac{5}{8}$, the numerator is 5 and the denominator is 8.

The fraction $\frac{3}{4}$ represents a part of a whole.

In the fraction $\frac{3}{4}$:

• The denominator, which is 4, indicates that the whole has been divided into 4 equal parts.
• The numerator, which is 3, indicates that we are considering 3 of those equal parts.

Therefore, the fraction $\frac{3}{4}$ represents three out of four equal parts of a whole.

For example, if you have a pizza cut into 4 equal slices, $\frac{3}{4}$ of the pizza means 3 of those 4 slices.

A fraction is a way of representing a part of a whole. It consists of a numerator and a denominator.

The numerator is the top number and tells us how many parts we have.

The denominator is the bottom number and tells us how many equal parts the whole is divided into.

The phrase "three-fifths" means we have three parts out of a whole that is divided into five equal parts.

So, the numerator is 3 and the denominator is 5.

The fraction for "three-fifths" is written as:

$\frac{\text{Numerator}}{\text{Denominator}} = \frac{3}{5}$

A number line is a straight line on which every point is assumed to correspond to a real number and every real number to a point.

To represent a fraction on a number line, we first determine the position of the fraction relative to whole numbers.

The given fraction is $\frac{1}{3}$.

Since the numerator (1) is less than the denominator (3), the fraction is a proper fraction.

Proper fractions are always greater than 0 and less than 1.

So, $\frac{1}{3}$ will lie between 0 and 1 on the number line.

To represent $\frac{1}{3}$ on the number line, we follow these steps:

1. Draw a number line and mark the points 0 and 1.

2. Look at the denominator of the fraction, which is 3.

3. Divide the segment between 0 and 1 into 3 equal parts.

4. Look at the numerator of the fraction, which is 1.

5. Starting from 0, mark the first point of division.

This first point represents $\frac{1}{3}$. The next point would represent $\frac{2}{3}$, and the last point (which is 1) represents $\frac{3}{3}$.

Visually, the number line segment from 0 to 1 is divided into three equal parts. The point marking the end of the first part, starting from 0, is the location of $\frac{1}{3}$.

A proper fraction is a fraction where the numerator (the top number) is less than the denominator (the bottom number).

In other words, for a fraction $\frac{a}{b}$ to be a proper fraction, the condition $a < b$ must be met.

Proper fractions represent a quantity that is less than one whole.

Example:

The fraction $\frac{2}{3}$ is a proper fraction.

Here, the numerator is 2 and the denominator is 3. Since $2 < 3$, it fits the definition of a proper fraction.

This fraction represents 2 parts out of 3 equal parts of a whole, which is less than one whole.

An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number).

In other words, for a fraction $\frac{a}{b}$ to be an improper fraction, the condition $a \geq b$ must be met.

Improper fractions represent a quantity that is equal to or greater than one whole.

Example:

The fraction $\frac{7}{4}$ is an improper fraction.

Here, the numerator is 7 and the denominator is 4. Since $7 > 4$, it fits the definition of an improper fraction.

This fraction represents 7 parts out of a whole that is divided into 4 equal parts, which is more than one whole ($\frac{4}{4}$ is one whole, and $\frac{7}{4} = \frac{4}{4} + \frac{3}{4} = 1 + \frac{3}{4}$).

Another example is $\frac{5}{5}$, which is also an improper fraction because the numerator is equal to the denominator. $\frac{5}{5}$ represents exactly one whole.

A mixed fraction (also called a mixed number) is a number consisting of a whole number and a proper fraction.

It represents a number that is greater than one whole.

A mixed fraction combines the two parts using addition, but the '+' sign is usually omitted when written.

For example, if we have 2 whole pizzas and $\frac{1}{4}$ of another pizza, we represent this as $2 \frac{1}{4}$. This means $2 + \frac{1}{4}$.

Example:

The number $3 \frac{2}{5}$ is a mixed fraction.

Here, 3 is the whole number part.

And $\frac{2}{5}$ is the proper fraction part (since the numerator 2 is less than the denominator 5).

This mixed fraction represents a value equivalent to 3 whole units plus an additional $\frac{2}{5}$ of a unit.

To convert an improper fraction to a mixed fraction, we divide the numerator by the denominator.

The quotient of the division becomes the whole number part of the mixed fraction.

The remainder of the division becomes the numerator of the fractional part.

The denominator remains the same as the original improper fraction.

Given improper fraction: $\frac{7}{3}$

We need to divide the numerator (7) by the denominator (3).

Let's perform the division:

$7 \div 3$

We find that:

$3 \times 2 = 6$

$7 - 6 = 1$

So, the quotient is 2 and the remainder is 1.

Using these results:

Whole number = Quotient = 2

Numerator of fractional part = Remainder = 1

Denominator of fractional part = Original denominator = 3

Thus, the mixed fraction is formed by combining the whole number and the fractional part:

Whole Number $\frac{\text{Remainder}}{\text{Denominator}} = 2 \frac{1}{3}$

So, the improper fraction $\frac{7}{3}$ is equivalent to the mixed fraction $2 \frac{1}{3}$.

To convert a mixed fraction into an improper fraction, we follow these steps:

1. Multiply the whole number part by the denominator of the fractional part.

2. Add the numerator of the fractional part to the result from step 1. This sum becomes the new numerator of the improper fraction.

3. The denominator of the improper fraction remains the same as the denominator of the fractional part in the mixed fraction.

Given mixed fraction: $2 \frac{1}{4}$

Here, the whole number is 2, the numerator of the fractional part is 1, and the denominator is 4.

Step 1: Multiply the whole number (2) by the denominator (4).

$2 \times 4 = 8$

Step 2: Add the numerator of the fractional part (1) to the result from Step 1 (8).

$8 + 1 = 9$

This sum, 9, is the new numerator.

Step 3: The denominator remains the same as the original denominator, which is 4.

So, the improper fraction is $\frac{9}{4}$.

The formula can be written as:

$\text{Mixed Fraction} = \text{Whole Number} \frac{\text{Numerator}}{\text{Denominator}}$

$\text{Improper Fraction} = \frac{(\text{Whole Number} \times \text{Denominator}) + \text{Numerator}}{\text{Denominator}}$

Applying this to $2 \frac{1}{4}$:

$\frac{(2 \times 4) + 1}{4} = \frac{8 + 1}{4} = \frac{9}{4}$

Thus, the mixed fraction $2\frac{1}{4}$ is equivalent to the improper fraction $\frac{9}{4}$.

Like fractions are fractions that have the same denominator.

The denominator is the bottom number of a fraction that indicates the total number of equal parts the whole is divided into.

When fractions have the same denominator, they are considered "like" because they are comparing parts of a whole that has been divided into the same number of equal sections.

Example:

Consider the fractions $\frac{1}{5}$ and $\frac{3}{5}$.

In the fraction $\frac{1}{5}$, the denominator is 5.

In the fraction $\frac{3}{5}$, the denominator is also 5.

Since both fractions have the same denominator (which is 5), they are like fractions.

Unlike fractions are fractions that have different denominators.

The denominator is the bottom number of a fraction that indicates the total number of equal parts the whole is divided into.

When fractions have different denominators, they are considered "unlike" because they are comparing parts of a whole that has been divided into a different number of equal sections. This makes direct comparison or addition/subtraction difficult without finding a common denominator first.

Example:

Consider the fractions $\frac{1}{2}$ and $\frac{2}{3}$.

In the fraction $\frac{1}{2}$, the denominator is 2.

In the fraction $\frac{2}{3}$, the denominator is 3.

Since the denominators are different (2 and 3), they are unlike fractions.

Equivalent fractions are fractions that represent the same value or the same part of a whole, even though they may have different numerators and denominators.

To check if $\frac{3}{7}$ and $\frac{6}{14}$ are equivalent fractions, we can use one of several methods.

Method 1: Simplifying Fractions

We can simplify both fractions to their lowest terms and see if they are equal.

The fraction $\frac{3}{7}$ is already in its lowest terms because the only common factor of 3 and 7 is 1.

Now let's simplify the fraction $\frac{6}{14}$. The greatest common divisor (GCD) of 6 and 14 is 2.

Divide both the numerator and the denominator by 2:

$\frac{6 \div 2}{14 \div 2} = \frac{3}{7}$

Since simplifying $\frac{6}{14}$ gives $\frac{3}{7}$, the two fractions are equivalent.

Method 2: Multiplying Numerator and Denominator

We can check if one fraction can be obtained from the other by multiplying the numerator and denominator by the same non-zero number.

Let's see if we can multiply the numerator and denominator of $\frac{3}{7}$ by some number to get $\frac{6}{14}$.

We observe that $3 \times 2 = 6$ and $7 \times 2 = 14$.

Since we can multiply both the numerator (3) and the denominator (7) of $\frac{3}{7}$ by the same number (2) to get the numerator (6) and denominator (14) of $\frac{6}{14}$, the two fractions are equivalent.

Method 3: Cross-Multiplication

We can cross-multiply the numerator of one fraction by the denominator of the other.

For fractions $\frac{a}{b}$ and $\frac{c}{d}$, they are equivalent if $a \times d = b \times c$.

For $\frac{3}{7}$ and $\frac{6}{14}$:

Numerator of first fraction = 3

Denominator of second fraction = 14

Product: $3 \times 14 = 42$

Numerator of second fraction = 6

Denominator of first fraction = 7

Product: $6 \times 7 = 42$}

Since the cross-products are equal ($42 = 42$), the fractions are equivalent.

Using any of these methods, we find that the fractions $\frac{3}{7}$ and $\frac{6}{14}$ are indeed equivalent.

Answer: Yes, $\frac{3}{7}$ and $\frac{6}{14}$ are equivalent fractions.

We are given the fraction $\frac{2}{5}$ and need to find an equivalent fraction with a denominator of 20.

Equivalent fractions are obtained by multiplying or dividing both the numerator and the denominator of a fraction by the same non-zero number.

We want to change the denominator from 5 to 20.

To find the factor by which we need to multiply the denominator 5 to get 20, we divide the target denominator by the original denominator:

Factor $= \frac{\text{Target Denominator}}{\text{Original Denominator}} = \frac{20}{5} = 4$

So, we need to multiply the denominator 5 by 4 to get 20.

To keep the fraction equivalent, we must also multiply the numerator (2) by the same factor (4).

New Numerator $= \text{Original Numerator} \times \text{Factor} = 2 \times 4 = 8$

The new denominator is 20, and the new numerator is 8.

The equivalent fraction is $\frac{8}{20}$.

We can write the process as:

$\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}$

Thus, the equivalent fraction of $\frac{2}{5}$ with the denominator 20 is $\frac{8}{20}$.

To reduce a fraction to its simplest form (also called lowest terms), we need to divide both the numerator and the denominator by their greatest common divisor (GCD).

The simplest form of a fraction is when the only common factor between the numerator and the denominator is 1.

The given fraction is $\frac{18}{24}$.

The numerator is 18 and the denominator is 24.

We need to find the greatest common divisor of 18 and 24.

Let's list the factors of each number:

Factors of 18: 1, 2, 3, 6, 9, 18

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

The common factors are 1, 2, 3, and 6.

The greatest common divisor (GCD) of 18 and 24 is 6.

Now, we divide both the numerator and the denominator by the GCD, which is 6.

New Numerator $= 18 \div 6 = 3$

New Denominator $= 24 \div 6 = 4$

So, the reduced fraction is $\frac{3}{4}$.

We can write the process as:

$\frac{18}{24} = \frac{18 \div 6}{24 \div 6} = \frac{3}{4}$

The fraction $\frac{3}{4}$ is in its simplest form because the only common factor of 3 and 4 is 1.

Thus, the simplest form of $\frac{18}{24}$ is $\frac{3}{4}$.

To compare fractions using a diagram, we can represent each fraction as a part of an identical whole.

We can use shapes like rectangles or circles to represent the whole.

Let's use a rectangular bar to represent one whole.

Diagram for $\frac{1}{2}$:

To represent $\frac{1}{2}$, we take a rectangular bar and divide it into 2 equal parts, because the denominator is 2.

We then shade 1 part out of these 2 equal parts, because the numerator is 1.

Conceptually, it looks like this:

A bar divided in half, with one half shaded.

Diagram for $\frac{1}{3}$:

To represent $\frac{1}{3}$, we take an identical rectangular bar (the same size as before) and divide it into 3 equal parts, because the denominator is 3.

We then shade 1 part out of these 3 equal parts, because the numerator is 1.

Conceptually, it looks like this:

A bar divided into three equal sections, with one section shaded.

Comparison:

Now, we compare the shaded areas in the two diagrams.

In the diagram for $\frac{1}{2}$, the shaded area covers one half of the bar.

In the diagram for $\frac{1}{3}$, the shaded area covers one third of the bar.

Visually, the shaded area representing $\frac{1}{2}$ is clearly larger than the shaded area representing $\frac{1}{3}$.

Therefore, based on the diagrammatic representation, we can conclude that $\frac{1}{2}$ is greater than $\frac{1}{3}$.

This is written mathematically as $\frac{1}{2} > \frac{1}{3}$.

We are asked to compare the fractions $\frac{3}{5}$ and $\frac{4}{5}$.

Observe that both fractions have the same denominator, which is 5.

Fractions that have the same denominator are called like fractions.

When comparing like fractions, we only need to compare their numerators.

The fraction with the larger numerator is the greater fraction.

The numerator of $\frac{3}{5}$ is 3.

The numerator of $\frac{4}{5}$ is 4.

Comparing the numerators, we see that $4 > 3$.

Since the numerator of $\frac{4}{5}$ is greater than the numerator of $\frac{3}{5}$, the fraction $\frac{4}{5}$ is greater than the fraction $\frac{3}{5}$.

Therefore, $\frac{4}{5}$ is greater than $\frac{3}{5}$.

This can be written as $\frac{4}{5} > \frac{3}{5}$.

We are asked to compare the fractions $\frac{2}{7}$ and $\frac{2}{9}$.

Observe that both fractions have the same numerator, which is 2.

When comparing fractions that have the same numerator but different denominators, the fraction with the larger denominator is the smaller fraction.

This is because if you divide a whole into more equal parts (a larger denominator), each individual part is smaller.

The denominator of $\frac{2}{7}$ is 7.

The denominator of $\frac{2}{9}$ is 9.

Comparing the denominators, we see that $9 > 7$.

Since the denominator of $\frac{2}{9}$ (which is 9) is larger than the denominator of $\frac{2}{7}$ (which is 7), the fraction $\frac{2}{9}$ is smaller than the fraction $\frac{2}{7}$.

Therefore, $\frac{2}{9}$ is smaller than $\frac{2}{7}$.

This can be written as $\frac{2}{9} < \frac{2}{7}$.

We need to add the fractions $\frac{3}{8}$ and $\frac{2}{8}$.

These are like fractions because they have the same denominator, which is 8.

To add like fractions, we follow these steps:

1. Add the numerators (the top numbers).

2. Keep the denominator (the bottom number) the same.

The numerators are 3 and 2.

Adding the numerators: $3 + 2 = 5$

The denominator is 8.

Keeping the denominator the same, the sum of the fractions is $\frac{5}{8}$.

So, the addition is performed as follows:

$\frac{3}{8} + \frac{2}{8} = \frac{3 + 2}{8} = \frac{5}{8}$

The result of the addition is $\frac{5}{8}$.

We need to subtract the fraction $\frac{1}{6}$ from $\frac{5}{6}$.

Observe that both fractions have the same denominator, which is 6.

These are like fractions.

To subtract like fractions, we follow these steps:

1. Subtract the numerators (the top numbers).

2. Keep the denominator (the bottom number) the same.

The numerators are 5 and 1.

Subtracting the numerators: $5 - 1 = 4$

The denominator is 6.

Keeping the denominator the same, the difference of the fractions is $\frac{4}{6}$.

So, the subtraction is performed as follows:

$\frac{5}{6} - \frac{1}{6} = \frac{5 - 1}{6} = \frac{4}{6}$

The resulting fraction is $\frac{4}{6}$. This fraction can be simplified.

To simplify, we find the greatest common divisor (GCD) of the numerator (4) and the denominator (6).

The factors of 4 are 1, 2, 4.

The factors of 6 are 1, 2, 3, 6.

The GCD of 4 and 6 is 2.

Divide both the numerator and the denominator by their GCD (2):

$\frac{4 \div 2}{6 \div 2} = \frac{2}{3}$

We can show the cancellation:

$\frac{\cancel{4}^2}{\cancel{6}_3} = \frac{2}{3}$

The simplified result of the subtraction is $\frac{2}{3}$.

To multiply a fraction by a whole number, we can write the whole number as a fraction with a denominator of 1.

The given problem is $\frac{2}{3} \times 5$.

We can write the whole number 5 as a fraction $\frac{5}{1}$.

So the multiplication becomes: $\frac{2}{3} \times \frac{5}{1}$.

To multiply two fractions, we multiply the numerators together and multiply the denominators together.

Multiply the numerators: $2 \times 5 = 10$

Multiply the denominators: $3 \times 1 = 3$

The product is the new numerator over the new denominator:

$\frac{2}{3} \times \frac{5}{1} = \frac{2 \times 5}{3 \times 1} = \frac{10}{3}$

The result is $\frac{10}{3}$. This is an improper fraction because the numerator (10) is greater than the denominator (3).

We can convert this improper fraction to a mixed fraction.

Divide 10 by 3:

$10 \div 3$

The quotient is 3 (since $3 \times 3 = 9$) and the remainder is 1 (since $10 - 9 = 1$).

The mixed fraction is Whole Number $\frac{\text{Remainder}}{\text{Denominator}} = 3 \frac{1}{3}$.

So, $\frac{2}{3} \times 5 = \frac{10}{3}$ or $3 \frac{1}{3}$.

We need to multiply the fractions $\frac{3}{4}$ and $\frac{1}{2}$.

To multiply two fractions, we multiply the numerators together and multiply the denominators together.

That is, for fractions $\frac{a}{b}$ and $\frac{c}{d}$, their product is $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$.

In this problem, the numerators are 3 and 1, and the denominators are 4 and 2.

Multiply the numerators: $3 \times 1 = 3$

Multiply the denominators: $4 \times 2 = 8$

The product of the fractions is the product of the numerators over the product of the denominators.

$\frac{3}{4} \times \frac{1}{2} = \frac{3 \times 1}{4 \times 2} = \frac{3}{8}$

The resulting fraction is $\frac{3}{8}$.

To check if it can be simplified, we find the common factors of the numerator (3) and the denominator (8).

Factors of 3: 1, 3

Factors of 8: 1, 2, 4, 8

The only common factor is 1. Therefore, the fraction $\frac{3}{8}$ is already in its simplest form.

The result of the multiplication is $\frac{3}{8}$.

The reciprocal of a fraction is found by flipping the fraction upside down.

This means we swap the positions of the numerator and the denominator.

For a fraction $\frac{a}{b}$, its reciprocal is $\frac{b}{a}$, provided that $a \neq 0$ and $b \neq 0$.

When you multiply a fraction by its reciprocal, the result is always 1.

The given fraction is $\frac{7}{9}$.

The numerator is 7.

The denominator is 9.

To find the reciprocal, we swap the numerator and the denominator.

The new numerator will be the original denominator (9).

The new denominator will be the original numerator (7).

The reciprocal of $\frac{7}{9}$ is $\frac{9}{7}$.

We can check this: $\frac{7}{9} \times \frac{9}{7} = \frac{7 \times 9}{9 \times 7} = \frac{63}{63} = 1$.

Thus, the reciprocal of $\frac{7}{9}$ is $\frac{9}{7}$.

To divide a fraction by a whole number, we multiply the fraction by the reciprocal of the whole number.

The given problem is $\frac{1}{5} \div 2$.

First, write the whole number 2 as a fraction: $2 = \frac{2}{1}$.

So the problem becomes: $\frac{1}{5} \div \frac{2}{1}$.

Next, find the reciprocal of the divisor, which is $\frac{2}{1}$.

The reciprocal of $\frac{2}{1}$ is found by swapping the numerator and the denominator, giving $\frac{1}{2}$.

Now, change the division problem into a multiplication problem using the reciprocal of the divisor.

$\frac{1}{5} \div \frac{2}{1} = \frac{1}{5} \times \frac{1}{2}$

To multiply fractions, multiply the numerators together and multiply the denominators together.

Multiply the numerators: $1 \times 1 = 1$

Multiply the denominators: $5 \times 2 = 10$

The product is the new numerator over the new denominator:

$\frac{1}{5} \times \frac{1}{2} = \frac{1 \times 1}{5 \times 2} = \frac{1}{10}$

The resulting fraction is $\frac{1}{10}$. This fraction is in its simplest form because the only common factor of 1 and 10 is 1.

The result of the division is $\frac{1}{10}$.

To divide a fraction by another fraction, we follow these steps:

1. Keep the first fraction as it is.

2. Change the division operation ($\div$) to a multiplication operation ($\times$).

3. Find the reciprocal of the second fraction (the divisor) by flipping it upside down.

4. Multiply the first fraction by the reciprocal of the second fraction.

The given problem is $\frac{3}{4} \div \frac{1}{3}$.

Step 1: Keep the first fraction: $\frac{3}{4}$.

Step 2: Change division to multiplication.

The problem becomes: $\frac{3}{4} \times (\text{reciprocal of } \frac{1}{3})$.

Step 3: Find the reciprocal of the second fraction $\frac{1}{3}$.

The reciprocal of $\frac{1}{3}$ is $\frac{3}{1}$.

Step 4: Multiply the first fraction by the reciprocal of the second fraction.

$\frac{3}{4} \times \frac{3}{1}$

To multiply fractions, multiply the numerators together and multiply the denominators together.

Multiply numerators: $3 \times 3 = 9$

Multiply denominators: $4 \times 1 = 4$}

The product is $\frac{9}{4}$.

So, the division is performed as follows:

$\frac{3}{4} \div \frac{1}{3} = \frac{3}{4} \times \frac{3}{1} = \frac{3 \times 3}{4 \times 1} = \frac{9}{4}$

The result is $\frac{9}{4}$. This is an improper fraction. The numerator (9) and denominator (4) have no common factors other than 1, so the fraction is in its simplest form.

The result of the division is $\frac{9}{4}$.

A fraction represents a part of a whole.

In this problem, the whole is the entire pizza, which is divided into 8 equal slices.

The part we are interested in is the number of slices eaten, which is 3.

The fraction is represented as:

Fraction eaten = $\frac{\text{Number of slices eaten}}{\text{Total number of equal slices}}$

Given:

Total number of equal slices = 8

Number of slices eaten = 3

Substituting these values into the fraction formula:

Fraction eaten = $\frac{3}{8}$

So, you ate $\frac{3}{8}$ of the pizza.

A fraction represents a part of a whole.

In this problem, the whole is the word "INDIA".

The parts we are interested in are the vowels within this word.

First, let's determine the total number of letters in the word "INDIA".

The letters are I, N, D, I, A.

Counting the letters, we find there are 5 letters in total.

Next, let's identify the vowels in the word "INDIA".

The vowels are A, E, I, O, U.

Looking at the letters in "INDIA", the vowels are I, I, A.

Counting the vowels, we find there are 3 vowels.

The fraction of vowels is represented as:

Fraction = $\frac{\text{Number of vowels}}{\text{Total number of letters}}$

Substituting the counts we found:

Fraction = $\frac{3}{5}$

So, the fraction of the letters in the word "INDIA" that are vowels is $\frac{3}{5}$.

A fraction is a mathematical term used to represent a part of a whole. It indicates how many equal parts of a whole are being considered.

Fractions are written with two numbers separated by a horizontal line (or sometimes a slash).

Let's use the example of sharing a cake among friends to illustrate the meaning of a fraction.

Imagine you have a whole cake that you want to share equally among 4 friends (including yourself). To do this, you would cut the cake into 4 equal slices.

In this scenario, the whole is the entire cake.

The whole cake is divided into 4 equal parts.

Now, suppose you take 1 slice of the cake for yourself.

The part you took is 1 slice out of the total 4 equal slices.

We can represent the amount of cake you took as a fraction:

Fraction = $\frac{\text{Number of parts taken}}{\text{Total number of equal parts the whole is divided into}}$

In this example:

Number of parts taken = 1 (the slice you took)

Total number of equal parts = 4 (the total number of slices the cake was cut into)

So, the fraction of the cake you took is $\frac{1}{4}$.

The fraction $\frac{1}{4}$ represents one out of four equal parts.

In any fraction $\frac{a}{b}$, the number on the top is called the numerator, and the number on the bottom is called the denominator.

In our example fraction $\frac{1}{4}$:

The numerator is 1.

The denominator is 4.

Significance of the Numerator:

The numerator tells us how many equal parts of the whole are being considered or taken.

In $\frac{1}{4}$, the numerator 1 signifies that we are considering 1 slice (part) of the cake.

Significance of the Denominator:

The denominator tells us the total number of equal parts the whole has been divided into.

In $\frac{1}{4}$, the denominator 4 signifies that the whole cake was divided into 4 equal slices (parts).

If, instead of taking 1 slice, you and one friend together took 2 slices, the fraction of the cake taken would be $\frac{2}{4}$. Here, the numerator 2 indicates that 2 parts are being considered, and the denominator 4 indicates the whole was divided into 4 equal parts.

Fractions like $\frac{2}{4}$ can often be simplified, but the concept remains the same: part over whole, where the whole is divided into equal pieces.

Fractions are broadly classified into different types based on the relationship between their numerator and denominator.

Proper Fractions:

A proper fraction is a fraction where the numerator is less than the denominator.

It represents a value that is less than one whole.

In a proper fraction $\frac{a}{b}$, we have $a < b$.

Examples:

$\frac{1}{2}$ (1 is less than 2)

$\frac{3}{4}$ (3 is less than 4)

$\frac{5}{8}$ (5 is less than 8)

Improper Fractions:

An improper fraction is a fraction where the numerator is greater than or equal to the denominator.

It represents a value that is equal to or greater than one whole.

In an improper fraction $\frac{a}{b}$, we have $a \geq b$.

Examples:

$\frac{5}{3}$ (5 is greater than 3)

$\frac{7}{7}$ (7 is equal to 7, representing one whole)

$\frac{10}{4}$ (10 is greater than 4)

Mixed Fractions:

A mixed fraction (or mixed number) is a number consisting of a whole number part and a proper fraction part.

It is another way of representing a value that is greater than one whole (which is also represented by improper fractions).

A mixed fraction is written in the form $W \frac{N}{D}$, where W is the whole number, and $\frac{N}{D}$ is the proper fraction ($N < D$).

Examples:

$1 \frac{1}{2}$ (1 whole and half)

$2 \frac{3}{4}$ (2 wholes and three quarters)

$3 \frac{1}{5}$ (3 wholes and one fifth)

Conversion between Improper Fractions and Mixed Fractions:

Improper fractions and mixed fractions are two different ways to write the same number when that number is greater than or equal to one.

Converting an Improper Fraction to a Mixed Fraction:

To convert an improper fraction $\frac{a}{b}$ into a mixed fraction, we divide the numerator ($a$) by the denominator ($b$).

The quotient of the division becomes the whole number part of the mixed fraction.

The remainder of the division becomes the numerator of the fractional part.

The denominator of the fractional part is the same as the original denominator ($b$).

Formula: $\frac{a}{b} = \text{Quotient } \frac{\text{Remainder}}{b}$

Example: Convert $\frac{10}{3}$ to a mixed fraction.

Divide 10 by 3.

$10 \div 3$

3 goes into 10 three times with a remainder of 1 ($3 \times 3 = 9$, $10 - 9 = 1$).

Quotient = 3

Remainder = 1

Denominator = 3

So, $\frac{10}{3} = 3 \frac{1}{3}$.

Converting a Mixed Fraction to an Improper Fraction:

To convert a mixed fraction $W \frac{N}{D}$ into an improper fraction, we follow these steps:

1. Multiply the whole number ($W$) by the denominator ($D$) of the fractional part.

2. Add the numerator ($N$) of the fractional part to the result from step 1. This sum becomes the new numerator of the improper fraction.

3. The denominator of the improper fraction is the same as the denominator ($D$) of the fractional part in the mixed fraction.

Formula: $W \frac{N}{D} = \frac{(W \times D) + N}{D}$

Example: Convert $2 \frac{3}{4}$ to an improper fraction.

Whole number ($W$) = 2

Numerator ($N$) = 3

Denominator ($D$) = 4

New Numerator $= (2 \times 4) + 3 = 8 + 3 = 11$

New Denominator = 4

So, $2 \frac{3}{4} = \frac{11}{4}$.

Equivalent fractions are different fractions that represent the same value or the same proportion of a whole.

They look different because the whole is divided into a different number of equal parts, but the total amount represented by the shaded or considered parts is the same.

How to Find Equivalent Fractions:

To find an equivalent fraction of a given fraction, you can multiply or divide both the numerator and the denominator by the same non-zero number.

Multiplying the numerator and denominator by the same number is like cutting each existing part of the whole into smaller, equal sub-parts. The total number of sub-parts changes (denominator), and the number of sub-parts you have changes (numerator), but the overall amount remains the same.

Dividing the numerator and denominator by the same number (which must be a common factor) is like grouping the existing parts together into larger, equal groups. This process is also known as simplifying or reducing a fraction.

For a fraction $\frac{a}{b}$, an equivalent fraction can be found as $\frac{a \times k}{b \times k}$ or $\frac{a \div m}{b \div m}$, where $k$ is any non-zero number and $m$ is a common factor of $a$ and $b$ ($m \neq 0$).

Finding Three Equivalent Fractions for $\frac{3}{7}$:

We can multiply the numerator and denominator by different non-zero integers (other than 1).

1. Multiply by 2:

$\frac{3}{7} = \frac{3 \times 2}{7 \times 2} = \frac{6}{14}$

2. Multiply by 3:

$\frac{3}{7} = \frac{3 \times 3}{7 \times 3} = \frac{9}{21}$

3. Multiply by 4:

$\frac{3}{7} = \frac{3 \times 4}{7 \times 4} = \frac{12}{28}$

So, three equivalent fractions for $\frac{3}{7}$ are $\frac{6}{14}$, $\frac{9}{21}$, and $\frac{12}{28}$. (Many other equivalent fractions exist).

Checking if $\frac{15}{20}$ and $\frac{9}{12}$ are Equivalent:

There are several ways to check for equivalence. Two common methods are simplifying both fractions or using cross-multiplication.

Method 1: Simplifying Both Fractions

We reduce both fractions to their simplest form. If the simplest forms are the same, the original fractions are equivalent.

For $\frac{15}{20}$:

Find the greatest common divisor (GCD) of 15 and 20. The factors of 15 are 1, 3, 5, 15. The factors of 20 are 1, 2, 4, 5, 10, 20. The GCD is 5.

Divide the numerator and denominator by 5:

$\frac{15 \div 5}{20 \div 5} = \frac{3}{4}$

We can show the cancellation:

$\frac{\cancel{15}^3}{\cancel{20}_4} = \frac{3}{4}$

For $\frac{9}{12}$:

Find the greatest common divisor (GCD) of 9 and 12. The factors of 9 are 1, 3, 9. The factors of 12 are 1, 2, 3, 4, 6, 12. The GCD is 3.

Divide the numerator and denominator by 3:

$\frac{9 \div 3}{12 \div 3} = \frac{3}{4}$

We can show the cancellation:

$\frac{\cancel{9}^3}{\cancel{12}_4} = \frac{3}{4}$

Since both fractions simplify to the same simplest form, $\frac{3}{4}$, they are equivalent.

Method 2: Cross-Multiplication

For two fractions $\frac{a}{b}$ and $\frac{c}{d}$, they are equivalent if the product of the numerator of the first fraction and the denominator of the second fraction ($a \times d$) is equal to the product of the denominator of the first fraction and the numerator of the second fraction ($b \times c$).

We are comparing $\frac{15}{20}$ and $\frac{9}{12}$.

Cross-product 1: Numerator of $\frac{15}{20}$ multiplied by Denominator of $\frac{9}{12}$.

$15 \times 12$

Let's calculate $15 \times 12$: $15 \times (10 + 2) = 15 \times 10 + 15 \times 2 = 150 + 30 = 180$.

Cross-product 2: Denominator of $\frac{15}{20}$ multiplied by Numerator of $\frac{9}{12}$.

$20 \times 9$

$20 \times 9 = 180$.

Since the cross-products are equal ($180 = 180$), the fractions are equivalent.

Using either method, we conclude that the fractions $\frac{15}{20}$ and $\frac{9}{12}$ are equivalent.

Answer: Yes, $\frac{15}{20}$ and $\frac{9}{12}$ are equivalent fractions.

Reducing a fraction to its simplest form (or lowest terms) means rewriting the fraction so that its numerator and denominator have no common factors other than 1.

This results in an equivalent fraction that is written using the smallest possible whole numbers for the numerator and denominator.

Process of Reducing a Fraction:

To reduce a fraction $\frac{a}{b}$ to its simplest form, you can use one of the following methods:

1. Using the Highest Common Factor (HCF): Find the greatest common divisor (GCD) or highest common factor (HCF) of the numerator and the denominator. Divide both the numerator and the denominator by their HCF.

2. Using Repeated Division: Find any common factor (greater than 1) of the numerator and the denominator. Divide both by this common factor. Repeat this process with the new numerator and denominator until they have no common factors other than 1.

Reducing $\frac{30}{45}$ to Simplest Form (Using HCF Method):

We need to find the HCF of the numerator 30 and the denominator 45.

Factors of 30: 1, 2, 3, 5, 6, 10, 15, 30

Factors of 45: 1, 3, 5, 9, 15, 45

The common factors are 1, 3, 5, and 15.

The greatest common factor (HCF) is 15.

Now, divide both the numerator and the denominator by the HCF (15).

$\frac{30}{45} = \frac{30 \div 15}{45 \div 15} = \frac{2}{3}$

We can show the cancellation:

$\frac{\cancel{30}^2}{\cancel{45}_3} = \frac{2}{3}$

The fraction $\frac{2}{3}$ is in its simplest form because the only common factor of 2 and 3 is 1.

Reducing $\frac{72}{108}$ to Simplest Form (Using Repeated Division Method):

We will repeatedly divide the numerator (72) and the denominator (108) by common factors until no more common factors exist.

Both 72 and 108 are even, so we can divide by 2.

$\frac{72}{108} = \frac{72 \div 2}{108 \div 2} = \frac{36}{54}$

Now consider $\frac{36}{54}$. Both are even, so divide by 2 again.

$\frac{36}{54} = \frac{36 \div 2}{54 \div 2} = \frac{18}{27}$

Now consider $\frac{18}{27}$. Both are divisible by 9.

$\frac{18}{27} = \frac{18 \div 9}{27 \div 9} = \frac{2}{3}$

Alternatively, from $\frac{18}{27}$, both are divisible by 3.

$\frac{18}{27} = \frac{18 \div 3}{27 \div 3} = \frac{6}{9}$

From $\frac{6}{9}$, both are divisible by 3.

$\frac{6}{9} = \frac{6 \div 3}{9 \div 3} = \frac{2}{3}$

In both repeated division paths, we arrive at $\frac{2}{3}$. The numerator 2 and the denominator 3 have no common factors other than 1, so $\frac{2}{3}$ is the simplest form.

We can show the cancellation for the overall process if we had used the HCF directly. The HCF of 72 and 108 is 36.

$\frac{72}{108} = \frac{72 \div 36}{108 \div 36} = \frac{2}{3}$

$\frac{\cancel{72}^{2}}{\cancel{108}_{3}} = \frac{2}{3}$

The simplest form of $\frac{30}{45}$ is $\frac{2}{3}$.

The simplest form of $\frac{72}{108}$ is $\frac{2}{3}$.

Unlike fractions are fractions that have different denominators.

Comparing unlike fractions directly can be difficult because they represent parts of wholes that are divided into different numbers of equal sections.

To compare unlike fractions, we convert them into like fractions (fractions with the same denominator). The easiest way to find a common denominator is to use the Least Common Multiple (LCM) of the original denominators.

Process of Comparing Unlike Fractions using LCM:

1. Find the Least Common Multiple (LCM) of the denominators of the given unlike fractions.

2. Convert each fraction into an equivalent fraction that has the LCM as its denominator.

3. Once the fractions have the same denominator (they are now like fractions), compare their numerators.

4. The fraction with the larger numerator is the greater fraction.

Comparing $\frac{2}{3}$ and $\frac{5}{7}$:

The given fractions are $\frac{2}{3}$ and $\frac{5}{7}$.

The denominators are 3 and 7.

Step 1: Find the LCM of the denominators 3 and 7.

Since 3 and 7 are prime numbers, their LCM is simply their product.

LCM$(3, 7) = 3 \times 7 = 21$

The common denominator we will use is 21.

Step 2: Convert each fraction to an equivalent fraction with a denominator of 21.

For $\frac{2}{3}$:

To change the denominator from 3 to 21, we multiply by $21 \div 3 = 7$.

We must multiply both the numerator and the denominator by 7 to get an equivalent fraction:

$\frac{2}{3} = \frac{2 \times 7}{3 \times 7} = \frac{14}{21}$

For $\frac{5}{7}$:

To change the denominator from 7 to 21, we multiply by $21 \div 7 = 3$.

We must multiply both the numerator and the denominator by 3 to get an equivalent fraction:

$\frac{5}{7} = \frac{5 \times 3}{7 \times 3} = \frac{15}{21}$

Step 3: Compare the new like fractions $\frac{14}{21}$ and $\frac{15}{21}$.

Now that the denominators are the same, we compare the numerators.

The numerator of $\frac{14}{21}$ is 14.

The numerator of $\frac{15}{21}$ is 15.

Comparing the numerators, we see that $15 > 14$.

Step 4: Conclude which original fraction is greater.

Since $\frac{15}{21}$ is greater than $\frac{14}{21}$, and $\frac{15}{21}$ is equivalent to $\frac{5}{7}$ and $\frac{14}{21}$ is equivalent to $\frac{2}{3}$, it means that $\frac{5}{7}$ is greater than $\frac{2}{3}$.

Comparison:

$\frac{14}{21} < \frac{15}{21}$

Therefore,

$\frac{2}{3} < \frac{5}{7}$

Answer: $\frac{5}{7}$ is greater than $\frac{2}{3}$.

Unlike fractions are fractions that have different denominators. To add or subtract unlike fractions, we must first convert them into like fractions (fractions with the same denominator).

Steps for Adding Two Unlike Fractions:

1. Find the Least Common Multiple (LCM) of the denominators of the two unlike fractions. This LCM will be the least common denominator (LCD) for the fractions.

2. Convert each unlike fraction into an equivalent fraction with the LCD as the new denominator. To do this, determine what factor you need to multiply the original denominator by to get the LCD. Then, multiply the numerator by the same factor.

3. Once both fractions have the same denominator (they are now like fractions), add the numerators of the new fractions.

4. Keep the common denominator the same.

5. Simplify the resulting fraction to its lowest terms if possible.

Calculate $\frac{1}{4} + \frac{2}{5}$:

The given fractions are $\frac{1}{4}$ and $\frac{2}{5}$. These are unlike fractions because the denominators (4 and 5) are different.

Step 1: Find the LCM of the denominators 4 and 5.

Multiples of 4: 4, 8, 12, 16, 20, 24, ...

Multiples of 5: 5, 10, 15, 20, 25, ...

The least common multiple of 4 and 5 is 20. So, the LCD is 20.

Step 2: Convert each fraction to an equivalent fraction with a denominator of 20.

For $\frac{1}{4}$:

We need to change the denominator from 4 to 20. We multiply 4 by 5 to get 20 ($4 \times 5 = 20$).

So, we multiply the numerator (1) by the same factor (5): $1 \times 5 = 5$.

The equivalent fraction is $\frac{5}{20}$.

$\frac{1}{4} = \frac{1 \times 5}{4 \times 5} = \frac{5}{20}$

For $\frac{2}{5}$:

We need to change the denominator from 5 to 20. We multiply 5 by 4 to get 20 ($5 \times 4 = 20$).

So, we multiply the numerator (2) by the same factor (4): $2 \times 4 = 8$.

The equivalent fraction is $\frac{8}{20}$.

$\frac{2}{5} = \frac{2 \times 4}{5 \times 4} = \frac{8}{20}$

Step 3 & 4: Add the numerators of the new like fractions and keep the common denominator.

Now we add $\frac{5}{20}$ and $\frac{8}{20}$.

Add the numerators: $5 + 8 = 13$.

Keep the common denominator: 20.

The sum is $\frac{13}{20}$.

$\frac{1}{4} + \frac{2}{5} = \frac{5}{20} + \frac{8}{20} = \frac{5 + 8}{20} = \frac{13}{20}$

Step 5: Simplify the resulting fraction $\frac{13}{20}$.

We find the greatest common divisor (GCD) of 13 and 20.

Factors of 13: 1, 13 (13 is a prime number)

Factors of 20: 1, 2, 4, 5, 10, 20

The only common factor is 1.

Therefore, the fraction $\frac{13}{20}$ is already in its simplest form.

The sum of $\frac{1}{4}$ and $\frac{2}{5}$ is $\frac{13}{20}$.

Unlike fractions are fractions that have different denominators. To subtract unlike fractions, just like adding them, we must first convert them into like fractions (fractions with the same denominator).

Steps for Subtracting Two Unlike Fractions:

1. Find the Least Common Multiple (LCM) of the denominators of the two unlike fractions. This LCM will serve as the least common denominator (LCD).

2. Convert each unlike fraction into an equivalent fraction that has the LCD as its new denominator. To do this, determine the factor by which the original denominator was multiplied to get the LCD. Multiply the original numerator by the same factor.

3. Once both fractions have the same denominator, subtract the numerator of the second fraction from the numerator of the first fraction.

4. Keep the common denominator the same for the result.

5. Simplify the resulting fraction to its lowest terms if possible.

Calculate $\frac{5}{6} - \frac{3}{4}$:

The given fractions are $\frac{5}{6}$ and $\frac{3}{4}$. These are unlike fractions because their denominators (6 and 4) are different.

Step 1: Find the LCM of the denominators 6 and 4.

Multiples of 6: 6, 12, 18, 24, ...

Multiples of 4: 4, 8, 12, 16, 20, 24, ...

The least common multiple of 6 and 4 is 12. So, the LCD is 12.

We can also find LCM using prime factorization:

$\begin{array}{c|cc} 2 & 6 \;, & 4 \\ \hline 3 & 3 \; , & 2 \\ \hline 2 & 1 \; , & 2 \\ \hline & 1 \; , & 1 \end{array}$

LCM$(6, 4) = 2 \times 3 \times 2 = 12$.

Step 2: Convert each fraction to an equivalent fraction with a denominator of 12.

For $\frac{5}{6}$:

To change the denominator from 6 to 12, we multiply by $12 \div 6 = 2$.

Multiply the numerator (5) by the same factor (2): $5 \times 2 = 10$.

The equivalent fraction is $\frac{10}{12}$.

$\frac{5}{6} = \frac{5 \times 2}{6 \times 2} = \frac{10}{12}$

For $\frac{3}{4}$:

To change the denominator from 4 to 12, we multiply by $12 \div 4 = 3$.

Multiply the numerator (3) by the same factor (3): $3 \times 3 = 9$.

The equivalent fraction is $\frac{9}{12}$.

$\frac{3}{4} = \frac{3 \times 3}{4 \times 3} = \frac{9}{12}$

Step 3 & 4: Subtract the numerators of the new like fractions and keep the common denominator.

Now we subtract $\frac{9}{12}$ from $\frac{10}{12}$.

Subtract the numerators: $10 - 9 = 1$.

Keep the common denominator: 12.

The difference is $\frac{1}{12}$.

$\frac{5}{6} - \frac{3}{4} = \frac{10}{12} - \frac{9}{12} = \frac{10 - 9}{12} = \frac{1}{12}$

Step 5: Simplify the resulting fraction $\frac{1}{12}$.

The numerator is 1. The only positive factor of 1 is 1.

The denominator is 12. Factors of 12 are 1, 2, 3, 4, 6, 12.

The only common factor of 1 and 12 is 1.

Therefore, the fraction $\frac{1}{12}$ is already in its simplest form.

The difference between $\frac{5}{6}$ and $\frac{3}{4}$ is $\frac{1}{12}$.

Multiplying a Fraction by a Whole Number:

To multiply a fraction by a whole number, you can follow these steps:

1. Write the whole number as a fraction with a denominator of 1. For example, the whole number $W$ can be written as $\frac{W}{1}$.

2. Multiply the numerator of the fraction by the whole number (or the numerator of the whole number written as a fraction).

3. Keep the denominator of the original fraction (or multiply the denominators, which will be the original denominator times 1).

4. Simplify the resulting fraction if possible.

Formula: $W \times \frac{a}{b} = \frac{W}{1} \times \frac{a}{b} = \frac{W \times a}{1 \times b} = \frac{W \times a}{b}$.

Multiplying a Fraction by Another Fraction:

To multiply a fraction by another fraction, you follow these steps:

1. Multiply the numerators of the two fractions together. The result is the numerator of the product.

2. Multiply the denominators of the two fractions together. The result is the denominator of the product.

3. Simplify the resulting fraction if possible.

Formula: $\frac{a}{b} \times \frac{c}{d} = \frac{a \times c}{b \times d}$.

Optionally, you can simplify *before* multiplying by cancelling out common factors between any numerator and any denominator.

Calculate $6 \times \frac{2}{9}$ and Reduce:

Problem: $6 \times \frac{2}{9}$

Write 6 as a fraction $\frac{6}{1}$.

$6 \times \frac{2}{9} = \frac{6}{1} \times \frac{2}{9}$

Multiply numerators: $6 \times 2 = 12$

Multiply denominators: $1 \times 9 = 9$

The product is $\frac{12}{9}$.

Now, reduce the fraction $\frac{12}{9}$ to its simplest form. We find the greatest common divisor (GCD) of 12 and 9.

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 9: 1, 3, 9

The GCD is 3.

Divide the numerator and denominator by 3:

$\frac{12}{9} = \frac{12 \div 3}{9 \div 3} = \frac{4}{3}$

We can show the cancellation (done before multiplication):

$6 \times \frac{2}{9} = \frac{\cancel{6}^2}{1} \times \frac{2}{\cancel{9}_3} = \frac{2 \times 2}{1 \times 3} = \frac{4}{3}$

The result in simplest form is $\frac{4}{3}$. This is an improper fraction and can also be written as a mixed fraction $1 \frac{1}{3}$.

Calculate $\frac{4}{5} \times \frac{3}{8}$ and Reduce:

Problem: $\frac{4}{5} \times \frac{3}{8}$

Multiply numerators: $4 \times 3 = 12$

Multiply denominators: $5 \times 8 = 40$

The product is $\frac{12}{40}$.

Now, reduce the fraction $\frac{12}{40}$ to its simplest form. Find the GCD of 12 and 40.

Factors of 12: 1, 2, 3, 4, 6, 12

Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40

The GCD is 4.

Divide the numerator and denominator by 4:

$\frac{12}{40} = \frac{12 \div 4}{40 \div 4} = \frac{3}{10}$

Alternatively, using cancellation before multiplication:

$\frac{4}{5} \times \frac{3}{8} = \frac{\cancel{4}^1}{5} \times \frac{3}{\cancel{8}_2} = \frac{1 \times 3}{5 \times 2} = \frac{3}{10}$

The fraction $\frac{3}{10}$ is in simplest form as the only common factor of 3 and 10 is 1.

The result of $6 \times \frac{2}{9}$ in simplest form is $\frac{4}{3}$.

The result of $\frac{4}{5} \times \frac{3}{8}$ in simplest form is $\frac{3}{10}$.

The Concept of the Reciprocal of a Fraction:

The reciprocal of a fraction is another fraction obtained by interchanging the numerator and the denominator of the original fraction.

If you have a fraction $\frac{a}{b}$, where $a \neq 0$ and $b \neq 0$, its reciprocal is $\frac{b}{a}$.

The product of any non-zero fraction and its reciprocal is always 1.

For example, the reciprocal of $\frac{2}{3}$ is $\frac{3}{2}$. When multiplied, $\frac{2}{3} \times \frac{3}{2} = \frac{6}{6} = 1$.

Similarly, the reciprocal of a whole number like 5 (which can be written as $\frac{5}{1}$) is $\frac{1}{5}$.

Relationship between Division and Multiplication by the Reciprocal:

Dividing by a fraction is the same as multiplying by the reciprocal of that fraction.

This is because division is the inverse operation of multiplication. Dividing by a number is equivalent to multiplying by its multiplicative inverse, which for fractions is the reciprocal.

So, to calculate $\frac{a}{b} \div \frac{c}{d}$, you perform the multiplication $\frac{a}{b} \times \frac{d}{c}$.

This rule is often remembered by the phrase "Keep, Change, Flip":

Keep the first fraction.

Change the division sign to a multiplication sign.

Flip the second fraction (find its reciprocal).

Calculate $\frac{5}{12} \div \frac{2}{3}$:

We need to calculate the division of $\frac{5}{12}$ by $\frac{2}{3}$.

Step 1: Keep the first fraction.

The first fraction is $\frac{5}{12}$.

Step 2: Change the division sign ($\div$) to a multiplication sign ($\times$).

The operation becomes multiplication.

Step 3: Flip the second fraction (find its reciprocal).

The second fraction is $\frac{2}{3}$. Its reciprocal is $\frac{3}{2}$.

Step 4: Multiply the first fraction by the reciprocal of the second fraction.

We now calculate $\frac{5}{12} \times \frac{3}{2}$.

Multiply numerators: $5 \times 3 = 15$

Multiply denominators: $12 \times 2 = 24$

The product is $\frac{15}{24}$.

$\frac{5}{12} \div \frac{2}{3} = \frac{5}{12} \times \frac{3}{2} = \frac{5 \times 3}{12 \times 2} = \frac{15}{24}$

Step 5: Simplify the resulting fraction $\frac{15}{24}$ if possible.

Find the greatest common divisor (GCD) of 15 and 24.

Factors of 15: 1, 3, 5, 15

Factors of 24: 1, 2, 3, 4, 6, 8, 12, 24

The GCD is 3.

Divide the numerator and denominator by 3:

$\frac{15 \div 3}{24 \div 3} = \frac{5}{8}$

We can also use cancellation before multiplying:

$\frac{5}{12} \div \frac{2}{3} = \frac{5}{\cancel{12}_4} \times \frac{\cancel{3}^1}{2} = \frac{5 \times 1}{4 \times 2} = \frac{5}{8}$

The fraction $\frac{5}{8}$ is in its simplest form because the only common factor of 5 and 8 is 1.

The result of the division is $\frac{5}{8}$.

Given:

Total length of the ribbon = $\frac{3}{4}$ meters

Length of each piece = $\frac{1}{8}$ meters

To Find:

The number of pieces that can be cut from the ribbon.

Solution:

To find the number of pieces of a certain length that can be cut from a total length, we need to divide the total length by the length of each piece.

Number of pieces = Total length $\div$ Length of each piece

Number of pieces = $\frac{3}{4} \div \frac{1}{8}$

To divide a fraction by another fraction, we multiply the first fraction by the reciprocal of the second fraction.

The reciprocal of $\frac{1}{8}$ is $\frac{8}{1}$.

So, the division becomes:

$\frac{3}{4} \div \frac{1}{8} = \frac{3}{4} \times \frac{8}{1}$

Now, multiply the numerators together and the denominators together:

Numerator: $3 \times 8 = 24$

Denominator: $4 \times 1 = 4$

The result of the multiplication is $\frac{24}{4}$.

$\frac{3}{4} \times \frac{8}{1} = \frac{3 \times 8}{4 \times 1} = \frac{24}{4}$

Simplify the resulting fraction $\frac{24}{4}$.

We can perform the division: $24 \div 4 = 6$.

Alternatively, reduce the fraction by dividing the numerator and denominator by their greatest common divisor (GCD), which is 4.

$\frac{24 \div 4}{4 \div 4} = \frac{6}{1}$

And $\frac{6}{1} = 6$.

We can show the cancellation during multiplication:

$\frac{3}{\cancel{4}^1} \times \frac{\cancel{8}^2}{1} = \frac{3 \times 2}{1 \times 1} = \frac{6}{1} = 6$

So, 6 pieces of ribbon, each $\frac{1}{8}$ meters long, can be cut from a ribbon that is $\frac{3}{4}$ meters long.

Answer:

6 pieces can be cut from the ribbon.

Given:

Weight of potatoes Rohan bought = $2\frac{1}{2}$ kg

Weight of onions Rohan bought = $1\frac{3}{4}$ kg

To Find:

Total weight of vegetables Rohan bought.

Solution:

To find the total weight of vegetables, we need to add the weight of potatoes and the weight of onions.

Total weight = Weight of potatoes + Weight of onions

Total weight = $2\frac{1}{2} + 1\frac{3}{4}$

First, convert the mixed fractions into improper fractions:

$2\frac{1}{2} = \frac{(2 \times 2) + 1}{2} = \frac{4 + 1}{2} = \frac{5}{2}$

$1\frac{3}{4} = \frac{(1 \times 4) + 3}{4} = \frac{4 + 3}{4} = \frac{7}{4}$

Now, add the improper fractions:

Total weight = $\frac{5}{2} + \frac{7}{4}$

These are unlike fractions (denominators are 2 and 4). We need to find a common denominator, which is the LCM of 2 and 4.

LCM$(2, 4) = 4$.

Convert the fractions to equivalent fractions with a denominator of 4.

For $\frac{5}{2}$: Multiply numerator and denominator by 2 (since $2 \times 2 = 4$).

$\frac{5}{2} = \frac{5 \times 2}{2 \times 2} = \frac{10}{4}$

The fraction $\frac{7}{4}$ already has the denominator 4.

Now, add the like fractions:

Total weight = $\frac{10}{4} + \frac{7}{4} = \frac{10 + 7}{4} = \frac{17}{4}$

The result is $\frac{17}{4}$. This is an improper fraction. We can convert it back to a mixed fraction.

Divide 17 by 4:

$17 \div 4$

The quotient is 4, and the remainder is 1.

So, $\frac{17}{4} = 4 \frac{1}{4}$.

Answer:

The total weight of vegetables Rohan bought is $4\frac{1}{4}$ kg.

Given:

Fraction of time spent studying Mathematics = $\frac{1}{4}$

Fraction of time spent studying Science = $\frac{1}{8}$

Fraction of time spent studying English = $\frac{1}{12}$

To Find:

The total fraction of time spent studying these three subjects.

Solution:

To find the total fraction of time spent studying, we need to add the fractions of time spent on each subject.

Total fraction of time = $\frac{1}{4} + \frac{1}{8} + \frac{1}{12}$

These are unlike fractions because they have different denominators (4, 8, and 12).

To add unlike fractions, we must first convert them into like fractions by finding a common denominator. The least common denominator (LCD) is the Least Common Multiple (LCM) of the denominators 4, 8, and 12.

Step 1: Find the LCM of 4, 8, and 12.

We can use prime factorization or listing multiples.

Using listing multiples:

Multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...

Multiples of 8: 8, 16, 24, 32, ...

Multiples of 12: 12, 24, 36, ...

The least common multiple is 24.

So, the LCD is 24.

Using prime factorization:

$4 = 2 \times 2 = 2^2$

$8 = 2 \times 2 \times 2 = 2^3$

$12 = 2 \times 2 \times 3 = 2^2 \times 3^1$

LCM is found by taking the highest power of all prime factors involved (2 and 3): $2^3 \times 3^1 = 8 \times 3 = 24$.

Step 2: Convert each fraction to an equivalent fraction with a denominator of 24.

For $\frac{1}{4}$:

To get a denominator of 24, multiply 4 by 6 ($4 \times 6 = 24$). Multiply the numerator by 6.

$\frac{1}{4} = \frac{1 \times 6}{4 \times 6} = \frac{6}{24}$

For $\frac{1}{8}$:

To get a denominator of 24, multiply 8 by 3 ($8 \times 3 = 24$). Multiply the numerator by 3.

$\frac{1}{8} = \frac{1 \times 3}{8 \times 3} = \frac{3}{24}$

For $\frac{1}{12}$:

To get a denominator of 24, multiply 12 by 2 ($12 \times 2 = 24$). Multiply the numerator by 2.

$\frac{1}{12} = \frac{1 \times 2}{12 \times 2} = \frac{2}{24}$

Step 3: Add the equivalent like fractions.

Total fraction of time = $\frac{6}{24} + \frac{3}{24} + \frac{2}{24}$

Add the numerators and keep the common denominator:

$\frac{6 + 3 + 2}{24} = \frac{11}{24}$

Step 4: Simplify the resulting fraction $\frac{11}{24}$.

The numerator is 11 (a prime number). The denominator is 24.

The only positive factors of 11 are 1 and 11.

The factors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.

The only common factor is 1. Therefore, the fraction $\frac{11}{24}$ is already in its simplest form.

Answer:

The student spent $\frac{11}{24}$ of his total time studying these three subjects.