Direct and Indirect Proportions

Direct and indirect proportions are fundamental concepts in mathematics with widespread applications in various fields. Understanding these relationships enables us to make predictions, solve problems and model real-world scenarios accurately. Whether analyzing financial data, exploring physical phenomena or
simply baking cookies, the principles of direct and indirect proportions provide valuable insights into the quantitative relationships that govern our world. This chapter explores these concepts, their mathematical representations and practical examples that showcase their significance.

Proportion

Proportion is when two sets of numbers are equal to each other.

There are two kinds of proportions.

a. Direct Proportion: If two numbers (x and y) increase or decrease together in such a way that the ratio (x : y) between them remains constant, then one number (x) is in direct proportion to another number (y).

The relationship is shown as ‘x ∝ y’ which means ‘x is directly proportional to y’.

If one number is double, then the other number will also be double.

b. Inverse Proportion: If two numbers (x and y) behave in such a way that one number (x) increases and the other number (y) decreases simultaneously and vice versa, then one number is in inverse proportion to another number. When one number (x) is multiplied by the other number (y), the result (x × y) remains constant.

The relationship is shown as ‘x ∝ 1/y’ which means ‘x is inversely proportional to y’.

If one number is double, then the other number will be half.

Unitary Method

The unitary method is a way of finding the value of a certain quantity by first figuring out the value of one unit of that quantity.

The unitary method is shown as:

Cost of x laptops = $y
Cost of 1 laptop = $(y/x)
Cost of z laptops = $(y/x) × z

Example: Cost of 3 laptops = $1800
Cost of 1 laptop = $(1800/3) = $600
Cost of 7 laptops = $(600 × 7) = $4200

Time and Work

If a certain number of people can complete a task (W1) in a specific number of days (D1) by working a certain number of hours each day (T1) and another group of people can complete a different task (W2) in a different number of days (D2) by working a different number of hours each day (T2), then there is a relationship between these quantities is shown as: