Direct proportion - KS3 Maths

Key points

The same table. To the right: C is directly proportional to n. C equals twenty n. — Image caption,
The cost of the pencils (C) is directly proportional to the number of pencils (n).

When two variables are , as one increases the other also increases at the same rate (proportionally). So if one doubles, the other also doubles.
Direct proportion is written using the (∝). For example, if two \(x\) and \(y\) are directly proportional to each other, then this statement can be represented as \(x\) ∝ \(y\).
When the proportionality sign (∝) is replaced with an equal sign (=), the equation is \(x = k y\). The constant value (often written as \(k\)) is known as the constant of proportionality and relates to the amounts that increase or decrease at the same rate.

To understand direct proportion, a good understanding of multiplication and division can be helpful. This is often referred to as .

What is direct proportion?

To decide whether two are , check for the following:

both variables have a value of zero at the same time
as one variable increases/decreases, the other variable increases/decreases in the same proportion

Examples

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The table shows the cost of buying a number of pencils. The more pencils bought, the greater the cost. The cost increases as the number of pencils increases. If no pencils are bought, there is no cost.
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The cost of the pencils (C) is directly proportional to the number of pencils (n). This can be written as C ∝ n. The constant of proportionality is 20, so C = 20n. For example, if n = 3, C = 20 × 3 If C = 20 × 3, the cost is 60p.
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Currency can be exchanged between pounds sterling (£) and euros (€).
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If two variables are directly proportional, both have a value of zero at the same time. This is true for the amounts of money in pounds and euros. The amounts of money in pounds and euros are directly proportional. £0 is the same as €0
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Two variables are directly proportional when as one variable increases/decreases, the other variable increases/decreases at the same rate. When £100 is doubled to £200, the amount of euros is also doubled (€120 × 2 = €240). The currencies are directly proportional. (Note: This is an example, exchange rates are variable).
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Temperatures may be measured in degrees Celsius (°C) or degrees Fahrenheit (°F).
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If two variables are directly proportional, both have a value of zero at the same time. This is not true for temperature. 0°C is 32°F. Degrees Celsius and degrees Fahrenheit are not directly proportional.

Question

Are these different scone mixes directly proportional?

A diagram showing two pieces of paper. First piece: Country scones. Thirty-four grams butter. One-hundred and twenty grams flour. Second piece: Classic scones: Eighty-five grams butter. Three-hundred grams flour.

Direct proportion and multiplicative reasoning

involves using multiplication and division to find other values.

Method:

Look for multiplicative links. Find a multiplication relationship (number of times greater) or division relationship (number of times smaller).
Repeat the same multiplication or division on the other variable.

Examples

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5 ice creams cost £4.20 Find the cost of 10 ice creams.
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The number of ice creams is directly proportional to how much they cost. 5 ice creams cost £4.20 10 is 5 doubled. If the number of ice creams is multiplied by 2 (5 × 2 = 10), then the cost will also be doubled (£4.20 × 2).
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£4.20 × 2 = £8.40 The cost of 10 ice creams is £8.40
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The distance in miles and the distance in kilometres are directly proportional. Find the missing distances in the table.
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Starting with the known fact (25 miles are the same as 40 km) look for multiplicative links. 25 × 5 = 125
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If 25 miles × 5 = 125 miles, then 40 km × 5 will give the number of kilometres. 40 km x 5 = 200 km. This means 125 miles are 200 kilometres.
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125 miles is 200 km. Look for multiplicative links. Divide 200 by 2 to get 100
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To find how many miles are the same as 100 km, divide by 2 (125 ÷ 2 = 62∙5). 100 kilometres is 62∙5 miles.

Question

30 pies are sold for £45. How much do 5 pies cost?

Game - Direct proportion

Try out these direct proportion puzzles from our Divided Islands game.

Play the full Divided Islands game.

Direct proportion problems

Direct proportion problems can be solved by using the unitary method (finding one):

Find the value of one by dividing the total value by the quantity given.
Multiply the value of one by the number required.

Examples

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Image caption,
Five ice creams cost £4.20. Find the cost of seven ice creams.
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The number of ice creams is directly proportional to their cost. The unitary method means finding the cost of one ice cream. To do this, divide the total cost (£4.20) by the quantity given (5). 4∙2 ÷ 5 = 0∙84. One ice cream costs £0.84 (84p).
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The cost of seven ice creams is the cost of one ice cream multiplied by 7. 0∙84 × 7 = 5∙88. The cost of seven ice creams is £5.88
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Distances measured in kilometres and miles are directly proportional. Find the missing values in the table.
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40 km are equal to 25 miles. Divide the number of km (40) by the number of miles (25) to find the number of kilometres equal to one mile. One mile is 1∙6 km. This fact can be used to find all the other values.
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One mile is 1∙6 km. The number of miles multiplied by 1∙6 will give the number of kilometres. 125 x 1∙6 = 200. This means 125 miles are 200 kilometres.
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One mile is 1∙6 km. The number of kilometres divided by 1∙6 will give the number of miles. 20 kilometres are 12∙5 miles. 100 kilometres are 62∙5 miles.

Question

30 pies are sold for £45. How much do 8 pies cost?

Practise ratio and direct proportion

Practise ratio and direct proportion with this quiz. You may need a pen and a piece of paper to help you work things out.

Quiz

Real-world maths

A person holding a fuel pump and adding fuel into a car fuel tank. — Image caption,
Fuel for a journey of 600 miles with fuel consumption of 40 miles per gallon would use 15 gallons.

In planning the cost of fuel for a journey, direct proportion is used. If a car has an average fuel consumption of 40 miles per gallon, a journey of 600 miles would be expected to use 15 gallons (600 ÷ 40 =15)

Currency exchange is another example of direct proportion being used in real life. The quantity of pounds sterling (£) is directly proportional to the quantity of euros (€). The two currencies increase and decrease in the same ratio. When the quantity of pounds is doubled, so is the quantity of euros. When the quantity of pounds is zero, the quantity of euros is also zero. If the exchange rate between pounds and euros is £1 = 1.19€, then £2000 would give 2380€