Chapter 8 Comparing Quantities (Concepts)

Ratio & its Related Terms

In many real-life situations, we need to compare different quantities. For example, we might compare the heights of two people, the cost of two items, or the number of boys and girls in a class. Simple comparison might tell us which quantity is larger or smaller. However, sometimes we need to understand the comparison in a relative sense, for instance, how many times one quantity is of another, or what part of one quantity is of another. This is where the concept of Ratio is useful.

In this chapter, we will learn about ratios and related terms, extend this idea to proportion, and use a technique called the unitary method to solve problems involving comparison of quantities.

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What is a Ratio?

A Ratio is a comparison of two quantities of the same kind and in the same units. This comparison is done by division. It tells us how many times one quantity is contained within or contains another quantity.

If $a$ and $b$ are two quantities of the same kind and in the same units (and $b$ is not zero), then the ratio of $a$ to $b$ is written as:

• $a : b$ (read as "a is to b")
• $\frac{a}{b}$ (written as a fraction, where $a$ is the numerator and $b$ is the denominator)

Example: Suppose there are 10 red apples and 15 green apples in a basket. We can find the ratio of the number of red apples to the number of green apples.

Number of red apples = 10.

Number of green apples = 15.

The quantities are of the same kind (number of apples) and in the same unit (count). The ratio of red apples to green apples is:

This fraction can be simplified by dividing the numerator and denominator by their HCF, which is 5.

So, the ratio of red apples to green apples is $2 : 3$. This means for every 2 red apples, there are 3 green apples in the basket.

Terms of a Ratio:

In the ratio $a : b$, $a$ and $b$ are called the Terms of the ratio.

• $a$ is the First Term or the Antecedent.
• $b$ is the Second Term or the Consequent.

Important Points about Ratios:

1. Quantities of the Same Kind and Unit: This is a fundamental requirement. You must compare quantities of the same type (lengths with lengths, volumes with volumes, counts with counts) and ensure they are in the same unit before forming the ratio.
2. Ratio Has No Units: Since the units of the quantities being compared are the same, they cancel out during the division. Hence, a ratio is a unitless number. Example: $\frac{50 \text{ cm}}{200 \text{ cm}} = \frac{50}{200} = \frac{1}{4}$ or $1:4$.
3. Order Matters: The ratio $a : b$ is generally different from $b : a$ (unless $a=b$). The order in which the quantities are stated determines the order of the terms in the ratio.
4. Simplest Form: Ratios should typically be expressed in their simplest form (lowest terms) by dividing both the antecedent and the consequent by their HCF. This is the same as reducing the fraction form of the ratio to its lowest terms.
5. Terms are Usually Positive: In the context of comparing quantities like lengths, weights, counts, etc., the terms of a ratio are generally positive numbers.

Example of Simplifying a Ratio:

Find the ratio of 50 cm to 2 m in simplest form.

Step 1: Ensure same units. Convert 2 m to cm. $1 \text{ m} = 100 \text{ cm}$, so $2 \text{ m} = 2 \times 100 \text{ cm} = 200 \text{ cm}$.

Step 2: Write the ratio. Ratio $= 50 \text{ cm} : 200 \text{ cm} = 50:200$.

Step 3: Simplify the ratio. Find HCF(50, 200). HCF is 50.

Divide both terms by 50: $50 \div 50 = 1$, $200 \div 50 = 4$.

The ratio in simplest form is $1 : 4$.

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Equivalent Ratios

Equivalent Ratios are ratios that represent the same comparison or proportion, even though the numbers (terms) might be different. They are similar to equivalent fractions.

You can obtain an equivalent ratio by multiplying or dividing both terms of a given ratio by the same non-zero number.

If $a : b$ is a ratio, and $k$ is any non-zero number, then $a \times k : b \times k$ is equivalent to $a : b$.

If $a : b$ is a ratio, and $m$ is a common factor of $a$ and $b$ ($m \neq 0$), then $a \div m : b \div m$ is equivalent to $a : b$.

Example: The ratio $2 : 3$ is equivalent to:

• $4 : 6$ (by multiplying both terms by 2)
• $6 : 9$ (by multiplying both terms by 3)
• $20 : 30$ (by multiplying both terms by 10)

All these ratios ($2:3, 4:6, 6:9, 20:30$) represent the same proportional relationship. If you write them as fractions ($\frac{2}{3}, \frac{4}{6}, \frac{6}{9}, \frac{20}{30}$), they all simplify to $\frac{2}{3}$.

Checking for Equivalence:

To check if two ratios $a : b$ and $c : d$ are equivalent, you can compare their fractional forms. They are equivalent if $\frac{a}{b} = \frac{c}{d}$.

Using cross-multiplication (as learned with fractions), $\frac{a}{b} = \frac{c}{d}$ if and only if $a \times d = b \times c$. So, you can check for equivalence by comparing the product of the extremes ($a \times d$) and the product of the means ($b \times c$).

Example: Are the ratios $2 : 5$ and $6 : 15$ equivalent?

Method 1 (Simplify to lowest terms):

$2 : 5$ is already in simplest form.

$6 : 15$. Find HCF(6, 15) = 3. Divide both terms by 3: $6 \div 3 = 2$, $15 \div 3 = 5$. The simplified ratio is $2:5$.

Since both ratios simplify to the same ratio $2:5$, they are equivalent.

Method 2 (Cross-product):

For $a:b :: c:d$, check if $a \times d = b \times c$. Here $a=2, b=5, c=6, d=15$.

Product of extremes $= a \times d = 2 \times 15 = 30$.

Product of means $= b \times c = 5 \times 6 = 30$.

Since the product of extremes equals the product of means ($30 = 30$), the ratios are equivalent.

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Comparing Ratios

To compare two or more ratios (to see which one is larger or smaller), we convert them into a format that allows for easy comparison.

1. Convert to Fractions with Common Denominator: Convert each ratio into its fractional form ($\frac{a}{b}$). Then, find the LCM of the denominators and convert all the fractions into equivalent like fractions (fractions with the same denominator). Compare the numerators of these like fractions; the fraction with the larger numerator corresponds to the larger ratio.
2. Convert to Decimals: Convert each ratio to its fractional form and then perform the division to get a decimal value (terminating or repeating). Compare the decimal values.

Example: Compare the ratios $2 : 3$ and $4 : 5$.

Method 1 (Common Denominator):

Ratio $2 : 3 = \frac{2}{3}$. Ratio $4 : 5 = \frac{4}{5}$.

LCM of the denominators 3 and 5 is $3 \times 5 = 15$. Convert both fractions to have denominator 15.

Now compare $\frac{10}{15}$ and $\frac{12}{15}$. Since $12 > 10$, we have $\frac{12}{15} > \frac{10}{15}$.

Therefore, $4 : 5 > 2 : 3$.

Method 2 (Decimals):

Convert ratios to decimal values:

$2 : 3 = \frac{2}{3}$. Perform division: $2 \div 3 \approx 0.666...$

$4 : 5 = \frac{4}{5}$. Perform division: $4 \div 5 = 0.8$.

Compare decimal values: $0.8$ and $0.666...$. Since $0.8 > 0.666...$, we have $4 : 5 > 2 : 3$.

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Dividing a Quantity in a Given Ratio

A common application of ratios is to divide a total quantity into parts according to a given ratio. For example, sharing an amount of money or a quantity of something in a specific ratio among a group of people.

Steps to Divide a Given Quantity in a Given Ratio ($m : n$):

1. Find the sum of the terms of the ratio ($m + n$). This sum represents the total number of equal shares the quantity is divided into.
2. Divide the given total quantity by the sum of the ratio terms (from Step 1). This gives you the value of one "share" or "part" in the ratio.
3. Multiply the value of one share (from Step 2) by each term of the ratio ($m$ and $n$) to find the value of each individual part.

Example: Divide $ \textsf{₹} 720 $ between Ram and Shyam in the ratio $4 : 5$.

The given ratio is Ram : Shyam = $4 : 5$. The total quantity to be divided is $ \textsf{₹} 720 $.

Step 1: Find the sum of the terms of the ratio. Sum $= 4 + 5 = 9$. This means the total amount is divided into 9 equal shares.

Step 2: Find the value of one share. Value of one share $= \frac{\text{Total Amount}}{\text{Sum of ratio terms}} = \frac{\textsf{₹} 720}{9}$.

Perform the division: $720 \div 9 = 80$. So, the value of one share is $ \textsf{₹} 80 $.

Step 3: Calculate the individual parts (Ram's share and Shyam's share) by multiplying the value of one share by each term of the ratio.

Ram's share $= 4 \times \textsf{₹} 80 = \textsf{₹} 320$.

Shyam's share $= 5 \times \textsf{₹} 80 = \textsf{₹} 400$.

Check: Add the shares to ensure they sum up to the total quantity: $ \textsf{₹} 320 + \textsf{₹} 400 = \textsf{₹} 720 $. The division is correct.

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Proportion & its Related Terms

In the previous section, we learned about Ratio as a way to compare two quantities using division. Now, we will extend this idea to compare two ratios. When two ratios are equal, we say they are in Proportion. Proportion is a powerful tool for solving problems involving quantities that are related to each other in a consistent way.

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What is Proportion?

Proportion is a statement that asserts the equality of two ratios. If two ratios are equal, the four quantities involved are said to be in proportion. This means that the relationship between the first two quantities is the same as the relationship between the last two quantities.

If four quantities $a, b, c,$ and $d$ are such that the ratio $a : b$ is equal to the ratio $c : d$, then $a, b, c,$ and $d$ are in proportion.

This is formally written using the symbol '::' (read as "as") between the two ratios:

Alternatively, since it's a statement of equality between ratios, we can use the equals sign:

In the proportion $a : b :: c : d$:

• The first term ($a$) and the fourth term ($d$) are located at the ends of the proportion statement. They are called the Extremes or Extreme Terms.
• The second term ($b$) and the third term ($c$) are located in the middle of the proportion statement. They are called the Means or Middle Terms.

For $a:b::c:d$ to be a valid proportion, $a, b$ must be of the same kind and unit (and $b \neq 0$), and $c, d$ must be of the same kind and unit (and $d \neq 0$). The quantities ($a, b$) can be of a different kind than ($c, d$), as long as the two ratios are numerically equal (e.g., ratio of lengths = ratio of costs).

Fundamental Property of Proportion:

A key property derived from the definition of proportion is that the product of the extreme terms is equal to the product of the middle terms.

If $a : b :: c : d$, which is equivalent to $\frac{a}{b} = \frac{c}{d}$, then by cross-multiplication:

This is known as the Property of Proportion:

This property is very useful for checking if four quantities are in proportion or for finding an unknown term when the other three are known.

Example: Are the numbers 2, 4, 6, 12 in proportion?

We need to check if $2:4 :: 6:12$ is a true statement. We can use the Product of Extremes and Product of Means test.

The extremes are the 1st and 4th terms: $a=2, d=12$. Product of extremes $= 2 \times 12 = 24$.

The means are the 2nd and 3rd terms: $b=4, c=6$. Product of means $= 4 \times 6 = 24$.

Since Product of extremes = Product of means ($24 = 24$), the numbers 2, 4, 6, and 12 are in proportion in that order.

We can write this as $2 : 4 :: 6 : 12$. This is true because $\frac{2}{4} = \frac{1}{2}$ and $\frac{6}{12} = \frac{1}{2}$, and $\frac{1}{2} = \frac{1}{2}$.

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Continued Proportion

Sometimes, we have a relationship between three quantities of the same kind, where the ratio of the first to the second is equal to the ratio of the second to the third. This is called Continued Proportion.

Three quantities $a, b,$ and $c$ (of the same kind) are in continued proportion if $a : b :: b : c$.

This can be written fractionally as $\frac{a}{b} = \frac{b}{c}$.

In this continued proportion $a : b :: b : c$:

• $a$ is the First Proportional.
• $c$ is the Third Proportional to $a$ and $b$.
• $b$ is the Mean Proportional (or geometric mean) between $a$ and $c$.

Applying the property of proportion ($a \times d = b \times c$) to $a : b :: b : c$, we get $a \times c = b \times b$, which is $ac = b^2$.

So, the mean proportional $b$ is the square root of the product of $a$ and $c$: $b = \sqrt{ac}$.

Example: Are the numbers 4, 6, 9 in continued proportion?

We need to check if $4 : 6 :: 6 : 9$ is true. Check if the ratio $4:6$ is equal to the ratio $6:9$.

Ratio $4 : 6 = \frac{4}{6}$. Simplify: $\frac{4 \div 2}{6 \div 2} = \frac{2}{3}$.

Ratio $6 : 9 = \frac{6}{9}$. Simplify: $\frac{6 \div 3}{9 \div 3} = \frac{2}{3}$.

Since both ratios are equal to $\frac{2}{3}$, the numbers 4, 6, and 9 are in continued proportion.

Here, 6 is the mean proportional between 4 and 9. Check $b^2 = ac$: $6^2 = 36$, and $4 \times 9 = 36$. The property holds.

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Finding an Unknown Term in a Proportion

If three of the four terms in a proportion are known, we can use the property "Product of extremes = Product of means" to find the value of the unknown term. We represent the unknown term with a variable and solve the resulting equation.

Example: Find the value of $x$ if $3 : 4 :: x : 8$.

The four terms in proportion are $a=3, b=4, c=x, d=8$.

Extremes are 3 and 8. Means are 4 and $x$.

Using the property: Product of extremes = Product of means.

Calculate the products:

We have a simple linear equation $4x = 24$. To find $x$, divide both sides by 4 (or transpose 4 which is multiplying $x$).

Perform the division:

So, the missing term is 6. The proportion is $3 : 4 :: 6 : 8$.

Check: $3:4 = \frac{3}{4}$. $6:8 = \frac{6 \div 2}{8 \div 2} = \frac{3}{4}$. Since $\frac{3}{4} = \frac{3}{4}$, the proportion is correct.

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Unitary Method

In the previous sections, we learned about ratios and proportions. Ratio is a comparison of two quantities, and proportion is an equality between two ratios. The concept of proportion is very useful for solving problems where quantities are related in a consistent way (e.g., if the cost per item is constant). The Unitary Method is a widely used technique that leverages the idea of finding the value of 'one unit' to solve such proportional problems efficiently.

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What is the Unitary Method?

The word 'unitary' comes from 'unit', which means 'one'. The Unitary Method is a method used to solve problems by first finding the value of a single unit and then using that value to find the value of the required number of units.

This method is particularly applicable to problems involving direct proportion, where an increase in one quantity leads to a proportional increase in another quantity, and a decrease in one leads to a proportional decrease in the other (e.g., more items cost more, less time covers less distance at constant speed).

The process essentially breaks down the problem into two manageable steps.

Steps in the Unitary Method:

Suppose you are given the total value (cost, distance, quantity, etc.) corresponding to a certain number of units, and you need to find the total value for a different number of units.

1. Step 1: Find the value of one unit. To do this, divide the given total value by the given number of units.

   Value of 1 unit $= \frac{\text{Total value of given number of units}}{\text{Given number of units}} $

   This step determines the rate or value associated with a single item or unit quantity (e.g., cost per pen, distance per hour).

2. Step 2: Find the value of the required number of units. Once you know the value of one unit, multiply this unit value by the required number of units to get the final answer.

   Value of required units $=$ (Value of 1 unit) $\times$ (Required number of units)

   This step scales up the unit value to the desired quantity.

The unitary method provides a clear and logical way to solve many real-life problems involving proportional relationships.

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Examples using Unitary Method (Direct Proportion)

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Inverse Proportion and Unitary Method (Brief Mention)

The unitary method as described above is designed for problems of direct proportion. However, there are also situations involving inverse proportion, where an increase in one quantity causes a proportional decrease in another quantity (e.g., if more people work on a task, the time taken to complete the task decreases). For inverse proportion problems, the product of the two quantities is constant ($x_1 y_1 = x_2 y_2$).

While a full treatment of inverse proportion with the unitary method is often covered in higher classes, the basic idea of finding the 'unit' value can be adapted:

For inverse proportion, if 'n' units of the first quantity correspond to a total value 'V' of the second quantity, then 1 unit of the first quantity corresponds to $V \times n$ units of the second quantity (e.g., 1 worker takes $10 \times 12$ days if 12 workers take 10 days).

This 'unit value' is then used to find the value for the required number of units, typically by division.

Example (Inverse Proportion): If 12 workers can build a wall in 10 days, how many days will 15 workers take to build the same wall? (Assuming all workers work at the same rate, this is inverse proportion: more workers, less time).

Total work done is constant. Work = Number of workers $\times$ Number of days.

Total work $= 12 \text{ workers} \times 10 \text{ days} = 120$ 'worker-days'.

Now, 15 workers need to complete 120 worker-days of work. Let D be the number of days 15 workers take.

Divide by 15:

So, 15 workers will take 8 days.

Using Unitary Method logic for Inverse Proportion:

12 workers take 10 days.

1 worker would take $10 \times 12 = 120$ days (If one worker does the whole job alone, it takes much longer). This is the value for 'one unit' (one worker).

15 workers would take $\frac{120 \text{ days}}{15} = 8$ days (If 15 workers share the work, it takes less time).

While the term 'unitary method' in Class 7 usually refers to direct proportion, it's good to be aware that similar logic applies with an adjustment for inverse relationships.

Percentage and its Conversion

In the previous sections, we looked at comparing quantities using ratios and proportions. Percentage is another very common and useful way to compare quantities, especially when the comparison needs to be made on a standard base of 100. The concept of percentage is widely used in various real-life scenarios, such as calculating discounts, expressing interest rates, reporting survey results, or comparing performance.

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What is Percentage?

The word "percent" comes from the Latin phrase "per centum", which means "by the hundred" or "for every hundred". A Percentage is a way of expressing a number or a ratio as a fraction of 100. The symbol used for percent is %.

When we say "$x$ percent" (written as $x\%$), it means $x$ parts out of 100 equal parts of a whole.

So, $x\% = \frac{x}{100}$.

Percentage is essentially a ratio with the consequent (second term) being 100.

Example: $50\%$ means 50 per hundred, which can be written as the fraction $\frac{50}{100}$.

Example: $75\%$ means 75 per hundred, or $\frac{75}{100}$.

Example: $100\%$ means 100 per hundred, or $\frac{100}{100} = 1$ (one whole).

Example: $10\%$ means 10 per hundred, or $\frac{10}{100} = \frac{1}{10}$.

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Converting Percentages

We can convert percentages into fractions or decimals, and vice-versa.

1. Converting Percentage to Fraction

To convert a percentage into a fraction, remember that $x\%$ means $\frac{x}{100}$. So, divide the number preceding the percent sign by 100, and then simplify the resulting fraction.

Steps:

1. Remove the % sign.
2. Write the number as the numerator and 100 as the denominator.
3. Simplify the fraction to its lowest terms, if possible.

Examples:

(a) Convert 50% to a fraction.

Simplify the fraction $\frac{50}{100}$ (divide numerator and denominator by HCF = 50).

So, $50\% = \frac{1}{2}$.

(b) Convert 75% to a fraction.

Simplify $\frac{75}{100}$ (divide by HCF = 25).

So, $75\% = \frac{3}{4}$.

(c) Convert $12.5\%$ to a fraction.

To remove the decimal from the numerator, multiply both the numerator and denominator by 10:

Simplify $\frac{125}{1000}$ (HCF is 125).

So, $12.5\% = \frac{1}{8}$.

(d) Convert $33\frac{1}{3}\%$ to a fraction.

First, convert the mixed number percentage to an improper fraction:

Now, convert this percentage fraction to a standard fraction by dividing by 100:

Multiply and simplify (cancel 100):

So, $33\frac{1}{3}\% = \frac{1}{3}$.

2. Converting Percentage to Decimal

To convert a percentage to a decimal, remember that $x\% = \frac{x}{100}$. Dividing by 100 involves shifting the decimal point two places to the left.

Steps:

1. Remove the % sign.
2. Move the decimal point two places to the left. If the number is a whole number, imagine the decimal point is at the end (e.g., $45 = 45.$). Add leading zeros if needed.

Examples:

(a) Convert 45% to a decimal.

So, $45\% = 0.45$.

(b) Convert 8% to a decimal.

So, $8\% = 0.08$.

(c) Convert 125% to a decimal.

So, $125\% = 1.25$.

(d) Convert $0.5\%$ to a decimal.

So, $0.5\% = 0.005$.

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Converting to Percentages

1. Converting Fraction to Percentage

To convert a fraction into a percentage, we need to express it as an equivalent fraction with a denominator of 100. A simple way to do this is to multiply the fraction by 100 and add the % sign.

Formula:

Examples:

(a) Convert $\frac{1}{4}$ to a percentage.

(b) Convert $\frac{3}{5}$ to a percentage.

(c) Convert $\frac{7}{20}$ to a percentage.

(d) Convert $1\frac{1}{2}$ to a percentage.

First, convert the mixed fraction to an improper fraction: $1\frac{1}{2} = \frac{(1 \times 2) + 1}{2} = \frac{3}{2}$.

Now, convert the improper fraction to a percentage:

Percentages can indeed be greater than 100%.

2. Converting Decimal to Percentage

To convert a decimal number to a percentage, we multiply the decimal by 100 and add the % sign. Multiplying by 100 means shifting the decimal point two places to the right.

Formula:

Steps:

1. Move the decimal point two places to the right. Add trailing zeros if needed.
2. Add the % sign at the end.

Examples:

(a) Convert $0.65$ to a percentage.

Move decimal in $0.65$ two places right: $65. \to 65$. Add % sign.

(b) Convert $0.9$ to a percentage.

Move decimal in $0.9$ two places right. Add a zero: $0.90 \to 90$. Add % sign.

(c) Convert $0.02$ to a percentage.

Move decimal in $0.02$ two places right: $0.02 \to 00.2 \to 2$. Add % sign.

(d) Convert $1.75$ to a percentage.

Move decimal in $1.75$ two places right: $1.75 \to 17.5 \to 175$. Add % sign.

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Interpreting Percentages

Percentages help us understand proportions and compare quantities easily by providing a common base of 100.

• Percentage as a Part of a Whole: A percentage represents a fraction of a whole where the whole is considered to be 100 units. If a student scores 80 marks out of a maximum of 100 marks, their score is 80%. This means they achieved 80 parts out of 100 total parts.
• Percentages Greater than 100%: A percentage greater than 100% represents a quantity that is more than one whole. For instance, if the population of a town was 1000 last year and is 1500 this year, the current population is $\frac{1500}{1000} \times 100\% = 1.5 \times 100\% = 150\%$ of last year's population. The increase in population is $50\%$.
• Using Percentages for Comparison: Percentages make it easy to compare different proportions. If Student A scores 40 out of 50 marks and Student B scores 70 out of 80 marks, simply comparing 40 and 70 doesn't tell us who performed better relative to the total marks. Converting to percentages helps: Student A's score is $\frac{40}{50} \times 100\% = 80\%$. Student B's score is $\frac{70}{80} \times 100\% = \frac{7}{8} \times 100\% = 0.875 \times 100\% = 87.5\%$. Comparing 80% and 87.5%, we see that Student B performed better relatively.

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Finding the Percentage of a Given Quantity

In the previous section, we learned what percentage is and how to convert between fractions, decimals, and percentages. A very practical application of percentages is to calculate a specific percentage of a given number or quantity. This is commonly used in calculating discounts, taxes, parts of a whole amount, or solving problems related to marks, money, and measurements.

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Method to Find Percentage of a Quantity

To find $x\%$ of a quantity $Q$, you can follow these steps:

1. Convert the percentage $x\%$ into its fractional form, which is $\frac{x}{100}$.
2. Multiply this fraction $\frac{x}{100}$ by the given quantity $Q$. Remember that the word "of" in such contexts means multiplication.

Formula:

Alternatively, you can convert the percentage to its decimal form and then multiply by the quantity:

The choice between using the fractional or decimal form depends on the numbers involved and which calculation seems easier.

Understanding "of" in Mathematics:

As mentioned, in mathematical phrases like "$x\%$ of $Q$" or "a fraction of a quantity", the word "of" is a keyword that indicates the operation of multiplication.

Example: $\frac{1}{2}$ of 100 = $\frac{1}{2} \times 100 = 50$.

Example: $25\%$ of 160 means $25\% \times 160$.

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Examples

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Finding What Percentage One Quantity is of Another

Sometimes, you are given two quantities and need to express the first quantity as a percentage of the second quantity (which serves as the base or the whole). To find what percentage a quantity A is of another quantity B:

1. Ensure that both quantities A and B are in the same units.
2. Form a fraction with quantity A as the numerator and quantity B as the denominator: $\frac{A}{B}$.
3. Multiply this fraction by 100.
4. Add the % sign at the end of the result.

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Profit and Loss

In the world of business and trade, buying and selling goods is a fundamental activity. Whenever an article is purchased and then sold, the prices involved determine whether the transaction results in a financial gain or a financial loss. Understanding these concepts of Profit and Loss is very important in mathematics and for managing personal finances.

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Basic Terms in Profit and Loss

1. Cost Price (C.P.)

The Cost Price (C.P.) is the total amount of money paid to acquire an article or commodity. This includes not only the price at which the article was initially bought, but also any additional expenses incurred to bring the article into a condition and location suitable for selling. These additional expenses are called Overhead Expenses.

Effective C.P. = Purchase Price + Overhead Expenses.

Examples of overhead expenses include transportation charges, packaging costs, repair costs to make an item usable, or installation charges.

2. Selling Price (S.P.)

The Selling Price (S.P.) is the price at which an article or commodity is sold to a customer.

3. Profit (or Gain)

When an article is sold for a price higher than its cost price, the difference is called Profit or Gain. A profit occurs when the seller earns more money than they spent to acquire the item.

4. Loss

When an article is sold for a price lower than its cost price, the difference is called a Loss. A loss occurs when the seller receives less money than they spent to acquire the item.

5. No Profit, No Loss

If the Selling Price (S.P.) of an article is exactly equal to its Cost Price (C.P.), then there is neither any profit nor any loss in the transaction. S.P. = C.P.

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Calculating Profit Percentage and Loss Percentage

Profit and loss are often expressed as a percentage to provide a standardised measure of performance relative to the initial investment (the cost price). The Cost Price (C.P.) is always used as the base for calculating profit or loss percentage, unless a different base (like selling price) is specifically mentioned (which is rare in basic problems).

1. Profit Percentage (Profit %)

The Profit Percentage (Profit %) tells us how much profit was made for every $\textsf{₹} 100$ of the cost price. It is calculated as the ratio of the profit to the cost price, multiplied by 100.

2. Loss Percentage (Loss %)

The Loss Percentage (Loss %) tells us how much loss was incurred for every $\textsf{₹} 100$ of the cost price. It is calculated as the ratio of the loss to the cost price, multiplied by 100.

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Formulas to Find S.P. or C.P.

Using the definitions of profit and loss percentages, we can derive formulas to find the Selling Price (S.P.) if the Cost Price (C.P.) and percentage profit or loss are known, or to find the C.P. if the S.P. and percentage profit or loss are known.

1. Finding S.P. when C.P. and Profit % are given:

If there is a profit of Profit %, the selling price is the cost price plus the profit amount. Profit amount is Profit % of C.P.

Profit Amount = Profit % of C.P. $= \frac{\text{Profit %}}{100} \times \text{C.P.}$.

S.P. = C.P. + Profit Amount = C.P. + $(\frac{\text{Profit %}}{100} \times \text{C.P.})$.

We can factor out C.P.:

Combine the terms inside the parenthesis by finding a common denominator (100):

2. Finding S.P. when C.P. and Loss % are given:

If there is a loss of Loss %, the selling price is the cost price minus the loss amount. Loss amount is Loss % of C.P.

Loss Amount = Loss % of C.P. $= \frac{\text{Loss %}}{100} \times \text{C.P.}$.

S.P. = C.P. $-$ Loss Amount = C.P. $-$ $(\frac{\text{Loss %}}{100} \times \text{C.P.})$.

We can factor out C.P.:

Combine the terms inside the parenthesis:

3. Finding C.P. when S.P. and Profit % are given:

We can rearrange the formula for S.P. when profit is made to find C.P.

S.P. = C.P. $\times (\frac{100 + \text{Profit %}}{100})$.

To isolate C.P., divide both sides by $(\frac{100 + \text{Profit %}}{100})$. Dividing by a fraction is the same as multiplying by its reciprocal.

4. Finding C.P. when S.P. and Loss % are given:

We rearrange the formula for S.P. when a loss is incurred to find C.P.

S.P. = C.P. $\times (\frac{100 - \text{Loss %}}{100})$.

To isolate C.P., divide both sides by $(\frac{100 - \text{Loss %}}{100})$. Multiply by its reciprocal.

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Simple Interest

In financial transactions involving borrowing or lending money, an additional amount is usually involved as a charge for using the money. This additional amount is called Interest. When you borrow money, you pay interest; when you lend money (or deposit it in a bank), you receive interest. There are different ways to calculate interest, and the most basic one is called Simple Interest (S.I.).

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Basic Terms in Simple Interest

1. Principal (P)

The Principal is the original sum of money that is borrowed, lent, or deposited. It is the initial amount on which interest is calculated. It is commonly denoted by the letter P.

2. Rate of Interest (R)

The Rate of Interest is the percentage at which the interest is calculated on the principal amount. It is usually specified as a percentage per annum (p.a.), which means "per year". It is denoted by R (or sometimes r).

Example: If the rate of interest is 10% p.a., it means that for every $\textsf{₹} 100$ of the principal amount, an interest of $\textsf{₹} 10$ will be charged or earned over a period of one year.

3. Time (T)

The Time is the duration for which the money is borrowed, lent, or deposited. For the Simple Interest formula, the time period is usually required in years. It is denoted by T (or sometimes n).

If the time is given in units other than years (like months or days), you must convert it into years before using the formula:

• If time is given in months: Time (in years) $= \frac{\text{Number of months}}{12}$. Example: 6 months $= \frac{6}{12}$ years $= \frac{1}{2}$ year. 18 months $= \frac{18}{12}$ years $= \frac{3}{2}$ years.
• If time is given in days: Time (in years) $= \frac{\text{Number of days}}{365}$ (for an ordinary year) or $\frac{\text{Number of days}}{366}$ (for a leap year). Unless specified otherwise, assume an ordinary year with 365 days. Example: 73 days $= \frac{73}{365}$ years $= \frac{1}{5}$ year (since 73 $\times$ 5 = 365).

4. Simple Interest (S.I.)

Simple Interest (S.I.) is the interest calculated solely on the original principal amount for the entire duration of the loan or deposit. In simple interest calculations, the principal amount remains constant over the entire time period.

5. Amount (A)

The Amount is the total money that is repaid by the borrower to the lender (or received by the lender/depositor from the bank) at the end of the agreed time period. It is the sum of the original Principal and the calculated Simple Interest.

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Formula for Simple Interest

The Simple Interest (S.I.) is directly proportional to the Principal (P), the Rate of Interest (R), and the Time (T). The formula for calculating Simple Interest is:

Where:

• S.I. is the Simple Interest amount (in Rupees).
• P is the Principal amount (in Rupees).
• R is the Rate of Interest (as a number, representing % per annum). For example, if the rate is 8%, use 8 in the formula.
• T is the Time period (in years).
• The division by 100 in the formula accounts for the 'percent' in the rate (R%).

Derivation of the S.I. Formula:

Let P be the principal amount, R be the rate of interest per annum (as R%), and T be the time in years.

By the definition of rate R%, the interest on $\textsf{₹} 100$ for 1 year is $\textsf{₹} R$.

Interest on $\textsf{₹} 1$ for 1 year $= \textsf{₹} \frac{R}{100}$.

Interest on $\textsf{₹} P$ for 1 year $$= (\text{Interest on } \textsf{₹} 1 \text{ for 1 year}) \times P = \textsf{₹} \frac{R}{100} \times P = \textsf{₹} \frac{PR}{100}$$

Simple Interest is calculated on the original principal for the entire time T. So, the interest on $\textsf{₹} P$ for T years $$= (\text{Interest on } \textsf{₹} P \text{ for 1 year}) \times T = \textsf{₹} (\frac{PR}{100}) \times T = \textsf{₹} \frac{P \times R \times T}{100}$$

Thus, S.I. $= \frac{P \times R \times T}{100}$.

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Calculating P, R, or T when other values are known

The Simple Interest formula $S.I. = \frac{P \times R \times T}{100}$ is an equation with four variables (S.I., P, R, T). If we know any three of these variables, we can use the formula to find the fourth one by rearranging the equation.

From $S.I. \times 100 = P \times R \times T$, we can derive:

1. To find Principal (P): Divide $(S.I. \times 100)$ by $(R \times T)$.

   P $= \frac{S.I. \times 100}{R \times T} $

2. To find Rate of Interest (R): Divide $(S.I. \times 100)$ by $(P \times T)$.

   R $= \frac{S.I. \times 100}{P \times T} $ (Result is in % p.a.)

3. To find Time (T): Divide $(S.I. \times 100)$ by $(P \times R)$.

   T $= \frac{S.I. \times 100}{P \times R} $ (Result is in years)

These derived formulas can be used directly, or you can always substitute the known values into the main formula $S.I. = \frac{P \times R \times T}{100}$ and solve the resulting equation for the unknown variable using algebraic methods (like transposition).

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