Year 6: Equivalent ratios and missing values
Part of the Year 6 ratio and proportion chapter. Chapter overview · whole chapter on one page
Equivalent ratios and missing values
Two ratios are equivalent when they describe the same mix. Look at 2:3. If you multiply both numbers by 2, you get 4:6. If you multiply both by 5, you get 10:15. All of these ratios are equivalent, because the blue part is always 1 and a half times the red part.
A ratio table shows several equivalent ratios in columns. Every column keeps the same rule. To find a missing number, find the scale first: divide the given number by the number it matches. Then multiply the other number in the ratio by the same scale.
| Red | 2 | 4 | 6 | 8 |
|---|---|---|---|---|
| Blue | 3 | 6 | 9 | 12 |
Each column is the ratio 2:3 multiplied by 1, 2, 3 and 4.
Key ideas
- To find an equivalent ratio, multiply both numbers by the same whole number.
- To find a missing number, divide the given number by the number it matches. That gives the scale.
- Then multiply the other number in the ratio by the scale.
- Adding the same number to both parts does not make an equivalent ratio.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Find the scale first, then multiply the other number of the ratio by the same scale.
- Write the ratio 2:3, and the first number of the ratio that matches 8. Keep the numbers in the order of the ratio.2:3, and 8 matches 2
- Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.8 ÷ 2 = 4
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.3 × 4 = 12
- Write the answer. 12
Check. 2 × 4 = 8, and the ratio 8:12 simplifies to 2:3. ✓
Watch out. Find the scale first. Do not add the same number to both parts.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Find the scale first, then multiply the other number of the ratio by the same scale.
- Write the ratio 3:4, and the first number of the ratio that matches 9. Keep the numbers in the order of the ratio.3:4, and 9 matches 3
- Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.9 ÷ 3 = 3
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.4 × 3 = 12
- Write the answer. 12
Check. 3 × 3 = 9, and the ratio 9:12 simplifies to 3:4. ✓
Watch out. Find the scale first. Do not add the same number to both parts.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Multiply both numbers by the same number, three times.
- Write the starting ratio. This is the ratio we start with.2:7
- Multiply both numbers by 2. This gives the first equivalent ratio.2 × 2 = 4 and 7 × 2 = 14, so 4:14
- Multiply both numbers by 3. This gives the next equivalent ratio.2 × 3 = 6 and 7 × 3 = 21, so 6:21
- Multiply both numbers by 4. This gives the third equivalent ratio.2 × 4 = 8 and 7 × 4 = 28, so 8:28
- Write the answer. 4:14, 6:21, 8:28
Check. Each ratio divided by its scale gives 2:7 again: 4 ÷ 2 = 2 and 14 ÷ 2 = 7. ✓
Watch out. Multiply both numbers by the same number. Adding 2 to both gives 4:9, which is not equivalent.
Watch out
Adding is not scaling. 2:3 plus 2 is 4:5, which is not equivalent to 2:3. Multiply both numbers by the same number instead.
Exam tip
Grown-up tip. Ask: “How many times bigger is the first number now?” The answer is the scale. Then ask the child to multiply the other number by it.
Exercise 7B Equivalent ratios and missing values
Non-calculatorKey words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Find the scale first, then multiply the other number of the ratio by the same scale.
- Write the ratio 2:5, and the first number of the ratio that matches 4. Keep the numbers in the order of the ratio.2:5, and 4 matches 2
- Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.4 ÷ 2 = 2
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.5 × 2 = 10
- Write the answer. 10
Check. 2 × 2 = 4, and the ratio 4:10 simplifies to 2:5. ✓
Watch out. Find the scale first. Do not add the same number to both parts.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Find the scale first, then multiply the other number of the ratio by the same scale.
- Write the ratio 3:4, and the second number of the ratio that matches 20. Keep the numbers in the order of the ratio.3:4, and 20 matches 4
- Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.20 ÷ 4 = 5
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.3 × 5 = 15
- Write the answer. 15
Check. 4 × 5 = 20, and the ratio 15:20 simplifies to 3:4. ✓
Watch out. Find the scale first. Do not add the same number to both parts.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Find the scale first, then multiply the other number of the ratio by the same scale.
- Write the ratio 6:7, and the first number of the ratio that matches 12. Keep the numbers in the order of the ratio.6:7, and 12 matches 6
- Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.12 ÷ 6 = 2
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.7 × 2 = 14
- Write the answer. 14
Check. 6 × 2 = 12, and the ratio 12:14 simplifies to 6:7. ✓
Watch out. Find the scale first. Do not add the same number to both parts.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Find the scale first, then multiply the other number of the ratio by the same scale.
- Write the ratio 5:2, and the first number of the ratio that matches 25. Keep the numbers in the order of the ratio.5:2, and 25 matches 5
- Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.25 ÷ 5 = 5
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.2 × 5 = 10
- Write the answer. 10
Check. 5 × 5 = 25, and the ratio 25:10 simplifies to 5:2. ✓
Watch out. Find the scale first. Do not add the same number to both parts.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Find the scale first, then multiply the other number of the ratio by the same scale.
- Write the ratio 3:8, and the first number of the ratio that matches 12. Keep the numbers in the order of the ratio.3:8, and 12 matches 3
- Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.12 ÷ 3 = 4
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.8 × 4 = 32
- Write the answer as a ratio, in the same order as the question. The ratio must keep its order.12:32
Check. 3 × 4 = 12, and the ratio 12:32 simplifies to 3:8. ✓
Watch out. Find the scale first. Do not add the same number to both parts.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Simplify both ratios. If the simplest forms are the same, the ratios are equivalent.
- Simplify 4:10. Divide both numbers by their HCF, 2.4:10 = 2:5
- Simplify 6:15. Divide both numbers by their HCF, 3.6:15 = 2:5
- Compare the two simplest forms. The same simplest form means the ratios are equivalent.2:5 and 2:5 are the same
- Write the answer. Yes
Check. 4 × 15 = 60 and 10 × 6 = 60, so the cross-products are equal. ✓
Watch out. Compare the simplest forms, not the first numbers.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Find the scale first, then multiply the other number of the ratio by the same scale.
- Write the ratio 3:2, and the first number of the ratio that matches 12. Keep the numbers in the order of the ratio.3:2, and 12 matches 3
- Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.12 ÷ 3 = 4
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.2 × 4 = 8
- Write the answer. 8
Check. 3 × 4 = 12, and the ratio 12:8 simplifies to 3:2. ✓
Watch out. Find the scale first. Do not add the same number to both parts.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Find the scale first, then multiply the other number of the ratio by the same scale.
- Write the ratio 4:7, and the second number of the ratio that matches 28. Keep the numbers in the order of the ratio.4:7, and 28 matches 7
- Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.28 ÷ 7 = 4
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.4 × 4 = 16
- Write the answer. 16
Check. 7 × 4 = 28, and the ratio 16:28 simplifies to 4:7. ✓
Watch out. Find the scale first. Do not add the same number to both parts.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Find the scale first, then multiply the other number of the ratio by the same scale.
- Write the ratio 5:2, and the second number of the ratio that matches 6. Keep the numbers in the order of the ratio.5:2, and 6 matches 2
- Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.6 ÷ 2 = 3
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.5 × 3 = 15
- Write the answer. 15
Check. 2 × 3 = 6, and the ratio 15:6 simplifies to 5:2. ✓
Watch out. Find the scale first. Do not add the same number to both parts.
| A | B | C | D | |
|---|---|---|---|---|
| Pens | 3 | ? | 9 | 12 |
| Pencils | 4 | 8 | ? | 16 |
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Find the scale first, then multiply the other number of the ratio by the same scale.
- Write the ratio 3:4, and the second number of the ratio that matches 8. Keep the numbers in the order of the ratio.3:4, and 8 matches 4
- Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.8 ÷ 4 = 2
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.3 × 2 = 6
- Write the answer. 6
Check. 4 × 2 = 8, and the ratio 6:8 simplifies to 3:4. ✓
Watch out. Find the scale first. Do not add the same number to both parts.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Find the scale first, then multiply the other number of the ratio by the same scale.
- Write the ratio 3:4, and the first number of the ratio that matches 9. Keep the numbers in the order of the ratio.3:4, and 9 matches 3
- Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.9 ÷ 3 = 3
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.4 × 3 = 12
- Write the answer. 12
Check. 3 × 3 = 9, and the ratio 9:12 simplifies to 3:4. ✓
Watch out. Find the scale first. Do not add the same number to both parts.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Find the scale first, then multiply the other number of the ratio by the same scale.
- Write the ratio 3:4, and the second number of the ratio that matches 20. Keep the numbers in the order of the ratio.3:4, and 20 matches 4
- Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.20 ÷ 4 = 5
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.3 × 5 = 15
- Write the answer as a ratio, in the same order as the question. The ratio must keep its order.15:20
Check. 4 × 5 = 20, and the ratio 15:20 simplifies to 3:4. ✓
Watch out. Find the scale first. Do not add the same number to both parts.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Check each statement by multiplying 3 and 5 by the same number.
- Check 9:15. Multiply 3 and 5 by 3.3 × 3 = 9 and 5 × 3 = 15, so 9:15 is equivalent
- Check 9:14. Find the number that 5 is multiplied by to give 14. 14 is not a multiple of 5.5 × ? = 14 has no whole-number answer
- Write the answer. No. Only 9:15 is equivalent to 3:5.
Check. 3 × 3 = 9 and 5 × 3 = 15, so 9:15 matches. ✓
Watch out. Both numbers must be multiplied by the same number. 14 is not 5 times anything.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Multiply 6 and 8 by 1, 2, 3, and so on, until a number reaches 30.
- Try multiplying by 1. Both numbers are below 30.6:8
- Try multiplying by 2. Both numbers are below 30.12:16
- Try multiplying by 3. Both numbers are below 30.18:24
- Try multiplying by 4. The second number, 32, is 30 or more, so this one is not allowed.6 × 4 = 24 and 8 × 4 = 32, and 32 is not below 30
- Write the answer. There are three ratios.3
Check. The three ratios are 6:8, 12:16 and 18:24. ✓
Watch out. Check both numbers. The second number 8 reaches 32 when multiplied by 4, which is too big.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Find the scale first, then multiply the other number of the ratio by the same scale.
- Write the ratio 3:5, and the second number of the ratio that matches 40. Keep the numbers in the order of the ratio.3:5, and 40 matches 5
- Find the scale: divide the given amount by the number it matches. The scale tells us how many times bigger (or smaller) this ratio is.40 ÷ 5 = 8
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.3 × 8 = 24
- Write the answer. 24
Check. 5 × 8 = 40, and the ratio 24:40 simplifies to 3:5. ✓
Watch out. Find the scale first. Do not add the same number to both parts.
Key words. A ratio compares two amounts of the same kind. We write it with a colon: 2:3 is read ‘2 to 3’. The order matters, because the first number is the first amount named.
Method. Find the scale from the red part, then multiply the green part by it.
- Find the scale for red: 15 ÷ 3. The number of red sweets matches the 3 in the ratio.15 ÷ 3 = 5
- Multiply the green part by 5. Green is 4 parts, so multiply it by the same scale.4 × 5 = 20
- Write the answer. Yes. 20 green sweets.
Check. The ratio 15:20 divides by 5 to give 3:4. ✓
Watch out. The ratio must hold for every part, so the green sweets must be 20.