Year 6: Missing values by division
Part of the Year 6 ratio and proportion chapter. Chapter overview · whole chapter on one page
Missing values by division
Sometimes the number you know is the bigger one in the ratio, or the scale is not a whole number of times. Then divide. Divide the known amount by the number it matches. The answer is the value of one part. Multiply the other number in the ratio by that value to find the missing amount.
Division also helps to simplify a ratio. Divide both numbers by the same number, such as the HCF, and the ratio stays the same. For example, 12:20 divided by 4 is 3:5.
Key ideas
- Divide the known amount by the number it matches, to find the scale or the value of one part.
- Multiply the other number of the ratio by the scale to find the missing number.
- A scale that is a fraction (such as 1/4) makes the other number smaller.
- Dividing both numbers by the same number simplifies a 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 3:5, and the first number of the ratio that matches 15. Keep the numbers in the order of the ratio.3:5, and 15 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.15 ÷ 3 = 5
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.5 × 5 = 25
- Write the answer. 25
Check. 3 × 5 = 15, and the ratio 15:25 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 first, then multiply the other number of the ratio by the same scale.
- Write the ratio 4:9, and the second number of the ratio that matches 27. Keep the numbers in the order of the ratio.4:9, and 27 matches 9
- 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.27 ÷ 9 = 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. 9 × 3 = 27, and the ratio 12:27 simplifies to 4:9. ✓
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 12:20, and the first number of the ratio that matches 3. Keep the numbers in the order of the ratio.12:20, and 3 matches 12
- 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.3 ÷ 12 = 1/4
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.20 × 1/4 = 5
- Write the answer. 5
Check. 12 × 1/4 = 3, and the ratio 3:5 simplifies to 12:20. ✓
Watch out. Find the scale first. Do not add the same number to both parts.
Watch out
When the scale is a fraction, the second number gets smaller. A pupil who multiplies 20 by 4 instead of dividing gets 80, which is far too big. Check: is the new ratio the same as the old one when simplified?
Exam tip
Grown-up tip. Ask: “How many equal parts is that?” Then: “What is one part worth?” Those two questions are enough for most of this section.
Exercise 7C Missing values by division
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 second number of the ratio that matches 30. Keep the numbers in the order of the ratio.2:5, and 30 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.30 ÷ 5 = 6
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.2 × 6 = 12
- Write the answer. 12
Check. 5 × 6 = 30, and the ratio 12:30 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 36. Keep the numbers in the order of the ratio.3:4, and 36 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.36 ÷ 4 = 9
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.3 × 9 = 27
- Write the answer. 27
Check. 4 × 9 = 36, and the ratio 27:36 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 7:3, and the second number of the ratio that matches 12. Keep the numbers in the order of the ratio.7:3, 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.7 × 4 = 28
- Write the answer. 28
Check. 3 × 4 = 12, and the ratio 28:12 simplifies to 7:3. ✓
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 5:8, and the second number of the ratio that matches 40. Keep the numbers in the order of the ratio.5:8, and 40 matches 8
- 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 ÷ 8 = 5
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.5 × 5 = 25
- Write the answer. 25
Check. 8 × 5 = 40, and the ratio 25:40 simplifies to 5:8. ✓
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. Find the scale first, then multiply the other number of the ratio by the same scale.
- Write the ratio 4:9, and the second number of the ratio that matches 45. Keep the numbers in the order of the ratio.4:9, and 45 matches 9
- 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.45 ÷ 9 = 5
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.4 × 5 = 20
- Write the answer. 20
Check. 9 × 5 = 45, and the ratio 20:45 simplifies to 4:9. ✓
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. Divide both numbers by their highest common factor (HCF).
- Write the ratio in the order the question gives. The order of the numbers must not change.16:24
- Find the highest common factor (HCF) of the two numbers. It is the biggest number that divides into both.HCF of 16 and 24 is 8
- Divide both numbers by the HCF. This keeps the same ratio but uses smaller numbers.16 ÷ 8 = 2, and 24 ÷ 8 = 3
- Write the answer. 2:3
Check. 2 × 8 = 16 and 3 × 8 = 24. ✓
Watch out. Divide both numbers by the same number. Keep the order the question gives.
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 35. Keep the numbers in the order of the ratio.5:2, and 35 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.35 ÷ 5 = 7
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.2 × 7 = 14
- Write the answer. 14
Check. 5 × 7 = 35, and the ratio 35:14 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 4:3, and the first number of the ratio that matches 24. Keep the numbers in the order of the ratio.4:3, and 24 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.24 ÷ 4 = 6
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.3 × 6 = 18
- Write the answer. 18
Check. 4 × 6 = 24, and the ratio 24:18 simplifies to 4:3. ✓
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 2:5, and the second number of the ratio that matches 40. Keep the numbers in the order of the ratio.2:5, 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.2 × 8 = 16
- Write the answer. 16
Check. 5 × 8 = 40, and the ratio 16:40 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 2:7, and the second number of the ratio that matches 14. Keep the numbers in the order of the ratio.2:7, and 14 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.14 ÷ 7 = 2
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.2 × 2 = 4
- Write the answer. 4
Check. 7 × 2 = 14, and the ratio 4:14 simplifies to 2:7. ✓
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 3:7, and the second number of the ratio that matches 42. Keep the numbers in the order of the ratio.3:7, and 42 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.42 ÷ 7 = 6
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.3 × 6 = 18
- Write the answer. 18
Check. 7 × 6 = 42, and the ratio 18:42 simplifies to 3:7. ✓
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. Simplify both ratios. If the simplest forms are the same, the ratios are equivalent.
- Simplify 12:18. Divide both numbers by their HCF, 6.12:18 = 2:3
- Simplify 6:8. Divide both numbers by their HCF, 2.6:8 = 3:4
- Compare the two simplest forms. The same simplest form means the ratios are equivalent.2:3 and 3:4 are different
- Write the answer. No
Check. 12 × 8 = 96 and 18 × 6 = 108, so the cross-products are not 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 5:6, and the second number of the ratio that matches 48. Keep the numbers in the order of the ratio.5:6, and 48 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.48 ÷ 6 = 8
- Multiply the other number in the ratio by the scale. Both numbers in a ratio are scaled in the same way.5 × 8 = 40
- Write the answer. 40
Check. 6 × 8 = 48, and the ratio 40:48 simplifies to 5:6. ✓
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 second number, then find the first number. Check the other option.
- Find the scale: 27 ÷ 9. The 27 matches the 9 in the ratio 4:9.27 ÷ 9 = 3
- Multiply the first number by 3. The first number is 4 parts.4 × 3 = 12
- Check the other option: could 27 be the first number? Then the second number would be 27 × 9 ÷ 4, which is not whole.27 × 9 ÷ 4 = 60.75
- Write the answer. The other number must be 12.No. The other number would be 12, not 13.
Check. 4:9 is the same as 12:27, because 12 ÷ 4 = 3 and 27 ÷ 9 = 3. ✓
Watch out. Check which number the given value matches before you find the other number.