Direct and inverse proportion

Learning outcome

Identify direct and inverse proportion from assumptions, solve constant-rate problems, and distinguish proportional relationships from trends.

Understanding the models

Positive quantities x and y are directly proportional when y/x is constant: y = k × x, with positive k. Multiplying x by a positive factor multiplies y by that factor because k stays fixed.

They are inversely proportional when x × y is constant: y = k/x. Multiplying x by a positive factor divides y by that factor. These changes compensate, keeping the product unchanged.

Choose the model from the conditions, not just the direction of change. Quantities increasing together are not automatically directly proportional; opposite movements do not establish inverse proportion. A trend or correlation alone proves neither equation.

A fixed unit price without extra charges supports direct proportion. A fixed job shared by equally productive machines supports inverse proportion when their rates remain constant and there is no interference. Match units within each measurement type before comparing ratios.

Worked example 1

At a fixed price per metre, 3.5 m of cloth costs ₹280. With no discounts or extra fees, find the cost of 6 m.

The unit price is 280/3.5 = ₹80 per metre. Thus 6 m costs 6 × 80 = ₹480. The unchanged cost-to-length quotient justifies direct proportion.

Worked example 2

6 identical machines complete a fixed job in 15 hours. They work at the same constant rate without interference. How long would 10 machines take?

The job requires 6 × 15 = 90 machine-hours. Therefore 10 machines need 90/10 = 9 hours. More machines reduce time because the work, represented by total machine-hours, stays fixed.

Worked example 3

Delivery costs a fixed ₹40 plus ₹12 per kilometre. Find the charges for 5 km and 10 km. Is cost directly proportional to distance?

The charges are 40 + 12 × 5 = ₹100 and 40 + 12 × 10 = ₹160. Distance doubles, but cost does not double to ₹200. Cost increases with distance, yet the fixed fee prevents direct proportion.

Common mistakes

Do not choose inverse proportion merely because one number decreases. Check the constant quantity and rate assumptions. A fixed fee is not a constant unit rate.

Practice questions

1. At a constant filling rate, 2.5 L fills in 40 seconds. How long will 7.5 L take?
2. 8 equally productive workers finish a fixed task in 18 days. How long would 12 such workers need, with the same daily hours and no interference?
3. A vehicle travels a fixed 150 km without stops. Find travel times at constant speeds of 30 km/h and 50 km/h.
4. Lamp rental costs ₹25 per day plus a fixed ₹60 fee. Find costs for 2 days and 4 days. Is total cost directly proportional to duration?

Worked answers

1. Volume increases by a factor of 7.5/2.5 = 3. At the unchanged rate, time is 40 × 3 = 120 seconds.
2. The task requires 8 × 18 = 144 worker-days. Thus 12 workers need 144/12 = 12 days under the stated conditions.
3. Time equals distance divided by speed: 150/30 = 5 hours and 150/50 = 3 hours. Speed multiplied by time remains 150 km.
4. Costs are 60 + 25 × 2 = ₹110 and 60 + 25 × 4 = ₹160. Doubling duration does not double cost to ₹220, so direct proportion does not apply.