Maths
What is direct proportion, and how do you solve it?
Quick answer
Two quantities are in direct proportion when one is always the same multiple of the other, so doubling one doubles the other. The rule is y = kx, where k is the constant of proportionality.
In one sentence
Two quantities are in direct proportion when they change at the same rate: double one and the other doubles, halve one and the other halves.
Explained
Cinema tickets cost $4 each. 1 ticket costs $4, 2 tickets cost $8, 5 tickets cost $20, and 0 tickets cost nothing. Whatever number of tickets you buy, the cost divided by the number of tickets is always 4. That is direct proportion: the cost is directly proportional to the number of tickets.
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The number that stays the same is called the constant of proportionality, written k. In a shopping problem it is the price of one item (in the US it is often called the unit rate). The rule is
y = kx, so k = y ÷ x
Mathematicians write "y is directly proportional to x" as y ∝ x. To turn that into an equation you replace ∝ with "= k": y = kx.
How do you know two quantities are in direct proportion? There are three tests, and they say the same thing in different ways:
- The ratio test: in a table, y ÷ x gives the same number every time.
- The graph test: on a line graph, the points lie on a straight line that passes through the origin (0, 0). The slope of that line is k.
- The zero test: when one quantity is 0, so is the other, and multiplying one by any number multiplies the other by the same number.
Not every straight line is direct proportion. A taxi that charges $9 for 2 miles and $17 for 6 miles has a fixed fee of $5 on top of $2 a mile. The graph is straight, but it starts at $5, not at 0, and 9 ÷ 2 is not the same as 17 ÷ 6. Doubling the distance does not double the fare.
The opposite idea is inverse proportion: when one quantity doubles, the other halves. If 4 workers take 6 hours to paint a fence, 8 workers working at the same rate take 3 hours. There the product stays the same (4 × 6 = 8 × 3 = 24), and the rule is y = k ÷ x.
Direct proportion is everywhere: a ratio in a recipe, exchange rates, map scales, distance at a steady speed, and percentages (10% of a price is always the price × 0.1).
Examples
| Situation | x | y | k | Rule |
|---|---|---|---|---|
| Tickets at $4 each | tickets | cost ($) | 4 | y = 4x |
| Car at a steady 60 km/h | hours | km | 60 | d = 60t |
| 50 g of flour per person | people | grams | 50 | f = 50p |
| Pay of $12 an hour | hours | pay ($) | 12 | P = 12h |
| Perimeter of a square | side | perimeter | 4 | P = 4s |
Worked example (the unitary method): 5 notebooks cost $7.50. How much do 8 cost? First find one: 7.50 ÷ 5 = $1.50. Then multiply: 8 × 1.50 = $12.00.
Worked example (with k): y is directly proportional to x, and y = 12 when x = 3. Then k = 12 ÷ 3 = 4, so y = 4x. When x = 10, y = 4 × 10 = 40. When y = 30, x = 30 ÷ 4 = 7.5.
Common mistakes
- Adding instead of multiplying. If 2 pizzas need 3 eggs, 4 pizzas do not need 5 eggs. You doubled the pizzas, so double the eggs: 6. Fix: ask "what did I multiply by?"
- Thinking every straight line is direct proportion. Fix: it must also pass through (0, 0). A fixed fee breaks direct proportion.
- Mixing up direct and inverse. More workers means less time, so that is inverse. Fix: ask whether the second quantity goes up or down.
- Dividing the wrong way round for k. Fix: k = y ÷ x. Check by putting your k back into y = kx.
Step by step
Step by step
- Check it is direct proportionDoes 0 give 0, and does doubling one quantity double the other?
- Find k (or the value of one)Divide a known y by its x: k = y ÷ x.
- Write the ruleY = kx, using your value of k.
- Use the ruleTo find y, multiply the new x by k. To find x, divide y by k.
- Sense-checkA bigger x must give a bigger y. If it does not, look again.
You can also write a proportion and cross-multiply. If 3 apples cost $6, then for 5 apples: 3 ÷ 6 = 5 ÷ y, so 3y = 30 and y = $10. It is the same as the unitary method, just written as one equation.
Practice questions
Direct proportion
Try each one, then check your answer.
- 1 pen costs $2. How much do 7 pens cost?
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7 × 2 = $14 - 4 apples cost $2. How much does 1 apple cost?
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2 ÷ 4 = $0.50 - 3 kg of potatoes cost $4.50. How much do 5 kg cost?
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1 kg costs 4.50 ÷ 3 = $1.50, so 5 kg cost 5 × 1.50 = $7.50 - A car travels 90 km on 6 litres of fuel. How far can it go on 10 litres?
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90 ÷ 6 = 15 km per litre, so 10 × 15 = 150 km - x = 2, 4, 6 and y = 10, 20, 30. Is this direct proportion?
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10 ÷ 2 = 20 ÷ 4 = 30 ÷ 6 = 5, so yes: y = 5x - x = 1, 2, 3 and y = 3, 5, 7. Is this direct proportion?
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3 ÷ 1 = 3 but 5 ÷ 2 = 2.5, so no - y ∝ x and y = 20 when x = 4. Find k.
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k = 20 ÷ 4 = 5 - y = 3x. Find y when x = 12.
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3 × 12 = 36 - y ∝ x and y = 18 when x = 6. Find y when x = 10.
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k = 18 ÷ 6 = 3, so y = 3 × 10 = 30 - y ∝ x and y = 14 when x = 4. Find x when y = 35.
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k = 14 ÷ 4 = 3.5, so x = 35 ÷ 3.5 = 10 - In a direct proportion table, x = 3 gives y = 12. Find y when x = 5, and x when y = 36.
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k = 4, so y = 20 and x = 9 - Does y = 2x + 3 show direct proportion?
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No. When x = 0, y = 3, so the line does not pass through the origin. - A straight-line graph passes through (0, 0) and (4, 10). Write its rule.
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k = 10 ÷ 4 = 2.5, so y = 2.5x - Solve the proportion 3 : 7 = x : 21.
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21 is 7 × 3, so x = 3 × 3 = 9 - Which is direct proportion: (a) the number of workers and the time to build a wall, (b) the number of identical bricks and their total mass?
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(b). (a) is inverse proportion.
Word problems
Direct proportion in real life
Try each one, then check your answer.
- Pancakes for 4 people need 200 g of flour. How much flour for 10 people?
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200 ÷ 4 = 50 g per person, so 10 × 50 = 500 g - Jo earns $12 an hour. How much does she earn for 7.5 hours?
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12 × 7.5 = $90 - On a map, 2 cm stands for 5 km. What real distance is 7 cm?
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5 ÷ 2 = 2.5 km per cm, so 7 × 2.5 = 17.5 km - A printer prints 45 pages in 3 minutes. How long does it take to print 120 pages?
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45 ÷ 3 = 15 pages a minute, so 120 ÷ 15 = 8 minutes - 5 identical books weigh 3 kg. How much do 12 of them weigh?
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3 ÷ 5 = 0.6 kg each, so 12 × 0.6 = 7.2 kg - 3 m of ribbon cost $4.80. How much do 5 m cost?
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4.80 ÷ 3 = $1.60 per metre, so 5 × 1.60 = $8.00
Questions people ask
What is the direct proportion formula?
y = kx, where k is the constant of proportionality. Find k by dividing any y-value by its matching x-value: k = y ÷ x.
What is the symbol for direct proportion?
The symbol is ∝. "y ∝ x" is read "y is directly proportional to x". To solve problems, replace ∝ with "= k" to get y = kx.
What does a direct proportion graph look like?
A straight line through the origin (0, 0). The steeper the line, the bigger k is, because k is the slope of the line.
What is the difference between direct and inverse proportion?
In direct proportion both quantities go up together and y ÷ x stays the same (y = kx). In inverse proportion one goes up while the other goes down and x × y stays the same (y = k ÷ x). Some textbooks call inverse proportion "indirect proportion".
Is direct proportion the same as direct variation?
Yes. In US algebra the same idea is called direct variation ("y varies directly with x"), and k is called the constant of variation. In grade 7 it is called a proportional relationship.
How do you know if a relationship is directly proportional?
Divide y by x for every pair in the table. If you always get the same number, it is direct proportion. On a graph, look for a straight line that goes through (0, 0).
Quick quiz
Direct proportion
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3 pens cost $6. At the same price, how much do 5 pens cost?
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Related topics
Ratio
A ratio compares two or more amounts, written with a colon. A ratio of 2:3 means for every 2 of one thing there are 3 of the other.
Percentages
A percentage is a number out of 100, written with the sign %. So 25% means 25 out of every 100, which is the same as one quarter.
Line graphs
A line graph shows how something changes over time. Points are plotted on a grid and joined by a line, so you can see rises, falls and patterns at a glance.
Speed, distance and time
Speed is distance divided by time. So speed = distance ÷ time, distance = speed × time, and time = distance ÷ speed. A car that goes 150 km in 2 hours travels at 75 km/h.
Compound interest and simple interest
Simple interest is paid only on the starting amount: I = P × r × t. Compound interest is also paid on earlier interest: A = P(1 + r)ⁿ. $1,000 at 10% for 2 years: $1,200 simple, $1,210 compound.
Profit and loss
Profit = selling price − cost price; loss = cost price − selling price. Profit percentage = profit ÷ cost price × 100. Buy at $20, sell at $25: profit $5, which is 25%.