Proportion:
find the missing value in a/b = c/d

What a proportion is

A proportion is a statement that two ratios are equal. If 5 kg of rice costs ₹300 and 8 kg costs ₹480, then 5 : 300 and 8 : 480 are the same ratio, and the quantities are in proportion.

Most real problems give you three of the four numbers and ask for the fourth. The skill is working out whether the quantities rise together (direct proportion) or one rises as the other falls (inverse proportion).

A ratio compares two quantities; a proportion says that two such comparisons match. The ratio guide covers simplifying and sharing ratios, and this guide is about solving for the unknown.

Proportion formula and cross-multiplication

Cross-multiplication works because multiplying both sides of a/b = c/d by b × d clears the fractions and leaves a × d = b × c. You never need to find a common denominator; you just multiply diagonally and divide by the number that sits next to the unknown.

1. 1Write the known pair first, then the pair that contains the unknown, keeping like quantities in the same position.
2. 2Decide whether the quantities rise together or one falls as the other rises.
3. 3Cross-multiply and divide to find the unknown.
4. 4Check that the answer is sensible: more items should cost more, and more workers should finish faster.

Example: price of 8 kg of rice

The unitary method does the same job in two steps. First find the value of one unit (₹60 per kg), then multiply by the number you need. Both methods give ₹480, so use whichever you find easier to check.

The unit price is also the quickest way to compare offers. If a 750 g pack costs ₹135 and a 1 kg pack costs ₹170, the rates are ₹180 and ₹170 per kg, so the larger pack is cheaper per kilogram even though it costs more in total.

Example: workers and days (inverse proportion)

Six workers take 12 days to finish a job. If you hire 9 workers, they take 6 × 12 ÷ 9 = 8 days. The total work, 72 worker-days, is fixed, so more workers means fewer days.

A fare of ₹50 plus ₹12 per km costs ₹110 for 5 km and ₹170 for 10 km. Doubling the distance did not double the fare, so a straight proportion would be wrong.

Speed, distance and time show both types at once. At a steady 60 km/h, distance and time are in direct proportion: 210 km takes 210 ÷ 60 = 3.5 hours. For a fixed 120 km trip, speed and time are in inverse proportion: at 40 km/h it takes 3 hours, and at 60 km/h it takes 2 hours.

Some problems chain several quantities together. Six workers on 8-hour days finish a job in 15 days, which is 6 × 8 × 15 = 720 worker-hours. With 10 workers on 6-hour days, the time is 720 ÷ (10 × 6) = 12 days. Find the constant product first and divide by everything else.

Proportion in maps, recipes and screens

• Map scale: on a 1 : 50,000 map, 3.2 cm on paper is 3.2 × 50,000 = 160,000 cm, which is 1.6 km on the ground.
• Recipe: a dal recipe uses 250 g for 4 people. For 10 people the quantity is 250 × 10 ÷ 4 = 625 g.
• Screen size: an image 1,280 pixels wide at 16 : 9 must be 1,280 × 9 ÷ 16 = 720 pixels high to keep its shape.

The aspect ratio calculator on this page reduces a width and height to their simplest ratio, which gives you the pair to put into the formula.

How to check a proportion answer

Substitute your answer back into the ratio. If 5 : 300 and 8 : 480 simplify to the same value (1 : 60), the proportion holds. For inverse proportion, multiply and compare: 6 × 12 = 72 and 9 × 8 = 72.

A second check is direction. In direct proportion, a larger input must give a larger output; in inverse proportion, it must give a smaller one. If the direction is wrong, the formula was chosen wrongly.

A third check is size. If you are scaling from 5 kg to 8 kg, you are multiplying by 1.6, so the price must also be 1.6 times the original: 300 × 1.6 = 480. This scale factor view is often the fastest way to solve a problem without writing an equation.

Percentages are proportions too. 18% of 2,500 means x ÷ 2,500 = 18 ÷ 100, so x = 450. The percentage-of-a-number calculator on this page does that step for you.

Common proportion mistakes

• Using direct proportion for a situation that is inverse, such as workers and days or speed and time.
• Putting the numbers in the wrong order, so the ratio compares kg to ₹ on one side and ₹ to kg on the other.
• Mixing units, such as hours on one side and minutes on the other.
• Assuming a fixed charge scales: a base fee or minimum order breaks pure proportion.
• Applying proportion to things that are not linear, such as age and height.

When in doubt, test with an easy case. If doubling one quantity should double the other, it is direct. If doubling one should halve the other, it is inverse.

Finally, resist the pull of round numbers. Real prices often include discounts for bulk, and real jobs include set-up time that does not shrink as the team grows. Proportion gives a clean estimate, and you adjust it for the exceptions you know about.