Class 6 — Mathematics (NCERT)

Ratio and Proportion

Class 6

✓By the end of this lesson students will be able to understand the concept of ratio and express it in its simplest form.
✓By the end of this lesson students will be able to compare two quantities using ratio, ensuring units are consistent.
✓By the end of this lesson students will be able to understand proportion as an equality of two ratios and apply the property of product of extremes and means.
✓By the end of this lesson students will be able to solve problems using the unitary method.
✓By the end of this lesson students will be able to apply the concepts of ratio, proportion, and unitary method to solve real-life problems.

Key concepts

Ratio

A ratio is a comparison of two quantities of the same kind by division. It shows how many times one quantity is of the other. For example, if there are 3 apples and 5 oranges, the ratio of apples to oranges is 3 to 5, written as 3:5 or 3/5. The first term of the ratio is called the antecedent and the second term is called the consequent. A ratio must always be expressed in its simplest form. Also, the two quantities being compared must have the same units.

a:b or a/b

Proportion

When two ratios are equal, they are said to be in proportion. For example, if the ratio of apples to oranges is 3:5 and the ratio of bananas to grapes is 6:10, then 3:5 and 6:10 are equal ratios (since 6:10 simplifies to 3:5). Thus, 3, 5, 6, 10 are in proportion. This is written as 3:5::6:10. In a proportion a:b::c:d, a and d are called the extreme terms (or extremes), and b and c are called the middle terms (or means). A key property of proportion is that the product of the extremes is equal to the product of the means, i.e., a × d = b × c.

If a:b::c:d, then a × d = b × c

Unitary Method

The unitary method is a technique used to find the value of a required number of units by first finding the value of a single unit. This method is particularly useful in solving problems involving direct variation. The steps generally involve: 1. Finding the value of one unit. 2. Multiplying the value of one unit by the number of units for which the value is to be found.

Key facts to remember

1A ratio compares two quantities of the same kind by division.
2Quantities must have the same units before finding their ratio.
3A ratio is always expressed in its simplest form.
4Ratios have no units.
5Proportion is an equality of two ratios.
6In a proportion a:b::c:d, the product of the extremes (a × d) is equal to the product of the means (b × c).
7The unitary method involves finding the value of a single unit first, then calculating the value for the required number of units.

Worked examples

Example 1

Find the ratio of 75 paise to ₹3.

IStep 1: Convert both quantities to the same unit. Since 1 Rupee (₹) = 100 paise, we convert ₹3 to paise.

II₹3 = 3 × 100 paise = 300 paise.

IIIStep 2: Write the ratio of the two quantities.

IVRatio = 75 paise : 300 paise

VStep 3: Simplify the ratio to its lowest terms by dividing both terms by their greatest common factor (GCF).

VIDivide both by 75: (75 ÷ 75) : (300 ÷ 75)

VIIRatio = 1 : 4

Answer

The ratio of 75 paise to ₹3 is 1:4.

Always ensure units are the same before finding a ratio.

Example 2

Check if the numbers 4, 12, 5, 15 are in proportion.

IStep 1: Form two ratios from the given numbers.

IIFirst ratio = 4:12

IIISecond ratio = 5:15

IVStep 2: Simplify each ratio to its lowest terms.

VFor 4:12, divide both by 4: (4 ÷ 4) : (12 ÷ 4) = 1:3

VIFor 5:15, divide both by 5: (5 ÷ 5) : (15 ÷ 5) = 1:3

VIIStep 3: Compare the simplified ratios.

VIIISince both simplified ratios are equal (1:3 = 1:3), the numbers are in proportion.

9Alternatively, using the product of extremes and means:

10Extremes are 4 and 15. Product of extremes = 4 × 15 = 60.

11Means are 12 and 5. Product of means = 12 × 5 = 60.

12Since Product of Extremes = Product of Means (60 = 60), the numbers are in proportion.

Answer

Yes, the numbers 4, 12, 5, 15 are in proportion.

The property 'product of extremes = product of means' is a quick way to check for proportion.

Example 3

If 7 pens cost ₹105, what will be the cost of 12 such pens?

IStep 1: Find the cost of one pen (unitary method).

IICost of 7 pens = ₹105

IIICost of 1 pen = ₹105 ÷ 7 = ₹15

IVStep 2: Find the cost of the required number of pens.

VCost of 12 pens = Cost of 1 pen × 12

VICost of 12 pens = ₹15 × 12 = ₹180

Answer

The cost of 12 pens will be ₹180.

This method is very useful for solving daily life problems involving quantities and costs.

Common mistakes

✗Not converting quantities to the same units before finding a ratio (e.g., comparing metres and centimetres directly).
✗Not simplifying ratios to their lowest terms.
✗Confusing the order of quantities in a ratio (e.g., ratio of A to B is not the same as ratio of B to A).
✗Incorrectly applying the product of extremes and means property in proportion problems.
✗Making calculation errors when finding the value of a single unit or multiplying in the unitary method.

Exam tips

★Always read the question carefully to identify the quantities to be compared and their units.
★For ratio problems, ensure all units are consistent before performing any calculations and simplify the ratio completely.
★When solving proportion problems, clearly state the ratios and then apply the property of product of extremes and means or simplify both ratios to check for equality.
★In unitary method problems, clearly show the step where you find the value of one unit, as this often carries marks.

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