Ratio and Proportion: CBSE Class 6 Maths

Welcome to the world of Ratio and Proportion! Have you ever wondered how a recipe tells you to use 2 cups of flour for every 1 cup of sugar? Or how we can say there are twice as many boys as girls in a line? This is all about ratios! A ratio is simply a way to compare two quantities. It helps us understand the relationship between them.

In this chapter, we'll go one step further and learn about proportion. Proportion is when two ratios are equal to each other. It's a powerful idea that helps us solve many real-life problems, like figuring out how much petrol a car needs for a long trip or how many ingredients are needed to bake a bigger cake. By the end of this lesson, you will master comparing quantities using ratios, checking if two ratios are in proportion, and solving problems using a handy technique called the unitary method. Let's start comparing!

What is a Ratio?

A ratio is a comparison of two or more quantities of the same kind and in the same units. It tells us how much of one thing there is compared to another. We use a colon (:) symbol to represent a ratio. For example, if there are 5 pencils and 3 pens in a box, the ratio of pencils to pens is written as 5 : 3. We read this as "5 is to 3".

Key Points about Ratios:

  1. Order Matters: The ratio 5 : 3 (pencils to pens) is different from 3 : 5 (pens to pencils). The order must match the question being asked.
  2. Same Units: Before finding a ratio, both quantities must be in the same unit. For instance, to find the ratio of 50 cm to 2 metres, you must first convert 2 metres to 200 cm. The ratio would then be 50 : 200.
  3. Simplest Form: Ratios are usually expressed in their simplest form, just like fractions. The ratio 50 : 200 can be simplified by dividing both parts by their highest common factor, which is 50. So, 50 ÷ 50 = 1 and 200 ÷ 50 = 4. The simplest form of the ratio is 1 : 4.

Understanding Proportion

Proportion is a statement that says two ratios are equal. Imagine you buy 2 chocolates for ₹10. The ratio of chocolates to cost is 2 : 10. Your friend buys 4 chocolates for ₹20. Their ratio is 4 : 20. Are these ratios equal? Let's simplify them.

Your ratio: 2 : 10 = 1 : 5 (dividing both by 2)
Your friend's ratio: 4 : 20 = 1 : 5 (dividing both by 4)

Since both ratios simplify to 1 : 5, they are equal. We can say that the four numbers 2, 10, 4, and 20 are in proportion. We write this using the symbol :: (which stands for 'as') or an equals sign =.

So, 2 : 10 :: 4 : 20 is read as "2 is to 10 as 4 is to 20".

In a proportion a : b :: c : d, the first and fourth terms (a and d) are called Extreme Terms (Extremes). The second and third terms (b and c) are called Middle Terms (Means).

A very important rule for proportion is: Product of Extremes = Product of Means.
For 2 : 10 :: 4 : 20, the product of extremes is 2 × 20 = 40, and the product of means is 10 × 4 = 40. Since they are equal, the numbers are in proportion.

How to Check if Numbers are in Proportion

  1. Example Problem — Let's check if the numbers 6, 18, 5, and 15 are in proportion. This means we are checking if the ratio 6 : 18 is equal to the ratio 5 : 15.
  2. Method 1: Simplifying Ratios — Step 1: Simplify the first ratio. The ratio is 6 : 18. The highest common factor of 6 and 18 is 6. 6 ÷ 6 = 1 18 ÷ 6 = 3 So, the simplified ratio is 1 : 3. Step 2: Simplify the second ratio. The ratio is 5 : 15. The highest common factor of 5 and 15 is 5. 5 ÷ 5 = 1 15 ÷ 5 = 3 So, the simplified ratio is 1 : 3. Step 3: Compare. Since both simplified ratios are the same (1 : 3 = 1 : 3), the numbers 6, 18, 5, and 15 are in proportion.
  3. Method 2: Product of Means and Extremes — Step 1: Identify the terms. In 6 : 18 :: 5 : 15: Extreme terms are 6 and 15. Middle terms are 18 and 5. Step 2: Calculate the product of extremes. Product of Extremes = 6 × 15 = 90. Step 3: Calculate the product of means. Product of Means = 18 × 5 = 90. Step 4: Compare the products. Since the Product of Extremes (90) is equal to the Product of Means (90), the numbers are in proportion.

Solving Problems with the Unitary Method

  • The Unitary Method is a technique to find the value of a single unit first, and then use that to find the value of the required number of units. It's very useful for ratio and proportion problems. Example: The cost of 5 kg of wheat is ₹150. What is the cost of 8 kg of wheat? Step 1: Find the value of one unit. Cost of 5 kg of wheat = ₹150 Cost of 1 kg of wheat = ₹150 ÷ 5 = ₹30 Step 2: Find the value of the required number of units. Now that we know the cost of 1 kg, we can find the cost of 8 kg. Cost of 8 kg of wheat = Cost of 1 kg × 8 = ₹30 × 8 = ₹240 Final Answer: The cost of 8 kg of wheat is ₹240.

Common Mistakes to Avoid

  • Ignoring Units: Always convert quantities to the same unit before finding a ratio. You cannot compare 1 metre to 50 centimetres directly. Convert 1 metre to 100 cm first.
  • Incorrect Order: The ratio of 'A to B' is different from the ratio of 'B to A'. Always write the numbers in the order they are asked.
  • Proportion Rule Mix-up: In a : b :: c : d, remember the rule is Product of Extremes (a × d) = Product of Means (b × c). Don't multiply the wrong pairs.
  • Unitary Method Error: In the unitary method, make sure you divide to find the value of one unit and then multiply to find the value of many. Doing it the other way around will give the wrong answer.

Practice Questions with Solutions

  • Q: Find the ratio of 90 cm to 1.5 m. A: Step 1: Convert both quantities to the same unit. It's easier to convert metres to centimetres. We know that 1 m = 100 cm. So, 1.5 m = 1.5 × 100 cm = 150 cm. Step 2: Write the ratio with the same units. The ratio is 90 cm to 150 cm, which we write as 90 : 150. Step 3: Simplify the ratio to its lowest terms. The highest common factor of 90 and 150 is 30. 90 ÷ 30 = 3 150 ÷ 30 = 5 Final answer: The ratio is 3 : 5.
  • Q: Do the ratios 15:20 and 24:32 form a proportion? A: Step 1: We will use the product of extremes and means method. The numbers in order are 15, 20, 24, 32. Extremes are the first and fourth terms: 15 and 32. Means are the second and third terms: 20 and 24. Step 2: Calculate the product of extremes. 15 × 32 = 480. Step 3: Calculate the product of means. 20 × 24 = 480. Step 4: Compare the products. Since the product of extremes (480) is equal to the product of means (480), the ratios form a proportion. Final answer: Yes, they form a proportion.
  • Q: If the cost of a dozen bananas (12 bananas) is ₹60, what is the cost of 7 bananas? A: Step 1: Use the unitary method to find the cost of one banana. We are given the cost of 12 bananas. Cost of 12 bananas = ₹60 Cost of 1 banana = ₹60 ÷ 12 = ₹5. Step 2: Now, calculate the cost of the required number of bananas (7). Cost of 7 bananas = Cost of 1 banana × 7 = ₹5 × 7 = ₹35. Final answer: The cost of 7 bananas is ₹35.
  • Q: Find the value of x if 8 : 12 :: x : 18. A: Step 1: We know that if numbers are in proportion, the product of extremes equals the product of means. Extremes are 8 and 18. Means are 12 and x. Step 2: Set up the equation based on the rule. Product of Extremes = Product of Means 8 × 18 = 12 × x 144 = 12 × x Step 3: Solve for x by dividing both sides by 12. x = 144 ÷ 12 x = 12. Final answer: The value of x is 12.

Frequently Asked Questions

What is the main difference between a ratio and a fraction?

A ratio compares two different quantities (like 3 girls to 4 boys, 3:4), whereas a fraction represents a part of a single whole (like 3 out of 7 total students, 3/7). While a ratio can be written as a fraction, its core meaning is about comparison.

Can a ratio have units?

No, a ratio is a pure number and has no units. This is because we compare quantities with the same units, so the units get cancelled out. For example, the ratio of 2 kg to 10 kg is 2:10 or 1:5, not 1:5 kg.

Why is the order so important in a ratio?

The order defines what is being compared to what. The ratio of teachers to students (e.g., 1:30) is very different from the ratio of students to teachers (30:1). Reversing the order changes the entire meaning of the comparison.

What is the unitary method used for?

The unitary method is a technique to find the value of a single unit from a given multiple, and then use that to find the value of any required multiple. It is very common for solving real-world problems involving cost, distance, and time.