Proportional Reasoning – 2 | Mathematics Grade 8 CBSE

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Six ways quantities relate

Some pairs rise together, some trade off against each other. Tap each idea to see whether it is direct, inverse or a mix — and the rule that ties the numbers together.

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The three big ideas

Direct proportion — both quantities grow (or shrink) in the same ratio; the quotient x/y stays constant. Example: 1 pen ₹10, 5 pens ₹50.
Inverse proportion — as one increases the other decreases so the product x × y stays constant. Example: more workers → fewer days.
The inverse rule: x₁ × y₁ = x₂ × y₂. Knowing three of the four values lets you find the fourth.
Spot the type first. Ask: "If this goes up, should the other go up or down?" Up → direct; down → inverse.

Worked example. 6 workers build a wall in 8 days. How long will 12 workers take, working at the same rate?

1. More workers should mean fewer days → this is inverse proportion.

2. Use x₁ × y₁ = x₂ × y₂ with workers × days: 6 × 8 = 12 × t.

3. So 48 = 12 × t, giving t = 48 ÷ 12 = 4 days.

4. Check: doubling the workers halved the time (8 → 4). ✓

Common mistake: using the direct rule (x₁/y₁ = x₂/y₂) for an inverse situation. For inverse problems you multiply, not divide across.

The relation: speed = distance ÷ time. Rearranged: distance = speed × time and time = distance ÷ speed.
Direct: at a fixed speed, more time means proportionally more distance.
Inverse: for a fixed distance, higher speed means proportionally less time (speed × time stays equal to the distance).
Units must match — km with hours gives km/h; m with seconds gives m/s. Convert before you calculate.

Worked example. A bus covers 180 km in 3 hours. (a) Find its speed. (b) How far does it go in 5 hours at the same speed?

1. Speed = distance ÷ time = 180 ÷ 3 = 60 km/h.

2. Distance in 5 hours = speed × time = 60 × 5 = 300 km.

Work rate — if a person finishes a job in n days, they do 1/n of it each day.
Working together — add the daily rates. If A does 1/6 a day and B does 1/3 a day, together they do 1/6 + 1/3 = 1/2 a day, so the job takes 2 days.
Compound proportion — when a result depends on two or more quantities at once (workers, hours, days), handle each relationship in turn using the unitary method.
Scaling — maps and models use a fixed scale (1 cm : 50 km). Map distance and real distance are in direct proportion.

Where you'll meet it

Proportions in daily life

Planning a journey

Leaving Jaipur for a town 270 km away, a driver who keeps 60 km/h will take 270 ÷ 60 = 4.5 hours. If she speeds up to 90 km/h, time drops to 3 hours — fixed distance, so speed and time trade off inversely.

Sharing the workload

A community hall needs painting in time for a festival. One painter would take 12 days; bringing in a second painter of equal speed splits the work, finishing in 6 days. Adding hands shortens the time inversely.

Stocking supplies

A hostel mess stores grain for 150 students for 30 days. If 30 students leave, the same grain feeds 120 for 150 × 30 ÷ 120 = 37.5 days — fewer mouths, more days, an inverse relationship the warden plans around.

Check yourself

Competency quiz

Modelled on the competency-based pattern — MCQ, assertion–reason and a case study, testing whether you can use the ideas, not just recall them.

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