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Direct and Inverse Proportion

Direct and Inverse Proportion

Direct and Inverse Proportion

Quantities that are proportional change at the same rate relative to each other.

If two quantities are directly proportional to each other, then increasing one increases the other by the same proportion.

For example, if you double one quantity, the other quantity will also double.

In a recipe for cooking pancakes you need to use 125g flour, 300ml milk and 1 egg to make 4 pancakes. How many grams of flour would you need to use to make 8 pancakes?

So, the proportion can be found by dividing the new quantity by the last. If we needed to make 16 pancakes, then we would need 4 times as many ingredients.


To boil rice you need 120g of rice and 250ml of water. How much water would you need for 600g of rice?

Sometimes you will have to work out a quantity that is not a whole number of times larger that the original amount you started with.

To work out how much larger, you can divide the new amount by the amount initially stated. This becomes your multiplier.

Let's have a look at an example. To boil rice, you need 120g of rice and 250ml of water. How much water do you need for 1kg of rice?


Remember that 1kg = 1000g

This will be useful in the next part


Find how much bigger 1kg is than 120g

1000120=8.33\dfrac{1000}{120}=8.33This is the proportion we need to multiply by.


Multiply 250ml by the multiplier

8.33×250=2083.333..ml8.33 \times


Sweet! 🙌

To the nearest 10ml, the answer is 2080ml2080ml

Yohann ran 100m in 9.8 seconds. If he maintained this speed, how many seconds would it take him to run 160m?

You may come across questions where there is a quantity and a cost to work out. It is important that you work through each part of the problem in stages, clearly labelling what you have worked out at each stage.

Butter costs £0.95 for 250g. Steph needs 900g of butter to make 3 cakes. How much will it cost for her to buy the butter needed for 8 cakes?


Find the butter needed for 1 cake

900g3=300g\dfrac{900g}{3}=300gbutter per cake.


Find the cost of butter for 1 cake

We know the cost of 250g is £0.95. As a proportion, the butter for one cake is 300250\dfrac{300}{250}.


What is the cost of butter for 1 cake?


Multiply to give the butter for 8 cakes

1 cake costs £1.14 in butter. Therefore, 8 cakes would cost 8 times this amount.


How much does butter cost for 8 cakes?


Great work! 🙌

The answer is £9.12\pounds9.12

Amaani uses 800g of lasagna sheets to make 2 lasagnas. a pack of 1kg of lasagna sheets costs £1.30. How much does she need to spend to buy enough sheets to make 5 lasagnas?

Sam wants to pave his 20m220m^2 garden. It costs £15 to buy 6 tiles which cover an area of 3m23m^2 altogether. How much does it cost to buy tiles to cover the whole garden?