Learning Objectives
- Understand the concept of ratio and how to express it
- Find equivalent ratios and simplest form of a ratio
- Understand proportion and check if two ratios are in proportion
- Solve problems using the unitary method
Key Concepts
What Is a Ratio?
A ratio compares two quantities of the same kind. It tells us how many times one quantity is contained in another. The ratio of a to b is written as a : b (read as "a is to b"). For example, if there are 4 boys and 6 girls, the ratio of boys to girls is 4 : 6, which simplifies to 2 : 3. The order of a ratio matters: a : b is not the same as b : a.
Simplest Form of a Ratio
A ratio is in its simplest form when the two numbers have no common factor other than 1 (they are co-prime). To simplify, divide both numbers by their HCF. For example, 12 : 18 → HCF of 12 and 18 is 6 → 12 ÷ 6 : 18 ÷ 6 = 2 : 3.
Equivalent Ratios
Just like equivalent fractions, we can find equivalent ratios by multiplying or dividing both terms by the same number. For example, 2 : 3 = 4 : 6 = 6 : 9 = 8 : 12. All of these are equivalent ratios.
What Is Proportion?
When two ratios are equal, they are said to be in proportion. We write a : b :: c : d (read as "a is to b as c is to d"). For example, 2 : 3 :: 4 : 6 because 2/3 = 4/6. In a proportion a : b :: c : d, a and d are called extremes, and b and c are called means. The product of extremes equals the product of means: a × d = b × c.
Unitary Method
The unitary method involves finding the value of one unit first, then finding the value of the required number of units. For example, if 5 pencils cost Rs 30, then 1 pencil costs Rs 30 ÷ 5 = Rs 6. So 8 pencils cost Rs 6 × 8 = Rs 48.
Summary
Ratios compare two quantities of the same kind. They can be simplified like fractions. When two ratios are equal, they form a proportion. The unitary method is a practical approach to solve problems by first finding the value of one unit. These concepts are widely used in everyday life for sharing, cooking, and comparing.
Important Terms
- Ratio: A comparison of two quantities of the same kind (a : b)
- Proportion: An equality of two ratios (a : b :: c : d)
- Extremes: The first and last terms in a proportion (a and d)
- Means: The second and third terms in a proportion (b and c)
- Unitary Method: Finding the value of one unit first, then multiplying for the required amount
- Equivalent Ratios: Ratios that simplify to the same value
Quick Revision
- Ratio a : b means a/b; both quantities must be in the same unit
- A ratio has no unit — it is a pure number
- The order matters: 3 : 5 is different from 5 : 3
- In proportion a : b :: c : d, product of extremes = product of means (a × d = b × c)
- Unitary method: find value of 1 unit first, then multiply
- To simplify a ratio, divide both terms by their HCF
- Ratios can only compare quantities of the same kind (length with length, not length with weight)