GCSE Maths: Direct and inverse proportion (Foundation)
Part of the GCSE Maths units and compound measures chapter (Foundation). Chapter overview · Higher version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Direct and inverse proportion
Two quantities are in direct proportion when they increase or decrease at the same rate: if one doubles, the other doubles; if one is halved, the other is halved. The cost of fuel and the number of litres bought is a good example: 20 litres cost twice as much as 10 litres. The key fact is that the ratio (y ÷ x) is always the same number.
That fixed number is the constant of proportionality, called k. Here, k is the price of one litre. The relationship is written y = kx, and we say “y is directly proportional to x” or y ∝ x. To test a table of values for direct proportion, work out y ÷ x for every column: they must all be equal.
| Litres, x | 10 | 20 | 35 | 50 |
|---|---|---|---|---|
| Cost (£), y | 14 | 28 | 49 | 70 |
| y ÷ x | 1.4 | 1.4 | 1.4 | 1.4 |
Two ways to solve a direct proportion problem
Unitary method: find the value for one, then multiply. Equation method: use a known pair to find k (k = y ÷ x), write y = kx, then substitute. The equation method is the best for algebra, and it is the one you need for Higher problems.
Inverse proportion
Two quantities are in inverse proportion when one increases as the other decreases in such a way that if one doubles the other is halved. More workers finish a job in fewer days; a faster speed gives a shorter time for the same distance; sharing a prize among more people gives each person less. In inverse proportion the product x × y is always the same number k. We write y = k ÷ x, and say “y is inversely proportional to x”, which means y is proportional to 1/x.
| Workers, n | 1 | 2 | 3 | 4 | 6 |
|---|---|---|---|---|---|
| Days, d | 12 | 6 | 4 | 3 | 2 |
| n × d | 12 | 12 | 12 | 12 | 12 |
Spot the difference
Direct: y = kx. Both go up together, and the graph is a straight line through the origin. Inverse: y = k/x. As one goes up the other goes down, and the graph is a curve. Be careful: not every pair that both increase is proportional. A taxi that charges a fixed fee plus a price per mile is not directly proportional to distance, because doubling the distance does not double the cost.
Key ideas
- Direct proportion: y = kx, so y ÷ x is constant; doubling x doubles y.
- Inverse proportion: y = k ÷ x, so x × y is constant; doubling x halves y.
- To solve: use a known pair to find k, write the equation, then substitute.
- A graph that does not go through the origin is not direct proportion.
Method. Test for directly proportional: check that y ÷ x is the same for every pair.
- Work out y ÷ x for every pair. In direct proportion this ratio is always the same number (the constant k).6 ÷ 2 = 3
9 ÷ 3 = 3
15 ÷ 5 = 3 - Compare the answers. If they are all equal the quantities are proportional; if even one is different they are not.The answers are all the same.
- State the conclusion. Yes, directly proportional (k = 3)
Check. A graph of a directly proportional relation is a straight line through (0, 0); an inverse one is a curve that never touches the axes.
Watch out. A table can go up steadily and still not be proportional. Check the ratio (or product) for every column.
Method. Test for directly proportional: check that y ÷ x is the same for every pair.
- Work out y ÷ x for every pair. In direct proportion this ratio is always the same number (the constant k).3 ÷ 1 = 3
5 ÷ 2 = 2.5
7 ÷ 3 = 73 - Compare the answers. If they are all equal the quantities are proportional; if even one is different they are not.The answers are not all the same.
- State the conclusion. No, not directly proportional
Check. A graph of a directly proportional relation is a straight line through (0, 0); an inverse one is a curve that never touches the axes.
Watch out. A table can go up steadily and still not be proportional. Check the ratio (or product) for every column.
Method. Unitary method: first find the value for ONE, then multiply to get the amount you need.
- Write down the rate you know. The same rate applies to every amount, so the amount for one can be found from any known amount.4 people goes with 300 g
- Find the flour for one person. Dividing by 4 gives the amount for 1.300 g ÷ 4 = 75 g
- Multiply by 10. We want 10 lots of one person.75 g × 10 = 750 g
Check. Compare with the original: 10 people is 2.5 times 4 people, so the flour is 2.5 times 300 g = 750 g. ✓
Watch out. Divide by the quantity you were given first, then multiply by the quantity you want.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "y is directly proportional to x" means y ∝ x. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).y = kx
- Use the known pair to find k. Put x = 4 and y = 20 into the equation, then solve for k.20 = k × 4
k = 20 ÷ 4 = 5 - Write the equation with the value of k. Now the equation links every pair of values.y = 5x
- Substitute x = 7. We want y when x is 7.y = 5 × 7 = 35
- Write the answer. y = 35
Check. For the new pair, 35 ÷ 7 = 5, the same constant k. ✓
Watch out. In direct proportion the ratio y ÷ x is always the same number, k.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "d is inversely proportional to n" means d ∝ 1n. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).d = kn
- Use the known pair to find k. Put n = 6 and d = 10 into the equation, then solve for k.10 = k6
k = 10 × 6 = 60 - Write the equation with the value of k. Now the equation links every pair of values.d = 60n
- Substitute n = 4. We want d when n is 4.d = 60 ÷ 4 = 15
- Write the answer. d = 15 days
Check. For the new pair, 4 × 15 = 60, the same constant k = 60. ✓
Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.
Watch out
A table can go up steadily and still not show proportion. Check the ratio (or product) in every column. And do not confuse the two types: in direct proportion you divide (y ÷ x) to find k; in inverse proportion you multiply (x × y).Exercise 16I Direct proportion
Calculator allowedMethod. Unitary method: first find the value for ONE, then multiply to get the amount you need.
- Write down the rate you know. The same rate applies to every amount, so the amount for one can be found from any known amount.5 pens goes with £3
- Find the cost for one pen. Dividing by 5 gives the amount for 1.£3 ÷ 5 = £0.60
- Multiply by 15. We want 15 lots of one pen.£0.60 × 15 = £9
Check. Compare with the original: 15 pens is 3 times 5 pens, so the cost is 3 times £3 = £9. ✓
Watch out. Divide by the quantity you were given first, then multiply by the quantity you want.
Method. Test for directly proportional: check that y ÷ x is the same for every pair.
- Work out y ÷ x for every pair. In direct proportion this ratio is always the same number (the constant k).6 ÷ 2 = 3
9 ÷ 3 = 3
15 ÷ 5 = 3 - Compare the answers. If they are all equal the quantities are proportional; if even one is different they are not.The answers are all the same.
- State the conclusion. Yes, directly proportional (k = 3)
Check. A graph of a directly proportional relation is a straight line through (0, 0); an inverse one is a curve that never touches the axes.
Watch out. A table can go up steadily and still not be proportional. Check the ratio (or product) for every column.
Method. Test for directly proportional: check that y ÷ x is the same for every pair.
- Work out y ÷ x for every pair. In direct proportion this ratio is always the same number (the constant k).4 ÷ 1 = 4
8 ÷ 2 = 4
11 ÷ 3 = 113 - Compare the answers. If they are all equal the quantities are proportional; if even one is different they are not.The answers are not all the same.
- State the conclusion. No, not directly proportional
Check. A graph of a directly proportional relation is a straight line through (0, 0); an inverse one is a curve that never touches the axes.
Watch out. A table can go up steadily and still not be proportional. Check the ratio (or product) for every column.
Method. Test for directly proportional: check that y ÷ x is the same for every pair.
- Work out y ÷ x for every pair. In direct proportion this ratio is always the same number (the constant k).10 ÷ 4 = 2.5
15 ÷ 6 = 2.5
25 ÷ 10 = 2.5 - Compare the answers. If they are all equal the quantities are proportional; if even one is different they are not.The answers are all the same.
- State the conclusion. Yes, directly proportional (k = 2.5)
Check. A graph of a directly proportional relation is a straight line through (0, 0); an inverse one is a curve that never touches the axes.
Watch out. A table can go up steadily and still not be proportional. Check the ratio (or product) for every column.
Method. Test for directly proportional: check that y ÷ x is the same for every pair.
- Work out y ÷ x for every pair. In direct proportion this ratio is always the same number (the constant k).5 ÷ 2 = 2.5
8 ÷ 4 = 2
11 ÷ 6 = 116 - Compare the answers. If they are all equal the quantities are proportional; if even one is different they are not.The answers are not all the same.
- State the conclusion. No, not directly proportional
Check. A graph of a directly proportional relation is a straight line through (0, 0); an inverse one is a curve that never touches the axes.
Watch out. A table can go up steadily and still not be proportional. Check the ratio (or product) for every column.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "y is directly proportional to x" means y ∝ x. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).y = kx
- Use the known pair to find k. Put x = 2 and y = 6 into the equation, then solve for k.6 = k × 2
k = 6 ÷ 2 = 3 - Write the equation with the value of k. Now the equation links every pair of values.y = 3x
- Substitute x = 5. We want y when x is 5.y = 3 × 5 = 15
- Write the answer. y = 15
Check. For the new pair, 15 ÷ 5 = 3, the same constant k. ✓
Watch out. In direct proportion the ratio y ÷ x is always the same number, k.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "y is directly proportional to x" means y ∝ x. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).y = kx
- Use the known pair to find k. Put x = 2 and y = 6 into the equation, then solve for k.6 = k × 2
k = 6 ÷ 2 = 3 - Write the equation with the value of k. Now the equation links every pair of values.y = 3x
- Substitute x = 8. We want y when x is 8.y = 3 × 8 = 24
- Write the answer. y = 24
Check. For the new pair, 24 ÷ 8 = 3, the same constant k. ✓
Watch out. In direct proportion the ratio y ÷ x is always the same number, k.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "y is directly proportional to x" means y ∝ x. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).y = kx
- Use the known pair to find k. Put x = 2 and y = 6 into the equation, then solve for k.6 = k × 2
k = 6 ÷ 2 = 3 - Write the equation with the value of k. Now the equation links every pair of values.y = 3x
- Substitute x = 10. We want y when x is 10.y = 3 × 10 = 30
- Write the answer. y = 30
Check. For the new pair, 30 ÷ 10 = 3, the same constant k. ✓
Watch out. In direct proportion the ratio y ÷ x is always the same number, k.
Method. Unitary method: first find the value for ONE, then multiply to get the amount you need.
- Write down the rate you know. The same rate applies to every amount, so the amount for one can be found from any known amount.4 people goes with 300 g
- Find the flour for one person. Dividing by 4 gives the amount for 1.300 g ÷ 4 = 75 g
- Multiply by 10. We want 10 lots of one person.75 g × 10 = 750 g
Check. Compare with the original: 10 people is 2.5 times 4 people, so the flour is 2.5 times 300 g = 750 g. ✓
Watch out. Divide by the quantity you were given first, then multiply by the quantity you want.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "y is directly proportional to x" means y ∝ x. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).y = kx
- Use the known pair to find k. Put x = 3 and y = 12 into the equation, then solve for k.12 = k × 3
k = 12 ÷ 3 = 4 - Write the equation with the value of k. Now the equation links every pair of values.y = 4x
- Substitute x = 10. We want y when x is 10.y = 4 × 10 = 40
- Write the answer. y = 40
Check. For the new pair, 40 ÷ 10 = 4, the same constant k. ✓
Watch out. In direct proportion the ratio y ÷ x is always the same number, k.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "y is directly proportional to x" means y ∝ x. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).y = kx
- Use the known pair to find k. Put x = 5 and y = 35 into the equation, then solve for k.35 = k × 5
k = 35 ÷ 5 = 7 - Write the equation with the value of k. Now the equation links every pair of values.y = 7x
- Substitute x = 8. We want y when x is 8.y = 7 × 8 = 56
- Write the answer. y = 56
Check. For the new pair, 56 ÷ 8 = 7, the same constant k. ✓
Watch out. In direct proportion the ratio y ÷ x is always the same number, k.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "y is directly proportional to x" means y ∝ x. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).y = kx
- Use the known pair to find k. Put x = 8 and y = 20 into the equation, then solve for k.20 = k × 8
k = 20 ÷ 8 = 2.5 - Write the equation with the value of k. Now the equation links every pair of values.y = 2.5x
- Substitute y = 35 and solve for x. We know y and want x, so rearrange the equation.35 = 2.5x
x = 35 ÷ 2.5 = 14 - Write the answer. x = 14
Check. For the new pair, 35 ÷ 14 = 2.5, the same constant k. ✓
Watch out. In direct proportion the ratio y ÷ x is always the same number, k.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "y is directly proportional to x" means y ∝ x. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).y = kx
- Use the known pair to find k. Put x = 6 and y = 9 into the equation, then solve for k.9 = k × 6
k = 9 ÷ 6 = 1.5 - Write the equation with the value of k. Now the equation links every pair of values.y = 1.5x
- Substitute x = 14. We want y when x is 14.y = 1.5 × 14 = 21
- Write the answer. y = 21
Check. For the new pair, 21 ÷ 14 = 1.5, the same constant k. ✓
Watch out. In direct proportion the ratio y ÷ x is always the same number, k.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "e is directly proportional to F" means e ∝ F. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).e = kF
- Use the known pair to find k. Put F = 12 and e = 3 into the equation, then solve for k.3 = k × 12
k = 3 ÷ 12 = 0.25 - Write the equation with the value of k. Now the equation links every pair of values.e = 0.25F
- Substitute F = 20. We want e when F is 20.e = 0.25 × 20 = 5
- Write the answer. e = 5 cm
Check. For the new pair, 5 ÷ 20 = 0.25, the same constant k. ✓
Watch out. In direct proportion the ratio y ÷ x is always the same number, k.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "e is directly proportional to F" means e ∝ F. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).e = kF
- Use the known pair to find k. Put F = 12 and e = 3 into the equation, then solve for k.3 = k × 12
k = 3 ÷ 12 = 0.25 - Write the equation with the value of k. Now the equation links every pair of values.e = 0.25F
- Substitute e = 7.5 and solve for F. We know e and want F, so rearrange the equation.7.5 = 0.25F
F = 7.5 ÷ 0.25 = 30 - Write the answer. F = 30 N
Check. For the new pair, 7.5 ÷ 30 = 0.25, the same constant k. ✓
Watch out. In direct proportion the ratio y ÷ x is always the same number, k.
Method. Compare the equation with y = kx.
- Match the equation. C = 12.5n has the form y = kx with k = 12.5.k = 12.5
- Interpret k. When n = 1, C = 12.5, so 12.5 is the cost of 1 ticket.The constant is the price of one ticket: £12.50
Check. n = 2 gives C = 25, twice as much. ✓
Method. Substitute n = 8.
- Put n = 8 into the equation. C = 12.5 × 8 = 100
- Write with a pound sign. £100
Check. 12.5 × 8 = 100 at £12.50 each: 10 tickets would be £125, so 8 is less. ✓
Method. Substitute C = 200 and solve for n.
- Put C = 200 into the equation. 200 = 12.5n
- Divide both sides by 12.5. This leaves n on its own.n = 200 ÷ 12.5 = 16
Check. 16 × 12.5 = 200. ✓
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "P is directly proportional to h" means P ∝ h. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).P = kh
- Use the known pair to find k. Put h = 15 and P = 127.5 into the equation, then solve for k.127.5 = k × 15
k = 127.5 ÷ 15 = 8.5 - Write the equation with the value of k. Now the equation links every pair of values.P = 8.5h
- Substitute h = 22. We want P when h is 22.P = 8.5 × 22 = 187
- Write the answer. P = 187 pounds
Check. For the new pair, 187 ÷ 22 = 8.5, the same constant k. ✓
Watch out. In direct proportion the ratio y ÷ x is always the same number, k.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "P is directly proportional to h" means P ∝ h. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).P = kh
- Use the known pair to find k. Put h = 15 and P = 127.5 into the equation, then solve for k.127.5 = k × 15
k = 127.5 ÷ 15 = 8.5 - Write the equation with the value of k. Now the equation links every pair of values.P = 8.5h
- Substitute P = 204 and solve for h. We know P and want h, so rearrange the equation.204 = 8.5h
h = 204 ÷ 8.5 = 24 - Write the answer. h = 24 hours
Check. For the new pair, 204 ÷ 24 = 8.5, the same constant k. ✓
Watch out. In direct proportion the ratio y ÷ x is always the same number, k.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "r is directly proportional to m" means r ∝ m. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).r = km
- Use the known pair to find k. Put m = 2 and r = 5 into the equation, then solve for k.5 = k × 2
k = 5 ÷ 2 = 2.5 - Write the equation with the value of k. Now the equation links every pair of values.r = 2.5m
- Substitute m = 7.5. We want r when m is 7.5.r = 2.5 × 7.5 = 18.75
- Write the answer. r = 18.75 km
Check. For the new pair, 18.75 ÷ 7.5 = 2.5, the same constant k. ✓
Watch out. In direct proportion the ratio y ÷ x is always the same number, k.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "r is directly proportional to m" means r ∝ m. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).r = km
- Use the known pair to find k. Put m = 2 and r = 5 into the equation, then solve for k.5 = k × 2
k = 5 ÷ 2 = 2.5 - Write the equation with the value of k. Now the equation links every pair of values.r = 2.5m
- Substitute r = 32.5 and solve for m. We know r and want m, so rearrange the equation.32.5 = 2.5m
m = 32.5 ÷ 2.5 = 13 - Write the answer. m = 13 cm
Check. For the new pair, 32.5 ÷ 13 = 2.5, the same constant k. ✓
Watch out. In direct proportion the ratio y ÷ x is always the same number, k.
Method. If cost were directly proportional, doubling the distance would double the cost. Test this with numbers.
- Cost for 1 mile. Fixed charge plus one mile.3 + 2 × 1 = £5
- Cost for 2 miles. Fixed charge plus two miles.3 + 2 × 2 = £7
- Compare. Direct proportion would give 2 × £5 = £10.£7 ≠ £10, so it is not proportional.
Check. Also cost ÷ distance is 5 for 1 mile and 3.5 for 2 miles: not constant. ✓
Method. Test for directly proportional: check that y ÷ x is the same for every pair.
- Work out y ÷ x for every pair. In direct proportion this ratio is always the same number (the constant k).3 ÷ 1 = 3
5 ÷ 2 = 2.5
7 ÷ 3 = 73 - Compare the answers. If they are all equal the quantities are proportional; if even one is different they are not.The answers are not all the same.
- State the conclusion. No, not directly proportional
Check. A graph of a directly proportional relation is a straight line through (0, 0); an inverse one is a curve that never touches the axes.
Watch out. A table can go up steadily and still not be proportional. Check the ratio (or product) for every column.
Method. Unitary method: first find the value for ONE, then multiply to get the amount you need.
- Write down the rate you know. The same rate applies to every amount, so the amount for one can be found from any known amount.250 leaflets goes with £17.50
- Find the cost for one leaflet. Dividing by 250 gives the amount for 1.£17.50 ÷ 250 = £0.07
- Multiply by 600. We want 600 lots of one leaflet.£0.07 × 600 = £42
Check. Compare with the original: 600 leaflets is 2.4 times 250 leaflets, so the cost is 2.4 times £17.50 = £42. ✓
Watch out. Divide by the quantity you were given first, then multiply by the quantity you want.
Method. Find the number of bottles filled in one minute, then work out the time for 2400.
- Bottles per minute. Rate = bottles ÷ minutes.360 ÷ 45 = 8 bottles per minute
- Minutes for 2400 bottles. Time = number of bottles ÷ rate.2400 ÷ 8 = 300 minutes
- Change to hours. Minutes to hours: divide by 60.300 ÷ 60 = 5 hours
Check. 2400 is 6⅔ times 360, and 6⅔ × 45 = 300 minutes. ✓
Exercise 16J Inverse proportion
Calculator allowedMethod. More people means less time. Find the total work in “person-hours”, then share it between the new number of people.
- Find the total work. 3 people for 12 hours is 36 person-hours.3 × 12 = 36
- Share between 6 people. Divide the total work by the number of people.36 ÷ 6 = 6 hours
Check. 6 people is double 3 people, so the time is halved: 12 ÷ 2 = 6. ✓
Method. Test for inversely proportional: check that x × y is the same for every pair.
- Work out x × y for every pair. In inverse proportion the product is always the same number (the constant k).1 × 24 = 24
2 × 12 = 24
3 × 8 = 24
4 × 6 = 24 - Compare the answers. If they are all equal the quantities are proportional; if even one is different they are not.The answers are all the same.
- State the conclusion. Yes, inversely proportional (k = 24)
Check. A graph of a directly proportional relation is a straight line through (0, 0); an inverse one is a curve that never touches the axes.
Watch out. A table can go up steadily and still not be proportional. Check the ratio (or product) for every column.
Method. Test for inversely proportional: check that x × y is the same for every pair.
- Work out x × y for every pair. In inverse proportion the product is always the same number (the constant k).2 × 10 = 20
4 × 5 = 20
5 × 3 = 15 - Compare the answers. If they are all equal the quantities are proportional; if even one is different they are not.The answers are not all the same.
- State the conclusion. No, not inversely proportional
Check. A graph of a directly proportional relation is a straight line through (0, 0); an inverse one is a curve that never touches the axes.
Watch out. A table can go up steadily and still not be proportional. Check the ratio (or product) for every column.
Method. Test for inversely proportional: check that x × y is the same for every pair.
- Work out x × y for every pair. In inverse proportion the product is always the same number (the constant k).2 × 15 = 30
5 × 6 = 30
10 × 3 = 30 - Compare the answers. If they are all equal the quantities are proportional; if even one is different they are not.The answers are all the same.
- State the conclusion. Yes, inversely proportional (k = 30)
Check. A graph of a directly proportional relation is a straight line through (0, 0); an inverse one is a curve that never touches the axes.
Watch out. A table can go up steadily and still not be proportional. Check the ratio (or product) for every column.
Method. Test for inversely proportional: check that x × y is the same for every pair.
- Work out x × y for every pair. In inverse proportion the product is always the same number (the constant k).1 × 10 = 10
2 × 8 = 16
3 × 6 = 18 - Compare the answers. If they are all equal the quantities are proportional; if even one is different they are not.The answers are not all the same.
- State the conclusion. No, not inversely proportional
Check. A graph of a directly proportional relation is a straight line through (0, 0); an inverse one is a curve that never touches the axes.
Watch out. A table can go up steadily and still not be proportional. Check the ratio (or product) for every column.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "d is inversely proportional to n" means d ∝ 1n. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).d = kn
- Use the known pair to find k. Put n = 6 and d = 10 into the equation, then solve for k.10 = k6
k = 10 × 6 = 60 - Write the equation with the value of k. Now the equation links every pair of values.d = 60n
- Substitute n = 4. We want d when n is 4.d = 60 ÷ 4 = 15
- Write the answer. d = 15 days
Check. For the new pair, 4 × 15 = 60, the same constant k = 60. ✓
Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "d is inversely proportional to n" means d ∝ 1n. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).d = kn
- Use the known pair to find k. Put n = 6 and d = 10 into the equation, then solve for k.10 = k6
k = 10 × 6 = 60 - Write the equation with the value of k. Now the equation links every pair of values.d = 60n
- Substitute n = 12. We want d when n is 12.d = 60 ÷ 12 = 5
- Write the answer. d = 5 days
Check. For the new pair, 12 × 5 = 60, the same constant k = 60. ✓
Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "d is inversely proportional to n" means d ∝ 1n. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).d = kn
- Use the known pair to find k. Put n = 6 and d = 10 into the equation, then solve for k.10 = k6
k = 10 × 6 = 60 - Write the equation with the value of k. Now the equation links every pair of values.d = 60n
- Substitute n = 15. We want d when n is 15.d = 60 ÷ 15 = 4
- Write the answer. d = 4 days
Check. For the new pair, 15 × 4 = 60, the same constant k = 60. ✓
Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "d is inversely proportional to n" means d ∝ 1n. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).d = kn
- Use the known pair to find k. Put n = 6 and d = 10 into the equation, then solve for k.10 = k6
k = 10 × 6 = 60 - Write the equation with the value of k. Now the equation links every pair of values.d = 60n
- Substitute n = 5. We want d when n is 5.d = 60 ÷ 5 = 12
- Write the answer. d = 12 days
Check. For the new pair, 5 × 12 = 60, the same constant k = 60. ✓
Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "t is inversely proportional to v" means t ∝ 1v. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).t = kv
- Use the known pair to find k. Put v = 60 and t = 3 into the equation, then solve for k.3 = k60
k = 3 × 60 = 180 - Write the equation with the value of k. Now the equation links every pair of values.t = 180v
- Substitute v = 90. We want t when v is 90.t = 180 ÷ 90 = 2
- Write the answer. t = 2 hours
Check. For the new pair, 90 × 2 = 180, the same constant k = 180. ✓
Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "t is inversely proportional to v" means t ∝ 1v. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).t = kv
- Use the known pair to find k. Put v = 60 and t = 3 into the equation, then solve for k.3 = k60
k = 3 × 60 = 180 - Write the equation with the value of k. Now the equation links every pair of values.t = 180v
- Substitute v = 45. We want t when v is 45.t = 180 ÷ 45 = 4
- Write the answer. t = 4 hours
Check. For the new pair, 45 × 4 = 180, the same constant k = 180. ✓
Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "t is inversely proportional to v" means t ∝ 1v. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).t = kv
- Use the known pair to find k. Put v = 60 and t = 3 into the equation, then solve for k.3 = k60
k = 3 × 60 = 180 - Write the equation with the value of k. Now the equation links every pair of values.t = 180v
- Substitute v = 120. We want t when v is 120.t = 180 ÷ 120 = 1.5
- Write the answer. t = 1.5 hours
Check. For the new pair, 120 × 1.5 = 180, the same constant k = 180. ✓
Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "y is inversely proportional to x" means y ∝ 1x. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).y = kx
- Use the known pair to find k. Put x = 1 and y = 24 into the equation, then solve for k.24 = k1
k = 24 × 1 = 24 - Write the equation with the value of k. Now the equation links every pair of values.y = 24x
- Substitute x = 2. We want y when x is 2.y = 24 ÷ 2 = 12
- Write the answer. y = 12
Check. For the new pair, 2 × 12 = 24, the same constant k = 24. ✓
Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "y is inversely proportional to x" means y ∝ 1x. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).y = kx
- Use the known pair to find k. Put x = 1 and y = 24 into the equation, then solve for k.24 = k1
k = 24 × 1 = 24 - Write the equation with the value of k. Now the equation links every pair of values.y = 24x
- Substitute x = 3. We want y when x is 3.y = 24 ÷ 3 = 8
- Write the answer. y = 8
Check. For the new pair, 3 × 8 = 24, the same constant k = 24. ✓
Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "y is inversely proportional to x" means y ∝ 1x. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).y = kx
- Use the known pair to find k. Put x = 1 and y = 24 into the equation, then solve for k.24 = k1
k = 24 × 1 = 24 - Write the equation with the value of k. Now the equation links every pair of values.y = 24x
- Substitute x = 4. We want y when x is 4.y = 24 ÷ 4 = 6
- Write the answer. y = 6
Check. For the new pair, 4 × 6 = 24, the same constant k = 24. ✓
Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "y is inversely proportional to x" means y ∝ 1x. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).y = kx
- Use the known pair to find k. Put x = 1 and y = 24 into the equation, then solve for k.24 = k1
k = 24 × 1 = 24 - Write the equation with the value of k. Now the equation links every pair of values.y = 24x
- Substitute x = 8. We want y when x is 8.y = 24 ÷ 8 = 3
- Write the answer. y = 3
Check. For the new pair, 8 × 3 = 24, the same constant k = 24. ✓
Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.
Method. Think about what happens when one doubles.
- Double the taps. Twice as many taps fill the bath twice as fast.The time is halved
- Decide. Doubling one halves the other.Inverse proportion
Check. taps × time stays the same. ✓
Method. Test with doubling.
- Age 4 and age 8. A child is not twice as tall at 8 as at 4, and not half as tall either.Doubling the age does not double or halve the height
- Decide. Neither
Check. Height depends on many things and the pattern is not constant. ✓
Method. Time = distance ÷ speed with a fixed distance.
- Write the formula. The distance is fixed at 100 km.time = 100 ÷ speed
- Decide. This has the form y = k/x.Inverse proportion
Check. Doubling the speed halves the time. ✓
Method. Each share = 60 ÷ number of people.
- Write the formula. share = 60 ÷ people
- Decide. This has the form y = k/x.Inverse proportion
Check. Twice the people, half the share each. ✓
Method. Cost = 4.50 × number of books.
- Write the formula. cost = 4.5 × books
- Decide. This has the form y = kx.Direct proportion
Check. Doubling the books doubles the cost. ✓
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "s is inversely proportional to n" means s ∝ 1n. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).s = kn
- Use the known pair to find k. Put n = 10 and s = 6 into the equation, then solve for k.6 = k10
k = 6 × 10 = 60 - Write the equation with the value of k. Now the equation links every pair of values.s = 60n
- Substitute n = 12. We want s when n is 12.s = 60 ÷ 12 = 5
- Write the answer. s = 5
Check. For the new pair, 12 × 5 = 60, the same constant k = 60. ✓
Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "s is inversely proportional to n" means s ∝ 1n. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).s = kn
- Use the known pair to find k. Put n = 10 and s = 6 into the equation, then solve for k.6 = k10
k = 6 × 10 = 60 - Write the equation with the value of k. Now the equation links every pair of values.s = 60n
- Substitute s = 4 and solve for n. We know s and want n, so rearrange the equation.4 = 60 ÷ n
n = 60 ÷ 4 = 15 - Write the answer. n = 15
Check. For the new pair, 15 × 4 = 60, the same constant k = 60. ✓
Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.
Method. Write the equation and replace x by 4x.
- Write the equation. Inverse proportion means y = k/x.y = kx
- Multiply x by 4. The new x is 4x.new y = k4x = 14 × kx
- Conclude. The new y is one quarter of the old y.y is divided by 4
Check. Example: k = 24, x = 2 gives y = 12. With x = 8, y = 3 = 12 ÷ 4. ✓
Key words. The constant of proportionality k is the fixed number that links the two quantities. ∝ means “is proportional to”.
Method. For a proportion problem: write the equation with k, use the known pair to find k, then use the equation to answer the question.
- Write the proportion as an equation with a constant k. "V is inversely proportional to P" means V ∝ 1P. A proportion sign always turns into an equals sign with a multiplier k (the constant of proportionality).V = kP
- Use the known pair to find k. Put P = 150 and V = 12 into the equation, then solve for k.12 = k150
k = 12 × 150 = 1800 - Write the equation with the value of k. Now the equation links every pair of values.V = 1800P
- Substitute P = 200. We want V when P is 200.V = 1800 ÷ 200 = 9
- Write the answer. V = 9 litres
Check. For the new pair, 200 × 9 = 1800, the same constant k = 1800. ✓
Watch out. For inverse proportion, y goes down as x goes up, and the product is constant.
Method. Count the work in “worker-days”, take off what has been done, then share the rest between the workers left.
- Total work. 10 workers for 12 days.10 × 12 = 120 worker-days
- Work done in 4 days. 10 workers for 4 days.10 × 4 = 40 worker-days
- Work left. Total minus done.120 − 40 = 80 worker-days
- Workers left. 2 workers leave.10 − 2 = 8 workers
- Days needed. Work left ÷ workers.80 ÷ 8 = 10 days
Check. 8 workers for 10 days is 8 × 10 = 80 worker-days, the work that was left. ✓
Watch out. Do not forget to subtract the work already done before sharing out what is left.