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Direct and Inverse Proportion Explained for Kids + 12 Worked Examples with Answers

Published: 04.09.2026·Updated: 04.09.2026

Nadia Rahmawati

Math Curriculum Team, Algonova

Direct and Inverse Proportion Explained for Kids + 12 Worked Examples with Answers

Direct proportion means that when one quantity rises the other rises too; inverse proportion means that when one rises the other falls. Children aged 10 to 13 usually know both formulas but pick the wrong one. Below is a single question that separates them instantly, plus 12 worked examples with full answers and working.

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One question that separates the two

Ask: if I increase the quantity, does the result become more or less?

If it becomes more, that is direct proportion. Buy more books, pay more money. Travel further, use more fuel.

If it becomes less, that is inverse proportion. Add more workers, the job takes less time. Increase the speed, the journey takes fewer hours.

A child can test this themselves with extreme numbers. If there were 1000 workers, would it finish sooner? Obviously yes, so the problem is inverse. This extreme-number trick works almost every time and is far more reliable than hunting for keywords.

Both rest on simple ratio. If that is shaky, start with Scale and Mixed Operations.

How to solve direct proportion

With direct proportion, find the value of one unit first, then multiply.

This is called the unitary method, and for primary-age children it is far easier to grasp than cross multiplication.

One example worked in full: 3 books cost Rp 45,000. What do 5 books cost?

Find the price of one book: Rp 45,000 divided by 3 equals Rp 15,000.
Multiply for 5 books: Rp 15,000 times 5 equals Rp 75,000.

Check the direction: the books went up from 3 to 5, and the price went up from 45,000 to 75,000. Same direction, so this is indeed direct proportion.

Indonesian children looking at fruit prices at a market stall with a parent

A market is the best practice ground for this topic. Ask your child to work out the price of 3 kg of oranges when 1 kg is shown on the price board. Practice like this sticks far better than ten textbook problems, because the child sees the point immediately.

6 direct proportion problems with full answers

Every problem below uses the unitary method: divide first, then multiply.

No	Problem	Unit value	Answer
1	3 books cost Rp 45,000. Cost of 5?	Rp 15,000 per book	Rp 75,000
2	2 kg rice costs Rp 24,000. Cost of 7 kg?	Rp 12,000 per kg	Rp 84,000
3	4 pencils cost Rp 12,000. Cost of 10?	Rp 3,000 per pencil	Rp 30,000
4	60 km uses 4 litres. 90 km uses?	15 km per litre	6 litres
5	5 shirts need 15 m of fabric. 8 shirts need?	3 m per shirt	24 m
6	12 cakes serve 4 children. 7 children need?	3 cakes per child	21 cakes

Problem 4 worked step by step. The car covers 60 km on 4 litres, so one litre covers 60 divided by 4, which is 15 km. For 90 km, the fuel needed is 90 divided by 15, which is 6 litres.

Check the direction: distance rose from 60 to 90 km, and fuel rose from 4 to 6 litres. Same direction, so the working is correct.

One error that shows up often in problems 4 and 5: the child divides and multiplies with the numbers swapped, so the answer shrinks instead. The direction check catches that in two seconds.

6 inverse proportion problems with full answers

With inverse proportion, multiply the two known numbers first, then divide. That product stays constant.

No	Problem	Constant product	Answer
7	6 workers finish in 12 days. 8 workers?	6 x 12 = 72	9 days
8	4 taps fill a pool in 6 hours. 3 taps?	4 x 6 = 24	8 hours
9	At 60 km/h the trip takes 3 hours. At 90 km/h?	60 x 3 = 180	2 hours
10	Food lasts 20 people 15 days. For 25 people?	20 x 15 = 300	12 days
11	5 machines finish in 8 hours. 10 machines?	5 x 8 = 40	4 hours
12	3 builders finish in 20 days. 5 builders?	3 x 20 = 60	12 days

An Indonesian teacher helping a small group of primary school children with exercises at a shared table

Problem 10 worked step by step. The food lasts 20 people for 15 days, so the total is 20 times 15, which is 300 person-days. With 25 people eating, 300 divided by 25 gives 12 days.

Check the direction: people rose from 20 to 25, and days fell from 15 to 12. Opposite directions, so this is indeed inverse and the working is right.

The most common error here is using the unitary method as though it were a direct problem, dividing then multiplying. The answer grows when it should shrink, and the direction check catches it immediately.

A comparison table: which one to use when

Keep this table on the study desk for two weeks. After that a child usually stops needing it.

	Direct proportion	Inverse proportion
Direction	Rise together, fall together	One rises, the other falls
Method	Divide first, then multiply	Multiply first, then divide
What stays constant	The value of one unit	The product of both values
Typical examples	Prices, material needed, distance and fuel	Workers and time, speed and time, people and supplies
Quick check	The result grows	The result shrinks

Three topics dominate exams: workers and time, speed and time, and food supplies. All three are inverse, and all three are routinely solved by children as though they were direct. If practice time is limited, drill these three patterns first.

For related material, see Speed, Distance and Time and Scale. Staged division practice is in How to Teach Long Division, and factor basics are in LCM and HCF. Year-by-year topics are at Year 6 and Year 7. Class options are at Maths for Secondary School.

Summary

Direct proportion: when one rises, the other rises. Inverse: when one rises, the other falls.
The fastest way to separate them is asking whether the result becomes more or less, then testing with extreme numbers.
Direct problems use the unitary method: divide to find one unit, then multiply.
Inverse problems use a constant product: multiply the two known numbers, then divide.
Workers and time, speed and time, and supplies against people are always inverse, and that is where children go wrong most often.

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Frequently Asked Questions About Direct and Inverse Proportion

What is the difference between direct and inverse proportion?

In direct proportion, when one quantity rises the other rises too. In inverse proportion, when one rises the other falls. A direct example: 3 books cost Rp 45,000, so 5 books cost Rp 75,000. More books, more money. An inverse example: 6 workers finish in 12 days, so 8 workers finish in 9 days. More workers, fewer days. The fastest way to separate them is asking whether the result becomes more or less when the quantity increases.

What is a fast way to tell which type a problem is?

Test it with extreme numbers. Imagine the quantity made enormous, then see whether the result grows or shrinks. Take the workers problem: with 1000 workers, would it finish sooner? Obviously yes, so the problem is inverse. Conversely with prices: buying 1000 books, would you pay more? Obviously yes, so it is direct. This trick works almost every time and is far more reliable than hunting keywords, because the same words appear in both types.

How do you solve a direct proportion problem?

Use the unitary method: divide to find the value of one unit, then multiply by what is asked. For example 3 books cost Rp 45,000. One book costs 45,000 divided by 3, which is Rp 15,000. For 5 books, multiply 15,000 by 5 to get Rp 75,000. This method is easier for primary-age children than cross multiplication, because every step carries a meaning you can explain. When finished, check the direction: quantity up and result up means the working is right.

How do you solve an inverse proportion problem?

Multiply the two known numbers first, then divide by the new number. That product stays constant. For example 6 workers finish in 12 days. Multiply 6 by 12 to get 72. For 8 workers, divide 72 by 8 to get 9 days. That 72 represents the total workload, and it does not change however many workers there are. Explaining what the number means helps a child remember the order. Check the direction afterwards: more workers and fewer days, so the answer makes sense.

At what age do children learn direct and inverse proportion?

Generally in Years 5 and 6 through Year 7, around ages 10 to 13. The prerequisite is fluency in long division and multiplication, because both methods rest on them. A child still stalling at division will stall here too, but the cause is not the proportion concept. Year 5 usually covers simple direct proportion, while inverse arrives in Year 6 and deepens in secondary school. The topic stays in use through senior school, particularly in speed and scale problems.

Which problems do children get wrong most often?

Three inverse patterns: workers and time, speed and time, and food supplies against the number of people. All three are inverse, yet children routinely solve them with the unitary method as though they were price problems. The answer then grows when it should shrink. For example food lasting 20 people 15 days. With 25 people eating, the answer is 12 days, not more than 15. If practice time is short, drill these three patterns first, because they carry heavy weight in exams.

Can it be solved with cross multiplication?

Yes, but it is better taught after the unitary method rather than as the first approach. Cross multiplication is faster but carries no meaning. A child who only memorises it gets confused by inverse problems, because the crossing rule differs and is easily mixed up. The unitary method is slower but every step can be explained, so the child can self-check when an answer feels wrong. From Year 7 onwards, cross multiplication is reasonable to use because the underlying understanding is in place.

How can we practise at home without a workbook?

Use everyday situations, especially the market and the kitchen, since both are full of direct proportion. At the market, ask your child to work out the price of 3 kg of fruit when the per-kilo price is displayed. In the kitchen, ask them to scale a recipe to double the servings. For inverse proportion, use housework. If sweeping one room alone takes 20 minutes, how long with two people? This kind of practice sticks better than ten textbook problems, because the child sees the point immediately.

How long until a child becomes fluent?

About two to three weeks at 15 minutes a day, provided problems are mixed between direct and inverse. What matters is the mixing rather than the volume. A child doing ten direct problems in a row will calculate fluently but still pick the wrong method on a shuffled set. A pattern that works: four problems a day, two direct and two inverse, in random order. Add one longer word problem every other day, because that is where the real difficulty lives.

My child calculates fine but picks the wrong method. What now?

Drill the method choice separately, with no calculation at all. Here is how: give ten problems and ask your child only to write direct or inverse, without solving anything. One round takes five minutes and attacks the root cause directly. Once they get ten out of ten on classification, then ask them to calculate. Separating the two skills is usually faster than adding more ordinary problems. If the same error persists, a diagnostic session with a teacher finds the cause faster.

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Direct and Inverse Proportion Explained for Kids + 12 Worked Examples with Answers

One question that separates the two

How to solve direct proportion

One example worked in full: 3 books cost Rp 45,000. What do 5 books cost?

6 direct proportion problems with full answers

6 inverse proportion problems with full answers

A comparison table: which one to use when

Summary

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Frequently Asked Questions About Direct and Inverse Proportion

How to solve direct proportion

One example worked in full: 3 books cost Rp 45,000. What do 5 books cost?

6 direct proportion problems with full answers

6 inverse proportion problems with full answers

A comparison table: which one to use when

Summary

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Online vs In-Person Coding Classes for Kids: Which Fits Better? (2026)

When Should Primary School Kids Start Coding: A Readiness Guide

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Frequently Asked Questions About Direct and Inverse Proportion