Proportional Reasoning For Class 8

Key ConceptsDirect proportion means two quantities change at the same constant rate. In exams, unit rates, tables, graphs, and equations help you find missing values accurately.

Unit rate = amount for 1 unit.

Direct proportion rule: y = kx, where k is the constant of proportionality.

Find k using: k = y ÷ x.

A proportional table has a constant value of y ÷ x.

Its graph is a straight line passing through (0, 0).

If one quantity doubles, the other also doubles.

Use matching units before calculating.

Before you continue: How can you check from a table whether two quantities are directly proportional?

Key TermsDirect proportion - A relationship where one quantity changes by the same factor as another.

Unit rate - The amount corresponding to 1 unit.

Constant of proportionality - The fixed number k in y = kx.

Ratio - A comparison of two quantities.

Proportional table - A table in which y ÷ x remains constant.

Origin - The point (0, 0) on a graph.

Active RecallDirect proportion__________

__________

Unit rate__________

__________

Constant of proportionality__________

__________

Ratio__________

__________

Proportional table__________

__________

Origin__________

__________

Worked ExamplesExample 1: Unit rateA car travels 180 km in 3 hours. Find its speed.

Speed = distance ÷ time.

180 ÷ 3 = 60.

The unit rate is 60 km/h.

Now you try: A cyclist travels 84 km in 4 hours. Find the unit rate.

__________

__________

__________

Example 2: Using direct proportionFive notebooks cost ₹150. Find the cost of 8 notebooks.

Cost of 1 notebook = 150 ÷ 5 = ₹30.

Cost of 8 notebooks = 8 × 30 = ₹240.

Therefore, 8 notebooks cost ₹240.

Now you try: Seven pens cost ₹105. Find the cost of 12 pens.

__________

__________

__________

Example 3: Table and equationThe table shows hours worked and wages earned.

Hours, x

2

5

8

Wages, y (₹)

360

900

1440

Calculate y ÷ x: 360 ÷ 2 = 180.

Check: 900 ÷ 5 = 180 and 1440 ÷ 8 = 180.

Thus, k = 180, so the equation is y = 180x.

For 7 hours: y = 180 × 7 = ₹1260.

Now you try: If y = 75x, find y when x = 6.

__________

__________

__________

Practice Questions(Easy) A tap fills 24 litres in 3 minutes. Find the unit rate.

__________

__________

__________

(Easy) Complete the proportional table: 4 kg costs ₹280; find the cost of 7 kg.

__________

__________

__________

(Medium) Is this table directly proportional? Give a reason: x = 2, 4, 6; y = 10, 20, 31.

__________

__________

__________

(Medium) A map uses 1 cm for 8 km. How far does 6.5 cm represent?

__________

__________

__________

(Hard) A machine produces 420 parts in 7 hours. How many parts will it produce in 12.5 hours?

__________

__________

__________

(Hard) The graph of a relationship is a straight line through (0, 0) and (4, 28). Find the equation and the value of y when x = 9.

__________

__________

__________

Answer Key8 litres per minute

₹490

No; 31 ÷ 6 is not 5

52 km

750 parts

y = 7x; y = 63

Exam Tips & Common MistakesDo not divide in the wrong order. Unit rate is always “amount ÷ number of units.”

Check that all units match, such as minutes with minutes or metres with metres.

Do not assume every straight line shows direct proportion; it must pass through the origin.

For tables, calculate y ÷ x in at least two columns.

...