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Direct proportion for AQA GCSE Maths

Direct proportion

This page covers the following topics:

1. Direct proportion
2. Converting units
3. Scale factors
4. Ratio fractions
5. Splitting quantities

Two quantities are directly proportional to each other when they are multiples of each other. The notation for direct proportionality is a โˆ b. Equations can be constructed using a constant of proportionality and evaluating it using given information.

Direct proportion

To convert a value from larger units to smaller units, multiply it by the ratio between the units. To convert a value from smaller units to larger units, divide it by the ratio between the units. Sometimes a conversion of units can achieved by multiplying or/and dividing a value in steps.

1 km = 1000 m
1 m = 100 cm
1 cm = 10 mm
1 kg = 1000 g
1 hour = 60 min = 3600 s

Converting units

A scale factor is a value which is multiplied or divided by quantities to scale them. Enlarging a shape can be done by multiplying by the scale factor, and reducing a shape can be done by dividing by the scale factor. When the scale factor of the side of a shape is found, it can be calculated for its area by squaring it and for its volume by cubing it.

Scale factors

Ratios can be expressed as fractions, by having the number of parts of a desired quantity in the numerator and the total number of parts in the denominator. To combine ratios together, fractions representing them can be multiplied together.

Ratio fractions

Quantities can be split using ratios by calculating the number of total parts and dividing this by the full amount of the quantity to find out how much each part is equivalent to. Multiplying the number of parts by this allows a quantity to be split up according to a ratio.

Splitting quantities

1

z is directly proportional to sยฒ. Given that when s = 3, z = 54, construct an equation for the relationship between z and sยฒ.

Since z is directly proportional to sยฒ, z = ksยฒ, where k is a constant of proportionality.
Substituting s = 3, z = 54 into this gives 54 = k(3ยฒ), thus k = 6.
Therefore, z = 6sยฒ.

z = 6sยฒ

z is directly proportional to sยฒ. Given that when s = 3, z = 54, construct an equation for the relationship between z and sยฒ.

2

Anna wants to split ยฃ300 between two of her accounts in the ratio 1 : 2. Calculate the amount that each bank account will get.

total number of parts = 1 + 2 = 3
ยฃ300 รท 3 = ยฃ100 per part
The first account will get 1 ร— 100 = ยฃ100.
The second account will get 2 ร— 100 = ยฃ200.

ยฃ100, ยฃ200

Anna wants to split ยฃ300 between two of her accounts in the ratio 1 : 2. Calculate the amount that each bank account will get.

3

For every dog in an animal shelter, there are 2 cats. State the fraction of cats in the animal shelter.

The ratio of dogs to cats is 1 : 2.
fraction of cats = 2/(1 + 2) = 2/3

2/3

For every dog in an animal shelter, there are 2 cats. State the fraction of cats in the animal shelter.

4

In a cake mixture, the ratio of cups of flour to sugar is given by 1 : 2. Given that Anna wants to make 6 cakes, calculate how many cups of flour and sugar she will need.

total number of parts = 1 + 2 = 3
6 รท 3 = 2 cups per part
cups of flour = 1 ร— 2 = 2
cups of sugar = 2 ร— 2 = 4

2, 4

In a cake mixture, the ratio of cups of flour to sugar is given by 1 : 2. Given that Anna wants to make 6 cakes, calculate how many cups of flour and sugar she will need.

5

The volume of a cylinder is enlarged from 120 cmยณ to 960 cmยณ. Given that the diameter of the smaller cylinder is 6 cm, calculate the diameter of the bigger cylinder.

volume scale factor = 960 cmยณ รท 120 cmยณ = 8
diameter scale factor = ยณโˆš8 = 2
diameter = 6 ร— 2 = 12 cm

12 cm

The volume of a cylinder is enlarged from 120 cmยณ to 960 cmยณ. Given that the diameter of the smaller cylinder is 6 cm, calculate the diameter of the bigger cylinder.

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