Ratio and Proportion Class 6 NCERT: Concepts, Solved Examples & Practice

Welcome, young mathematician! Have you ever compared your height with your friend's height, or wondered how to share a box of chocolates fairly? In everyday life, we often compare quantities. Sometimes we subtract to see who has more, but when we want to see how many times one quantity is of another, we use a powerful math tool called Ratio. In this chapter of ratio and proportion class 6 ncert, we will master how to express ratios, simplify them, understand what equal ratios (proportions) are, and solve real-world problems using the Unitary Method. This chapter is super important because it lays the foundation for percentages, fractions, and practical scaling in higher classes. By the end of this guide, you will be solving CBSE class 6 ratio and proportion problems with absolute confidence. Let's grab our digital sketchpad and start learning!

Understanding Ratio: Comparing by Division

Comparing two quantities of the same unit by division is called finding their ratio. We use the symbol ':' (read as 'is to') to represent a ratio. For example, if there are 20 girls and 40 boys in a classroom, we can compare them by division: 20/40 = 1/2. We write this ratio as 1 : 2.

There are two crucial rules to remember when writing ratios:

  1. The order of terms matters: The ratio of girls to boys is 1 : 2, but the ratio of boys to girls is 2 : 1. You must write the first mentioned term in the numerator and the second in the denominator.
  2. Units must be identical: You cannot find the ratio between 50 paise and 2 rupees directly. You must first convert rupees to paise (2 rupees = 200 paise) and then calculate the ratio as 50 : 200, which simplifies to 1 : 4.

Key Mathematical Terms

Ratio
A comparison of two mathematical quantities of the same kind and with the same unit, obtained by dividing the first quantity by the second.
Proportion
An equation that states that two given ratios are equal. If a : b = c : d, then a, b, c, d are in proportion.
Unitary Method
A method of solving problems where we first find the value of a single unit (by division) and then find the value of the required number of units (by multiplication).

How to Simplify and Find Equivalent Ratios

  1. Step 1: Write the ratio as a fraction — Express the ratio a : b in the fraction form a/b.
  2. Step 2: Find the Highest Common Factor (HCF) — Find the HCF of the numerator and the denominator. For example, for 12/18, the HCF of 12 and 18 is 6.
  3. Step 3: Divide both terms by the HCF — Divide both the numerator and denominator by their HCF to get the simplest form: (12 / 6) / (18 / 6) = 2/3 or 2 : 3.
  4. Step 4: Find equivalent ratios — To find other equivalent ratios, multiply or divide both terms by the same non-zero number. For example, (2 2) / (3 2) = 4/6 (4 : 6 is equivalent to 2 : 3).

Proportion Rule: Extremes and Means

When four numbers a, b, c, d are in proportion, we write them as a : b :: c : d. Here, the first and fourth terms (a and d) are called the extreme terms (or extremes), while the second and third terms (b and c) are called the middle terms (or means).

The Golden Rule:
Product of Extremes = Product of Means (a d = b c)

If this equation holds true, the terms are in proportion. If not, they are not! Use this trick to solve 'find the missing term' questions instantly on your exam paper.

Practice Questions with Solutions

  • Q: Find the ratio of 30 minutes to 1.5 hours.
    A: Step 1: Identify the units. The units are different (minutes and hours). We must convert them to be the same.
    Step 2: Convert hours to minutes. Since 1 hour = 60 minutes, 1.5 hours = 1.5 * 60 = 90 minutes.
    Step 3: Write the ratio as a fraction: 30/90.
    Step 4: Simplify the fraction by dividing both terms by their HCF, which is 30. (30 / 30) / (90 / 30) = 1/3.
    Final answer: The ratio of 30 minutes to 1.5 hours is 1 : 3.

  • Q: Determine if the ratios 25 g : 30 g and 40 kg : 48 kg form a proportion.
    A: Step 1: Simplify the first ratio: 25/30. Dividing by their HCF (5), we get 5/6 or 5 : 6.
    Step 2: Simplify the second ratio: 40/48. Dividing by their HCF (8), we get 5/6 or 5 : 6.
    Step 3: Compare both simplified ratios. Since both ratios are equal to 5 : 6, they form a proportion.
    Final answer: Yes, the ratios 25 g : 30 g and 40 kg : 48 kg form a proportion.

  • Q: If the cost of 6 cans of juice is Rs. 210, what will be the cost of 4 cans of juice?
    A: Step 1: Identify that this is a Unitary Method problem.
    Step 2: Find the cost of 1 can of juice. Cost of 1 can = Total Cost / Total Cans = 210 / 6 = Rs. 35.
    Step 3: Find the cost of 4 cans of juice. Cost of 4 cans = Cost of 1 can 4 = 35 4 = Rs. 140.
    Final answer: The cost of 4 cans of juice is Rs. 140.

  • Q: Divide Rs. 1,200 between Amit and Rahul in the ratio 5 : 7.
    A: Step 1: Find the sum of the ratio terms. Sum of parts = 5 + 7 = 12.
    Step 2: Calculate Amit's share. Amit gets 5/12 of the total money. Amit's share = (5/12) 1200 = 5 100 = Rs. 500.
    Step 3: Calculate Rahul's share. Rahul gets 7/12 of the total money. Rahul's share = (7/12) 1200 = 7 100 = Rs. 700.
    Step 4: Check if the sum is correct. Rs. 500 + Rs. 700 = Rs. 1200.
    Final answer: Amit gets Rs. 500 and Rahul gets Rs. 700.

Frequently Asked Questions

What is the main difference between ratio and proportion?

A ratio compares two quantities of the same unit by dividing them. A proportion is a mathematical statement showing that two given ratios are equal.

Why doesn't a ratio have any units?

A ratio has no units because it is a comparison of two quantities of the same kind. When we divide them, the similar units cancel each other out, leaving a pure number.

How do you identify if four numbers are in proportion?

Four numbers are in proportion if the product of the first and fourth numbers (extremes) equals the product of the second and third numbers (means).