NCERT Solutions for Class 6 Maths: Ratio Proportion Ex 12.3

Welcome, students! In this chapter on Ratio and Proportion, we've learned how to compare quantities. Now, in Exercise 12.3, we will learn a very powerful and practical technique called the Unitary Method. Have you ever been to a shop and wondered, 'If 5 pencils cost ₹25, how much would 8 pencils cost?' The unitary method helps you solve exactly these kinds of real-life problems!

This method is all about finding the value of 'one unit' first (like the cost of one pencil) and then using that to find the value of the number of units you need. It’s a simple, two-step process: first divide, then multiply. By the end of this lesson, you will be able to confidently solve word problems involving cost, distance, time, and more using this super useful method.

Understanding the Unitary Method

The Unitary Method is a technique used to solve problems involving variation. The core idea is to first calculate the value of a single unit and then use that value to find the value of the required number of units.

The word 'unitary' comes from the word 'unit', which means one. So, in every problem, our first goal is to find the value of one item.

Let's take a simple example. Imagine you buy 6 chocolate bars for ₹60.

  • Given: The cost of 6 bars is ₹60.
  • Goal: Find the cost of, say, 4 chocolate bars.

To do this using the unitary method, we first find the cost of one chocolate bar.

  • Cost of 6 bars = ₹60
  • Cost of 1 bar = ₹60 ÷ 6 = ₹10

Now that we know the cost of one bar, we can easily find the cost of any number of bars. To find the cost of 4 bars:

  • Cost of 4 bars = Cost of 1 bar × 4 = ₹10 × 4 = ₹40.

This two-step process (finding the value of one, then finding the value of many) is the essence of the unitary method.

How to Use the Unitary Method: A Step-by-Step Guide

  1. Step 1: Find the Value of One Unit — Read the problem carefully. You will be given the value of a certain number of items (like the cost of 10 pens). To find the value of one unit, you will almost always need to divide. For example, if 10 pens cost ₹50, the cost of 1 pen is ₹50 ÷ 10 = ₹5.
  2. Step 2: Find the Value of the Required Number of Units — Once you have the value of one unit, you can find the value of any number of units. To do this, you will need to multiply. For example, to find the cost of 7 pens, you would multiply the cost of one pen by 7: ₹5 × 7 = ₹35.

Worked Examples from Ex 12.3

  • Problem 1: If the cost of 7 m of cloth is ₹ 1470, find the cost of 5 m of cloth. Solution: Step 1: Find the cost of 1 m of cloth. Cost of 7 m of cloth = ₹ 1470 Cost of 1 m of cloth = ₹ 1470 ÷ 7 To calculate 1470 ÷ 7: 14 ÷ 7 = 2 7 ÷ 7 = 1 0 ÷ 7 = 0 So, Cost of 1 m of cloth = ₹ 210 Step 2: Find the cost of 5 m of cloth. Now, we multiply the cost of 1 m by 5. Cost of 5 m of cloth = Cost of 1 m × 5 Cost of 5 m of cloth = ₹ 210 × 5 = ₹ 1050 Final Answer: The cost of 5 m of cloth is ₹ 1050.
  • Problem 2: Ekta earns ₹ 3000 in 10 days. How much will she earn in 30 days? Solution: Step 1: Find how much Ekta earns in 1 day. Earning in 10 days = ₹ 3000 Earning in 1 day = ₹ 3000 ÷ 10 = ₹ 300 Step 2: Find how much she will earn in 30 days. Now, we multiply her earning for 1 day by 30. Earning in 30 days = Earning in 1 day × 30 Earning in 30 days = ₹ 300 × 30 = ₹ 9000 Final Answer: Ekta will earn ₹ 9000 in 30 days.

Important Tips for Accuracy

Always find the value of ONE first! Some students try to find a shortcut and get confused. The safest way is to always find the value of a single unit. Remember the two magic words: Divide, then Multiply.

Check your units. Make sure you are comparing the same things. If a question gives you a price in rupees and asks for a price, your answer should be in rupees. If it talks about kilograms, your answer should be in kilograms. Don't mix them up!

Practice Questions with Solutions

  • Q: The cost of a dozen soaps is ₹ 153.60. What will be the cost of 15 such soaps? A: Step 1: Find the cost of one soap. A dozen means 12. Cost of 12 soaps = ₹ 153.60 Cost of 1 soap = ₹ 153.60 ÷ 12 = ₹ 12.80 Step 2: Find the cost of 15 soaps. Cost of 15 soaps = Cost of 1 soap × 15 Cost of 15 soaps = ₹ 12.80 × 15 = ₹ 192 Final answer: The cost of 15 such soaps is ₹ 192.
  • Q: A car travels 90 km in 2.5 hours. How much time is required to cover 30 km with the same speed? A: Step 1: Find the time required to cover 1 km. Time taken for 90 km = 2.5 hours Time taken for 1 km = 2.5 ÷ 90 hours Step 2: Find the time required to cover 30 km. Time taken for 30 km = (2.5 ÷ 90) × 30 hours Time taken for 30 km = 2.5 × (30 / 90) = 2.5 × (1 / 3) = 2.5 / 3 hours. To simplify, 2.5 hours is 2.5 60 = 150 minutes. Time for 90 km = 150 mins. Time for 1 km = 150/90 mins. Time for 30 km = (150/90) 30 = 50 minutes. Final answer: 50 minutes are required to cover 30 km.
  • Q: The weight of 72 books is 9 kg. What is the weight of 40 such books? A: Step 1: Find the weight of one book. Weight of 72 books = 9 kg Weight of 1 book = 9 ÷ 72 kg = 1/8 kg Step 2: Find the weight of 40 books. Weight of 40 books = Weight of 1 book × 40 Weight of 40 books = (1/8) × 40 kg = 40/8 kg = 5 kg Final answer: The weight of 40 such books is 5 kg.
  • Q: A truck requires 108 litres of diesel for covering a distance of 594 km. How much diesel will be required by the truck to cover a distance of 1650 km? A: Step 1: Find the diesel required to cover 1 km. Diesel for 594 km = 108 litres Diesel for 1 km = 108 ÷ 594 litres Step 2: Find the diesel required for 1650 km. Diesel for 1650 km = (108 ÷ 594) × 1650 litres Simplify the fraction: 108/594 = 2/11 (dividing both by 54). Diesel required = (2/11) × 1650 = 2 × 150 = 300 litres. Final answer: 300 litres of diesel will be required to cover 1650 km.

Frequently Asked Questions

What is the unitary method in maths?

The unitary method is a problem-solving technique where you first find the value of a single unit (one item) from the value of multiple units. Then, you use the value of that single unit to find the value of the required number of units.

Why is it called the 'unitary method'?

It's called the 'unitary method' because the word 'unitary' is derived from 'unit', which means one. The entire method is based on the central step of finding the value of one unit before proceeding.

When should I divide and when should I multiply in the unitary method?

It's simple! To find the value of ONE unit from many, you DIVIDE. To find the value of MANY units from one, you MULTIPLY.

Is the unitary method related to ratio and proportion?

Yes, absolutely! The unitary method is a practical application of the concept of proportion. When you say 'if 10 pens cost ₹50, then 7 pens cost ₹35', you are working with quantities that are in direct proportion.