We often need to compare two quantities. We can compare them by finding the difference or by finding how many times one quantity is of the other. When we compare quantities by division, we get a ratio. For example, if there are 8 boys and 12 girls in a class, the number of boys is 8/12 or two-thirds of the number of girls. This comparison by division is called a ratio.
A ratio is written as a : b (read as "a is to b"). The two numbers in a ratio are called its terms. The first term is called the antecedent and the second term is called the consequent. Ratios compare quantities of the same kind, and the quantities must be in the same units. For example, we cannot directly compare 2 kg and 500 g; we must first convert them to the same unit.
When two ratios are equal, we say they are in proportion. Proportion is used in many real-life situations: mixing paint, scaling maps, cooking recipes and sharing money fairly. In this chapter, we learn how to compare quantities using ratios, express ratios in their simplest form, and solve problems involving proportion.
The ratio of two quantities a and b (b is not equal to 0) is the fraction a/b, written as a : b. A ratio tells us how many times one quantity is of the other.
A ratio can be reduced to its simplest form by dividing both terms by their HCF, just like simplifying a fraction. The ratio 8 : 12 simplifies to 2 : 3 because the HCF of 8 and 12 is 4.
To compare quantities of the same kind but different units, first convert them to the same unit. For example, to find the ratio of 2 kg to 500 g, convert 2 kg to 2000 g, then the ratio is 2000 : 500 = 4 : 1.
Four quantities are said to be in proportion if the ratio of the first two equals the ratio of the last two. If a : b = c : d, then a, b, c and d are in proportion, written as a : b :: c : d.
In a proportion a : b :: c : d, the product of the extremes equals the product of the means.
Product of extremes = a x d Product of means = b x c
For a proportion, a x d = b x c. For example, in 2 : 3 :: 4 : 6, we have 2 x 6 = 12 and 3 x 4 = 12.
If the product of the means equals the product of the extremes, the four quantities are in proportion.
The unitary method is a technique for solving problems by first finding the value of one unit, and then finding the value of the required number of units.
If 5 pens cost 75 rupees, then the cost of 1 pen is 75 divided by 5 = 15 rupees. Therefore, the cost of 8 pens is 8 x 15 = 120 rupees.
The unitary method is a very important problem-solving tool and is used in ratio, proportion, profit and loss, and many other topics.
To compare two ratios, we can convert them to like fractions and compare, or use the cross multiplication method.
For example, to compare 3 : 4 and 4 : 5, compare the fractions 3/4 and 4/5. Cross multiply: 3 x 5 = 15 and 4 x 4 = 16. Since 15 < 16, we have 3/4 < 4/5, so the ratio 4 : 5 is greater.
A ratio is in its simplest form when the two terms have no common factor other than 1.
To divide a quantity in the ratio a : b, we divide it into (a + b) equal parts, give a parts to the first share and b parts to the second share.
For example, to divide 50 rupees in the ratio 2 : 3, the total number of parts is 2 + 3 = 5. One part is 50 / 5 = 10 rupees. The first share is 2 x 10 = 20 rupees and the second share is 3 x 10 = 30 rupees.
To find the missing term in a proportion, use the cross product rule. For example, in 2 : 5 :: 6 : x, we have 2 x x = 5 x 6 = 30, so x = 30 / 2 = 15.
| Term | Meaning | Example |
|---|---|---|
| Ratio | Comparison of two quantities by division | 8 : 12 = 2 : 3 |
| Antecedent | First term of a ratio | In 2 : 3, antecedent is 2 |
| Consequent | Second term of a ratio | In 2 : 3, consequent is 3 |
| Proportion | Equality of two ratios | 2 : 3 :: 4 : 6 |
| Means | Middle terms of a proportion | 3 and 4 in 2 : 3 :: 4 : 6 |
| Extremes | Outer terms of a proportion | 2 and 6 in 2 : 3 :: 4 : 6 |
| Proportion Rule | Result |
|---|---|
| Product of extremes | a x d |
| Product of means | b x c |
| In proportion | a x d = b x c |
Consider a problem where the ratio of two numbers is 3 : 5 and their sum is 48. To solve it, we first add the parts of the ratio to find the total number of parts: 3 + 5 = 8. Each part is therefore 48 divided by 8, which is 6. The first number is 3 parts, so it equals 3 x 6 = 18, and the second number is 5 parts, so it equals 5 x 6 = 30. We can check the answer by adding 18 and 30, which gives 48, and by simplifying the ratio 18 : 30 to 3 : 5, which matches the given ratio. This method of finding the value of one part is the direct application of the unitary method to ratio problems.
A second example shows how the cross product rule finds a missing term. Suppose we are told that 6 : 9 :: 10 : x and asked to find x. Setting the product of the extremes equal to the product of the means gives 6 x x = 9 x 10, that is, 6x = 90, so x = 90 divided by 6 = 15. We can verify the result by checking that the ratio 6 : 9 simplifies to 2 : 3 and the ratio 10 : 15 also simplifies to 2 : 3, so the four numbers are indeed in proportion. This two-step habit of solving and then verifying is an excellent way to avoid careless mistakes in the examination.
A third common application is converting units before writing a ratio. For example, to find the ratio of 3 minutes to 45 seconds, we first convert 3 minutes to 180 seconds, and then write the ratio 180 : 45, which simplifies to 4 : 1. Problems of this kind remind us that a ratio can only compare quantities of the same kind and the same unit, and that careful conversion is the first and most important step. Students who always ask whether the two quantities are in the same unit before writing a ratio will rarely lose marks on such questions.
Ratio and proportion help us compare quantities meaningfully and solve a wide variety of everyday problems. We learnt to express comparisons as ratios, simplify them, check proportions using the cross product rule, and apply the unitary method. These tools are used in maps, recipes, sharing money and many other situations. Ratio and proportion also form the basis for percentages and direct variation in higher classes.