Chapter 12 Ratio And Proportion (Concepts)

Introduction to Ratio

In earlier chapters, we learned about numbers, integers, fractions, decimals, and how to perform operations on them. We also looked at how to compare numbers and quantities. This chapter introduces a new and powerful way to compare quantities: using Ratio. Understanding ratios will help us relate different amounts and solve various real-life problems. We will then extend this concept to Proportion and the Unitary Method.

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What is a Ratio?

In everyday life, we often compare quantities. For example, we might say one person is taller than another, or one bag of rice is heavier than another. Ratio provides a specific way to compare two quantities. A Ratio is a comparison of two quantities of the same kind and in the same units by using division.

Example: Suppose a class has 20 girls and 15 boys. We can compare the number of girls to the number of boys. The ratio of the number of girls to the number of boys is the number of girls divided by the number of boys.

This fraction can be simplified by dividing both the numerator and the denominator by their Highest Common Factor (HCF), which is 5.

A ratio is usually written using a colon ($:$). The ratio of 20 to 15 is written as $20:15$. The simplified ratio is $4:3$. This means for every 4 girls in the class, there are 3 boys.

The ratio $a$ to $b$ is written as $a:b$ or as a fraction $\frac{a}{b}$ (where $b \neq 0$).

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Terms of a Ratio

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

• The first term, $a$, is called the Antecedent. It is the numerator when the ratio is written as a fraction.
• The second term, $b$, is called the Consequent. It is the denominator when the ratio is written as a fraction.

Example: In the ratio $4:3$ (from the girls to boys example), $4$ is the antecedent and $3$ is the consequent.

Example: In the ratio $3:5$, $3$ is the antecedent and $5$ is the consequent.

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Important Points about Ratio

To correctly understand and use ratios, keep the following points in mind:

• Order Matters: The order of the terms in a ratio is very important. The ratio of quantity A to quantity B is written as A:B, which is different from the ratio of quantity B to quantity A, written as B:A (unless A and B are equal). In the class example, the ratio of girls to boys is $4:3$. The ratio of boys to girls is $15:20$, which simplifies to $3:4$. Notice that $4:3$ is different from $3:4$.

• Quantities of the Same Kind: A ratio can only compare quantities that are of the same kind. You can compare lengths with lengths, weights with weights, or numbers of students with numbers of students. You cannot find the ratio of someone's height to their weight, or the ratio of the number of apples to the price of oranges.

• Quantities in the Same Units: Before finding a ratio, the two quantities must be expressed in the same unit. If the units are different, you must convert one quantity so that both are in the same unit.

  Example: Find the ratio of 1 metre to 50 cm.

  The quantities are length, but the units are different (metres and centimetres).

  Convert 1 metre to centimetres: $1 \text{ metre} = 100 \text{ cm}$.

  Now find the ratio of 100 cm to 50 cm:

  Ratio $= 100 \text{ cm} : 50 \text{ cm} = 100:50$.

  Simplify the ratio: $100:50$. The HCF of 100 and 50 is 50.

  Divide both terms by 50: $100 \div 50 = 2$, and $50 \div 50 = 1$.

  The simplified ratio is $2:1$.

• Ratio Has No Units: When you find the ratio of two quantities in the same units, the units cancel out during the division. Therefore, a ratio itself does not have any units.

  Example: $\frac{100 \text{ cm}}{50 \text{ cm}} = \frac{100}{50} = \frac{2}{1}$. The 'cm' unit cancels out.

  The ratio $2:1$ is just a number, not 2 cm : 1 cm or 2 : 1 cm.

• Simplifying Ratios: Just like fractions, ratios should usually be expressed in their simplest form. This is done by dividing both terms of the ratio by their Highest Common Factor (HCF).

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Finding Ratio in Simplest Form

To express a ratio in its simplest form, find the HCF of the antecedent and the consequent, and divide both terms by the HCF. This is equivalent to reducing the fraction $\frac{\text{Antecedent}}{\text{Consequent}}$ to its lowest terms.

Example: Find the ratio of 40 minutes to 1 hour in simplest form.

Step 1: Ensure the quantities are in the same units.

Convert 1 hour to minutes: $1 \text{ hour} = 60 \text{ minutes}$.

The quantities are 40 minutes and 60 minutes.

Step 2: Write the ratio.

Ratio $= 40 \text{ minutes} : 60 \text{ minutes} = 40:60$.

Step 3: Simplify the ratio by finding the HCF of the terms (40 and 60) and dividing both terms by the HCF.

Factors of 40: 1, 2, 4, 5, 8, 10, 20, 40

Factors of 60: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60

The HCF of 40 and 60 is 20.

Divide both terms by 20:

The ratio in simplest form is $2:3$.

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Introduction to Proportion

In the previous section, we learned about ratio as a way to compare two quantities of the same kind and in the same units by division. Now, we will extend this concept to situations where we compare two ratios. When two ratios are equal, we say they are in Proportion.

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What is Proportion?

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

We use the symbol '::' or '=' to indicate that two ratios are in proportion.

For example, consider the ratio $2:3$. The fraction form is $\frac{2}{3}$. Now consider the ratio $4:6$. The fraction form is $\frac{4}{6}$. If we simplify the fraction $\frac{4}{6}$ by dividing the numerator and denominator by 2, we get $\frac{2}{3}$. Since $\frac{2}{3}$ is equal to $\frac{4}{6}$, the ratio $2:3$ is equal to the ratio $4:6$.

Therefore, the numbers 2, 3, 4, and 6 are in proportion. This can be written in two ways:

• Using the '::' symbol: $2:3 :: 4:6$
• Using the '=' symbol: $2:3 = 4:6$

Both notations mean the same thing: the ratio of 2 to 3 is equal to the ratio of 4 to 6. When we write $2:3 :: 4:6$, we usually read it as "2 is to 3 as 4 is to 6".

In general, four numbers $a, b, c,$ and $d$ are in proportion if the ratio $a:b$ is equal to the ratio $c:d$. That is, $a:b :: c:d \iff \frac{a}{b} = \frac{c}{d}$.

Note: For the proportion $a:b :: c:d$ to be meaningful, $a, b, c,$ and $d$ should be non-zero quantities, and $a$ and $b$ must be of the same kind and in the same units, and $c$ and $d$ must also be of the same kind and in the same units. The kind of quantities $a, b$ can be different from the kind of quantities $c, d$ (e.g., ratio of lengths = ratio of costs), but within each ratio, the quantities must be of the same kind and unit.

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Terms of a Proportion

When four quantities $a, b, c,$ and $d$ are in proportion, written as $a:b :: c:d$, these four quantities are called the Terms of the Proportion.

These terms have specific names based on their position:

• The first term ($a$) and the fourth term ($d$) are called the Extreme Terms or simply the Extremes.
• The second term ($b$) and the third term ($c$) are called the Middle Terms or the Means.

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Rule of Proportion (Product of Extremes and Means)

There is a fundamental rule that holds true for any proportion. If four quantities are in proportion, the product of the extreme terms is equal to the product of the middle terms.

If $a:b :: c:d$, then it means $\frac{a}{b} = \frac{c}{d}$.

By cross-multiplication (multiplying the numerator of the first fraction by the denominator of the second and vice versa), we get:

This is the Rule of Proportion:

$\text{Product of Extremes} = \text{Product of Means}$

This rule is extremely important. It allows us to check if two ratios are equal and to find a missing term in a proportion if the other three terms are known.

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Checking for Proportion

To check if four numbers $a, b, c,$ and $d$ are in proportion (in that order, i.e., if $a:b :: c:d$), we can use the Rule of Proportion. Calculate the product of the extremes ($a \times d$) and the product of the means ($b \times c$). If the two products are equal, then the numbers are in proportion; otherwise, they are not.

Example: Are the numbers 3, 4, 6, and 8 in proportion?

We need to check if $3:4 :: 6:8$ is true.

• Identify the extremes: $a=3, d=8$. Product of extremes $= 3 \times 8 = 24$.
• Identify the means: $b=4, c=6$. Product of means $= 4 \times 6 = 24$.

Since the Product of Extremes ($24$) is equal to the Product of Means ($24$), the numbers 3, 4, 6, and 8 are in proportion.

Example: Are the numbers 2, 3, 5, and 6 in proportion?

We need to check if $2:3 :: 5:6$ is true.

• Product of extremes $= 2 \times 6 = 12$.
• Product of means $= 3 \times 5 = 15$.

Since the Product of Extremes ($12$) is NOT equal to the Product of Means ($15$), the numbers 2, 3, 5, and 6 are NOT in proportion.

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Finding a Missing Term in Proportion

The Rule of Proportion ($a \times d = b \times c$) is very useful when we know three terms of a proportion and need to find the fourth, unknown term. We can represent the unknown term by a variable (like $x$) and then use the rule to form an equation and solve for the variable.

Example: Find the missing term in the proportion $5:10 :: x:30$.

The four terms are $a=5, b=10, c=x, d=30$.

The extremes are the first and fourth terms: 5 and 30.

The means are the second and third terms: 10 and $x$.

Using the Rule of Proportion: Product of Extremes = Product of Means.

Calculate the product on the left side:

To find the value of $x$, we need to isolate $x$. Since $x$ is being multiplied by 10, we do the inverse operation, which is division. Divide both sides of the equation by 10 to maintain the equality:

Perform the division:

So, the missing term is 15. The proportion is $5:10 :: 15:30$.

We can check our answer: Product of extremes $= 5 \times 30 = 150$. Product of means $= 10 \times 15 = 150$. Since $150 = 150$, the proportion is true.

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Unitary Method

In the previous sections, we learned about ratios and proportions, which are used to compare quantities and state the equality of ratios. Now, we will learn about a practical method for solving problems involving quantities that vary in direct proportion. This method is called the Unitary Method.

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What is the Unitary Method?

The term 'unitary' is derived from the word 'unit', meaning 'one'. The Unitary Method is a technique used to find the value of a desired number of units when the value of a given number of units is known. The core idea is to first find the value of a single, individual unit, and then use that value to calculate the value for the required quantity.

This method is based on the principle of direct proportion, where if you increase the number of items, the total cost (or quantity, distance, etc.) also increases proportionally, and if you decrease the number of items, the total cost decreases proportionally. This means the cost per item (value per unit) remains constant.

The unitary method typically involves two key steps:

1. Step 1: Find the value of one unit. This is done by dividing the given total value by the number of units.
2. Step 2: Find the value of the required number of units. This is done by multiplying the value of one unit (found in Step 1) by the required number of units.

This method is widely used in everyday situations and problems involving cost and quantity, distance and time (at a constant speed), work and time (at a constant rate), etc.

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Steps of the Unitary Method (General Procedure)

Let's formalise the steps. Suppose you are given that 'n' items (or units) have a total value of 'V'. You need to find the value of 'm' items (or units).

1. Step 1: Find the value of one item (the unit value). Divide the total given value (V) by the number of items (n) to get the value per item.

   $\text{Value of 1 item} = \frac{\text{Total Value (V)}}{\text{Number of items (n)}}$

   The result of this step tells you the value corresponding to a single unit quantity.

2. Step 2: Find the value of the required number of items. Multiply the value of 1 item (found in Step 1) by the required number of items (m).

   $\text{Value of m items} = (\text{Value of 1 item}) \times \text{Number of required items (m)}$

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

The success of the unitary method relies on the relationship between the quantities being directly proportional, meaning the ratio between the value and the number of units remains constant.

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