GCSE Maths: Proportion as equality of ratios; ratio and linear functions (Higher)
Part of the GCSE Maths ratio and proportion chapter (Higher). Chapter overview · Foundation version of this lesson · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Proportion as equality of ratios; ratio and linear functions
Two ratios are equal (or “in proportion”) when they simplify to the same ratio. 2 : 3 and 6 : 9 are equal because 6 : 9 = 2 : 3 after dividing both parts by 3. We write 2 : 3 = 6 : 9. The second ratio is the first multiplied by 3. To find a missing number in equal ratios, find this multiplier from the pair where both numbers are known, and use it on the other pair.
Checking whether ratios are equal. Simplify both ratios and compare, or cross-multiply: the ratios a : b and c : d are equal if a × d = b × c. For 6 : 9 and 10 : 15 we get 6 × 15 = 90 and 9 × 10 = 90, so they are equal. Remember that equal ratios come from multiplying or dividing both parts, never from adding the same number to both.
Direct proportion and tables. If y is in direct proportion to x, then y ÷ x is the same number every time. Call this number k. For example, if cost is directly proportional to tickets and 2 tickets cost £7, then k = 7 ÷ 2 = 3.5 and every pair has y ÷ x = 3.5. The ratio y : x is always k : 1.
Ratio and linear functions. Since y ÷ x = k, we have y = kx. This is a linear function. Its graph is a straight line through the origin (0, 0), and the gradient of the line is k (Chapter 12). For a conversion graph from miles to kilometres, k = 1.6 and the line is y = 1.6x. A line such as y = 2x + 3 is also straight but does not go through the origin, so it does not show direct proportion: doubling x does not double y.
Key ideas
- Equal ratios: multiply or divide both parts by the same number. Find the multiplier from the known pair.
- Test equality by simplifying both ratios, or by cross-multiplying: a : b = c : d if ad = bc.
- Direct proportion: y ÷ x is always the same number k. y = kx.
- The graph of y = kx is a straight line through the origin with gradient k.
- A straight line that does not pass through the origin is not direct proportion.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Equal ratios are multiples of each other. Find the multiplier from the pair you know, then apply it to the other number.
- Write the two ratios as equal. Equal ratios (proportion) means one ratio is the other multiplied by the same number.3 : 5 = 12 : ?
- Find the multiplier on the side where both numbers are known. Divide the new number by the old number in the same position. This tells us how much the ratio has been scaled.12 ÷ 3 = 4
- Multiply the other number by the same multiplier. Both parts of an equal ratio must be scaled by the same number.5 × 4 = 20
Check. Divide the new ratio by 4: 12 ÷ 4 = 3 and 20 ÷ 4 = 5, giving 3 : 5 again. ✓
Watch out. The multiplier is found by division and applied by multiplication to the other part. It need not be a whole number.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. Simplify both ratios. If they are the same, the ratios are equal (the quantities are in proportion).
- Simplify the first ratio. Two ratios are equal when they simplify to the same ratio.6 : 9 = 2 : 3
- Simplify the second ratio. Use the same method: divide by the highest common factor.10 : 15 = 2 : 3
- Compare. Look at the two simplest forms.2 : 3 and 2 : 3 are the same, so the ratios are equal.
Check. Cross-multiply as a check: 6 × 15 = 90 and 9 × 10 = 90. They are equal. ✓
Watch out. Adding the same number to both parts changes a ratio. Equal ratios come from multiplying or dividing both parts, not adding.
| Tickets (x) | 2 | 5 | 8 |
|---|---|---|---|
| Cost in £ (y) | 7 | 17.5 | 28 |
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. For direct proportion, y ÷ x must be the same for every pair of values. Work out y ÷ x for each column.
- Divide y by x for each pair. If they are all the same number, the quantities are in direct proportion.7 ÷ 2 = 3.5
17.5 ÷ 5 = 3.5
28 ÷ 8 = 3.5 - State the constant k. All three results are the same, so the cost is directly proportional to the number of tickets, and k = 3.5.k = 3.5
- Write the rule. Use y = kx.y = 3.5x
Check. Substitute x = 8: y = 3.5 × 8 = 28, which matches the table. ✓
Watch out. A table with a constant difference in y is not necessarily proportional. It must have a constant ratio y ÷ x.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. For direct proportion, write y = kx, use the known pair to find k, then use the equation.
- Write down the rule for direct proportion. If y is directly proportional to x, then y is always the same multiple of x. That multiple is called k, the constant of proportionality.y = kx
- Use the pair of values we know to find k. Substitute x and y into y = kx, then divide to get k on its own.10 = k × 4
k = 10 ÷ 4 = 2.5 - Write the rule with the value of k. Now we have an equation we can use for any pair of values.y = 2.5x
- Substitute x = 14. Replace x by the number and multiply.y = 2.5 × 14 = 35
Check. 14 is 3.5 times 4, so y should be 3.5 times 10 = 35. ✓
Watch out. k is found by dividing y by x (y ÷ x), not x by y.
Watch out
A straight-line graph is not enough to show direct proportion. It must pass through the origin (0, 0). Also, y = kx means y ÷ x = k. Do not work out x ÷ y by mistake.Exercise 15H Equal ratios and direct proportion
Non-calculatorKey words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Equal ratios are multiples of each other. Find the multiplier from the pair you know, then apply it to the other number.
- Write the two ratios as equal. Equal ratios (proportion) means one ratio is the other multiplied by the same number.2 : 3 = 8 : ?
- Find the multiplier on the side where both numbers are known. Divide the new number by the old number in the same position. This tells us how much the ratio has been scaled.8 ÷ 2 = 4
- Multiply the other number by the same multiplier. Both parts of an equal ratio must be scaled by the same number.3 × 4 = 12
Check. Divide the new ratio by 4: 8 ÷ 4 = 2 and 12 ÷ 4 = 3, giving 2 : 3 again. ✓
Watch out. The multiplier is found by division and applied by multiplication to the other part. It need not be a whole number.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Equal ratios are multiples of each other. Find the multiplier from the pair you know, then apply it to the other number.
- Write the two ratios as equal. Equal ratios (proportion) means one ratio is the other multiplied by the same number.5 : 4 = 20 : ?
- Find the multiplier on the side where both numbers are known. Divide the new number by the old number in the same position. This tells us how much the ratio has been scaled.20 ÷ 5 = 4
- Multiply the other number by the same multiplier. Both parts of an equal ratio must be scaled by the same number.4 × 4 = 16
Check. Divide the new ratio by 4: 20 ÷ 4 = 5 and 16 ÷ 4 = 4, giving 5 : 4 again. ✓
Watch out. The multiplier is found by division and applied by multiplication to the other part. It need not be a whole number.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Equal ratios are multiples of each other. Find the multiplier from the pair you know, then apply it to the other number.
- Write the two ratios as equal. Equal ratios (proportion) means one ratio is the other multiplied by the same number.3 : 7 = ? : 21
- Find the multiplier on the side where both numbers are known. Divide the new number by the old number in the same position. This tells us how much the ratio has been scaled.21 ÷ 7 = 3
- Multiply the other number by the same multiplier. Both parts of an equal ratio must be scaled by the same number.3 × 3 = 9
Check. Divide the new ratio by 3: 21 ÷ 3 = 7 and 9 ÷ 3 = 3, giving 3 : 7 again. ✓
Watch out. The multiplier is found by division and applied by multiplication to the other part. It need not be a whole number.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Equal ratios are multiples of each other. Find the multiplier from the pair you know, then apply it to the other number.
- Write the two ratios as equal. Equal ratios (proportion) means one ratio is the other multiplied by the same number.4 : 9 = ? : 36
- Find the multiplier on the side where both numbers are known. Divide the new number by the old number in the same position. This tells us how much the ratio has been scaled.36 ÷ 9 = 4
- Multiply the other number by the same multiplier. Both parts of an equal ratio must be scaled by the same number.4 × 4 = 16
Check. Divide the new ratio by 4: 36 ÷ 4 = 9 and 16 ÷ 4 = 4, giving 4 : 9 again. ✓
Watch out. The multiplier is found by division and applied by multiplication to the other part. It need not be a whole number.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Equal ratios are multiples of each other. Find the multiplier from the pair you know, then apply it to the other number.
- Write the two ratios as equal. Equal ratios (proportion) means one ratio is the other multiplied by the same number.6 : 4 = 9 : ?
- Find the multiplier on the side where both numbers are known. Divide the new number by the old number in the same position. This tells us how much the ratio has been scaled.9 ÷ 6 = 1.5
- Multiply the other number by the same multiplier. Both parts of an equal ratio must be scaled by the same number.4 × 1.5 = 6
Check. Divide the new ratio by 1.5: 9 ÷ 1.5 = 6 and 6 ÷ 1.5 = 4, giving 6 : 4 again. ✓
Watch out. The multiplier is found by division and applied by multiplication to the other part. It need not be a whole number.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Equal ratios are multiples of each other. Find the multiplier from the pair you know, then apply it to the other number.
- Write the two ratios as equal. Equal ratios (proportion) means one ratio is the other multiplied by the same number.10 : 6 = 4 : ?
- Find the multiplier on the side where both numbers are known. Divide the new number by the old number in the same position. This tells us how much the ratio has been scaled.4 ÷ 10 = 0.4
- Multiply the other number by the same multiplier. Both parts of an equal ratio must be scaled by the same number.6 × 0.4 = 2.4
Check. Divide the new ratio by 0.4: 4 ÷ 0.4 = 10 and 2.4 ÷ 0.4 = 6, giving 10 : 6 again. ✓
Watch out. The multiplier is found by division and applied by multiplication to the other part. It need not be a whole number.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. Simplify both ratios. If they are the same, the ratios are equal (the quantities are in proportion).
- Simplify the first ratio. Two ratios are equal when they simplify to the same ratio.4 : 6 = 2 : 3
- Simplify the second ratio. Use the same method: divide by the highest common factor.10 : 15 = 2 : 3
- Compare. Look at the two simplest forms.2 : 3 and 2 : 3 are the same, so the ratios are equal.
Check. Cross-multiply as a check: 4 × 15 = 60 and 6 × 10 = 60. They are equal. ✓
Watch out. Adding the same number to both parts changes a ratio. Equal ratios come from multiplying or dividing both parts, not adding.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. Simplify both ratios. If they are the same, the ratios are equal (the quantities are in proportion).
- Simplify the first ratio. Two ratios are equal when they simplify to the same ratio.3 : 5 = 3 : 5
- Simplify the second ratio. Use the same method: divide by the highest common factor.9 : 20 = 9 : 20
- Compare. Look at the two simplest forms.3 : 5 and 9 : 20 are different, so the ratios are not equal.
Check. Cross-multiply as a check: 3 × 20 = 60 and 5 × 9 = 45. They are not equal. ✓
Watch out. Adding the same number to both parts changes a ratio. Equal ratios come from multiplying or dividing both parts, not adding.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. Simplify both ratios. If they are the same, the ratios are equal (the quantities are in proportion).
- Simplify the first ratio. Two ratios are equal when they simplify to the same ratio.8 : 12 = 2 : 3
- Simplify the second ratio. Use the same method: divide by the highest common factor.6 : 9 = 2 : 3
- Compare. Look at the two simplest forms.2 : 3 and 2 : 3 are the same, so the ratios are equal.
Check. Cross-multiply as a check: 8 × 9 = 72 and 12 × 6 = 72. They are equal. ✓
Watch out. Adding the same number to both parts changes a ratio. Equal ratios come from multiplying or dividing both parts, not adding.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. Simplify both ratios. If they are the same, the ratios are equal (the quantities are in proportion).
- Simplify the first ratio. Two ratios are equal when they simplify to the same ratio.5 : 7 = 5 : 7
- Simplify the second ratio. Use the same method: divide by the highest common factor.15 : 22 = 15 : 22
- Compare. Look at the two simplest forms.5 : 7 and 15 : 22 are different, so the ratios are not equal.
Check. Cross-multiply as a check: 5 × 22 = 110 and 7 × 15 = 105. They are not equal. ✓
Watch out. Adding the same number to both parts changes a ratio. Equal ratios come from multiplying or dividing both parts, not adding.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Equal ratios are multiples of each other. Find the multiplier from the pair you know, then apply it to the other number.
- Write the two ratios as equal. Equal ratios (proportion) means one ratio is the other multiplied by the same number.3 : 5 = 12 : ?
- Find the multiplier on the side where both numbers are known. Divide the new number by the old number in the same position. This tells us how much the ratio has been scaled.12 ÷ 3 = 4
- Multiply the other number by the same multiplier. Both parts of an equal ratio must be scaled by the same number.5 × 4 = 20
Check. Divide the new ratio by 4: 12 ÷ 4 = 3 and 20 ÷ 4 = 5, giving 3 : 5 again. ✓
Watch out. The multiplier is found by division and applied by multiplication to the other part. It need not be a whole number.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Equal ratios are multiples of each other. Find the multiplier from the pair you know, then apply it to the other number.
- Write the two ratios as equal. Equal ratios (proportion) means one ratio is the other multiplied by the same number.5 : 3 = 450 : ?
- Find the multiplier on the side where both numbers are known. Divide the new number by the old number in the same position. This tells us how much the ratio has been scaled.450 ÷ 5 = 90
- Multiply the other number by the same multiplier. Both parts of an equal ratio must be scaled by the same number.3 × 90 = 270
Check. Divide the new ratio by 90: 450 ÷ 90 = 5 and 270 ÷ 90 = 3, giving 5 : 3 again. ✓
Watch out. The multiplier is found by division and applied by multiplication to the other part. It need not be a whole number.
| Tickets (x) | 2 | 5 | 8 |
|---|---|---|---|
| Cost in £ (y) | 7 | 17.5 | 28 |
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Work out y ÷ x for each pair. If it is always the same number, the quantities are in direct proportion.
- Divide y by x for each pair. Look for the same answer each time.7 ÷ 2 = 3.5
17.5 ÷ 5 = 3.5
28 ÷ 8 = 3.5 - State the constant k. All three are 3.5, so y is directly proportional to x.k = 3.5
- Write the rule. Use y = kx.y = 3.5x
Check. Check one more value: x = 10 would give y = 35, and 35 ÷ 10 = 3.5. ✓
Watch out. The constant of proportionality is y ÷ x.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. For direct proportion, write y = kx, use the known pair to find k, then use the equation.
- Write down the rule for direct proportion. If y is directly proportional to x, then y is always the same multiple of x. That multiple is called k, the constant of proportionality.y = kx
- Use the pair of values we know to find k. Substitute x and y into y = kx, then divide to get k on its own.10 = k × 4
k = 10 ÷ 4 = 2.5 - Write the rule with the value of k. Now we have an equation we can use for any pair of values.y = 2.5x
- Substitute x = 14. Replace x by the number and multiply.y = 2.5 × 14 = 35
Check. 14 is 3.5 times 4, so y should be 3.5 times 10 = 35. ✓
Watch out. k is found by dividing y by x (y ÷ x), not x by y.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. For direct proportion, write y = kx, use the known pair to find k, then use the equation.
- Write down the rule for direct proportion. If y is directly proportional to x, then y is always the same multiple of x. That multiple is called k, the constant of proportionality.y = kx
- Use the pair of values we know to find k. Substitute x and y into y = kx, then divide to get k on its own.15 = k × 6
k = 15 ÷ 6 = 2.5 - Write the rule with the value of k. Now we have an equation we can use for any pair of values.y = 2.5x
- Substitute x = 10. Replace x by the number and multiply.y = 2.5 × 10 = 25
Check. 10 is 123 times 6, so y should be 123 times 15 = 25. ✓
Watch out. k is found by dividing y by x (y ÷ x), not x by y.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. For direct proportion, write y = kx, use the known pair to find k, then use the equation.
- Write down the rule for direct proportion. If y is directly proportional to x, then y is always the same multiple of x. That multiple is called k, the constant of proportionality.y = kx
- Use the pair of values we know to find k. Substitute x and y into y = kx, then divide to get k on its own.2 = k × 5
k = 2 ÷ 5 = 0.4 - Write the rule with the value of k. Now we have an equation we can use for any pair of values.y = 0.4x
- Substitute y = 8 and solve for x. Replace y by the number, then divide both sides by k to get x on its own.8 = 0.4x
x = 8 ÷ 0.4 = 20
Check. Put it back: 0.4 × 20 = 8. ✓
Watch out. k is found by dividing y by x (y ÷ x), not x by y.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. For direct proportion, write y = kx, use the known pair to find k, then use the equation.
- Write down the rule for direct proportion. If y is directly proportional to x, then y is always the same multiple of x. That multiple is called k, the constant of proportionality.y = kx
- Use the pair of values we know to find k. Substitute x and y into y = kx, then divide to get k on its own.36 = k × 8
k = 36 ÷ 8 = 4.5 - Write the rule with the value of k. Now we have an equation we can use for any pair of values.y = 4.5x
- Substitute y = 54 and solve for x. Replace y by the number, then divide both sides by k to get x on its own.54 = 4.5x
x = 54 ÷ 4.5 = 12
Check. Put it back: 4.5 × 12 = 54. ✓
Watch out. k is found by dividing y by x (y ÷ x), not x by y.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. For direct proportion, write y = kx, use the known pair to find k, then use the equation.
- Write down the rule for direct proportion. If y is directly proportional to x, then y is always the same multiple of x. That multiple is called k, the constant of proportionality.y = kx
- Use the pair of values we know to find k. Substitute x and y into y = kx, then divide to get k on its own.39 = k × 5
k = 39 ÷ 5 = 7.8 - Write the rule with the value of k. Now we have an equation we can use for any pair of values.y = 7.8x
- Substitute x = 8. Replace x by the number and multiply.y = 7.8 × 8 = 62.4
Check. 8 is 1.6 times 5, so y should be 1.6 times 39 = 62.4. ✓
Watch out. k is found by dividing y by x (y ÷ x), not x by y.
Method. The gradient is the rise divided by the run. Use two points on the line.
- Read two points. Choose points where the line crosses grid lines clearly.(0, 0) and (8, 20)
- Work out rise and run. Rise is the change in y and run is the change in x.rise = 20 − 0 = 20, run = 8 − 0 = 8
- Divide. Gradient = rise ÷ run.20 ÷ 8 = 2.5
Check. The point (4, 10): 10 ÷ 4 = 2.5. ✓
Watch out. The gradient of y = kx is k, the cost for 1 hour.
Method. For direct proportion y = kx where k is the gradient.
- Use the gradient from part (a). k = 2.5.k = 2.5
- Write the equation. Replace k by 2.5.y = 2.5x
Check. x = 4: y = 2.5 × 4 = 10, which is on the graph. ✓
Watch out. The equation has no “+ number” because the line goes through the origin.
Method. Substitute x = 10 into the equation.
- Write the equation. From part (b).y = 2.5x
- Substitute x = 10. Replace x by 10 and multiply.y = 2.5 × 10 = 25
Check. 10 hours is 10 ÷ 4 = 2.5 times 4 hours, and 2.5 × £10 = £25. ✓
Watch out. The answer is a cost, so write £.
Method. Substitute y = 35 and solve for x.
- Write the equation with y = 35. The cost is 35.35 = 2.5x
- Divide both sides by 2.5. This leaves x on its own.x = 35 ÷ 2.5 = 14
Check. 2.5 × 14 = 35. ✓
Watch out. To find x from y, divide by k.
Method. Direct proportion means y = kx, with no number added or subtracted. Check each equation.
- y = 3x. This is y = kx with k = 3.Yes
- y = x + 3. There is a +3. When x = 0, y = 3, so the line does not pass through the origin.No
- y = x/2. This is y = kx with k = 1/2.Yes
- y = 5x − 1. There is a −1. When x = 0, y = −1, so it does not pass through the origin.No
Check. Test: for y = 3x, doubling x from 1 to 2 doubles y from 3 to 6. For y = x + 3, doubling x from 1 to 2 changes y from 4 to 5, which is not double. ✓
Watch out. Direct proportion needs y = 0 when x = 0.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Work out y ÷ x for each pair in each table. They must all be equal for direct proportion.
- Table P. Divide y by x for each column.12 ÷ 5 = 2.4
24 ÷ 10 = 2.4
36 ÷ 15 = 2.4 - Table Q. Do the same.12 ÷ 5 = 2.4
22 ÷ 10 = 2.2
32 ÷ 15 = 2.133... - Decide. Table P has the same answer every time. Table Q does not.Table P
Check. In Table P, doubling x from 5 to 10 doubles y from 12 to 24. In Table Q, y goes from 12 to 22, which is not double. ✓
Watch out. In Table Q, y goes up by 10 each time (a constant difference) but that is not enough for proportion.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Use y = kx, with y the dollars and x the pounds.
- Find k. £1 is worth $1.25.k = 1.25
- Write the rule. dollars = k × pounds.y = 1.25x
- Substitute x = 40. Multiply.y = 1.25 × 40 = 50
Check. 50 ÷ 1.25 = 40. ✓
Watch out. Multiply by 1.25 to change pounds to dollars.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Use y = 1.25x again. This time y = 100 and we want x.
- Write the rule. dollars = 1.25 × pounds.y = 1.25x
- Substitute y = 100. Replace y by 100.100 = 1.25x
- Divide both sides by 1.25. This gets x on its own.x = 100 ÷ 1.25 = 80
Check. 1.25 × 80 = 100. ✓
Watch out. Divide by 1.25 to change dollars to pounds.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Work out the fare for 1 km for each journey. If they are equal, the fare is directly proportional to the distance.
- Fare per km for 8 km. Divide the fare by the distance.£5.60 ÷ 8 = £0.70
- Fare per km for 20 km. Do the same.£14 ÷ 20 = £0.70
- Compare. Both are £0.70 per km, so the fare is directly proportional to the distance and k = 0.70.0.70 = 0.70
Check. Check: 20 km is 2.5 times 8 km, and 2.5 × £5.60 = £14. ✓
Watch out. Showing proportion needs the same ratio for both journeys, not just one.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Use the fare per km found in part (a).
- Write the rule. Fare = 0.70 × distance.y = 0.70x
- Substitute x = 35. Multiply.y = 0.70 × 35 = 24.50
Check. 35 km is 35 ÷ 20 = 1.75 times 20 km, and 1.75 × £14 = £24.50. ✓
Watch out. Write money as £24.50, with two decimal places.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. If the cost were directly proportional to the distance, doubling the distance would double the cost. Test this with numbers.
- Find the cost of 2 km. Fixed charge plus the cost per km.£3 + 2 × £2 = £7
- Find the cost of 4 km. Double the distance.£3 + 4 × £2 = £11
- Compare with double. If it were proportional, 4 km would cost 2 × £7 = £14.£11 is not £14
- Give the reason. The cost is not directly proportional because of the £3 fixed charge: when the distance is 0 the cost is already £3, so the graph does not pass through the origin.No.
Check. Cost ÷ distance: 7 ÷ 2 = 3.5 and 11 ÷ 4 = 2.75. These are different, so not proportional. ✓
Watch out. A fixed charge means the graph does not start at (0, 0).
Method. Think about what y = kx looks like on a graph.
- Recall the rule. For direct proportion, y = kx.y = kx
- Find where it starts. When x = 0, y = k × 0 = 0, so the graph passes through the origin (0, 0).Passes through (0, 0)
- Find its shape. The rule is linear (a straight-line equation), so the graph is a straight line. The gradient is k.Straight line
- State the answer. The graph is a straight line through the origin.
Check. Compare with y = 2x + 3. It is straight but passes through (0, 3), so it is not direct proportion. ✓
Watch out. Both conditions are needed: straight and through the origin.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Equal ratios are multiples of each other. Find the multiplier from the pair you know, then apply it to the other number.
- Write the two ratios as equal. Equal ratios (proportion) means one ratio is the other multiplied by the same number.15 : 10 = 24 : ?
- Find the multiplier on the side where both numbers are known. Divide the new number by the old number in the same position. This tells us how much the ratio has been scaled.24 ÷ 15 = 1.6
- Multiply the other number by the same multiplier. Both parts of an equal ratio must be scaled by the same number.10 × 1.6 = 16
Check. Divide the new ratio by 1.6: 24 ÷ 1.6 = 15 and 16 ÷ 1.6 = 10, giving 15 : 10 again. ✓
Watch out. The multiplier is found by division and applied by multiplication to the other part. It need not be a whole number.
Key words. Two quantities are in direct proportion when one is a fixed multiple of the other: if one doubles, so does the other. The unitary method means finding the value of one unit first.
Method. Equal ratios are multiples of each other. Find the multiplier from the pair you know, then apply it to the other number.
- Write the two ratios as equal. Equal ratios (proportion) means one ratio is the other multiplied by the same number.15 : 10 = ? : 25
- Find the multiplier on the side where both numbers are known. Divide the new number by the old number in the same position. This tells us how much the ratio has been scaled.25 ÷ 10 = 2.5
- Multiply the other number by the same multiplier. Both parts of an equal ratio must be scaled by the same number.15 × 2.5 = 37.5
Check. Divide the new ratio by 2.5: 25 ÷ 2.5 = 10 and 37.5 ÷ 2.5 = 15, giving 15 : 10 again. ✓
Watch out. The multiplier is found by division and applied by multiplication to the other part. It need not be a whole number.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. Simplify both ratios. If they are the same, the ratios are equal (the quantities are in proportion).
- Simplify the first ratio. Two ratios are equal when they simplify to the same ratio.15 : 10 = 3 : 2
- Simplify the second ratio. Use the same method: divide by the highest common factor.24 : 20 = 6 : 5
- Compare. Look at the two simplest forms.3 : 2 and 6 : 5 are different, so the ratios are not equal.
Check. Cross-multiply as a check: 15 × 20 = 300 and 10 × 24 = 240. They are not equal. ✓
Watch out. Adding the same number to both parts changes a ratio. Equal ratios come from multiplying or dividing both parts, not adding.