Year 6 Arithmetic (SATs): Missing numbers and two unknowns
Part of the Year 6 Arithmetic (SATs) algebra and formulae chapter. Chapter overview · whole chapter on one page
Missing numbers and two unknowns
Rule
A letter stands for a missing number. In a + b = 10, if a = 3, then b must be 7.
When there are two unknowns, there can be several pairs of numbers that work. Test each pair, or solve the two equations together by adding or subtracting them.
Key ideas
- A letter stands for one missing number.
- If the sum is fixed, one number decides the other: b = 10 - a.
- Two equations can be solved together by adding or subtracting them.
Method. Replace a with 3, then find the missing number.
- Substitute. 3 + b = 10.3 + b = 10
- Find the missing number. b = 10 - 3.b = 7
- Check. Adding 3 and 7 gives 10.3 + 7 = 10
- Write the answer. 7
Check. 3 + 7 = 10. ✓
Method. Add the two equations together to remove y.
- Add the equations. (x + y) + (x - y) = 12 + 4, so 2x = 16.2x = 16
- Find x. x = 16 ÷ 2 = 8.x = 8
- Find y. Use x + y = 12: 8 + y = 12, so y = 4.y = 4
- Write the answer. x = 8 and y = 4
Check. 8 + 4 = 12 and 8 - 4 = 4. ✓
Method. Try each value of a, then check the other number.
- Try a = 1. b = 8, and 1 < 8.(1, 8)
- Try a = 2, 3 and 4. b = 7, 6 and 5, all bigger than a.(2, 7), (3, 6), (4, 5)
- List the pairs. Stop at a = 4, since a must be smaller than b.(1, 8), (2, 7), (3, 6), (4, 5)
- Write the answer. (1, 8), (2, 7), (3, 6), (4, 5)
Check. Each pair adds to 9 with the smaller number first. ✓
Exercise 6D Practice: missing numbers and two unknowns
Non-calculatorMethod. Take a from the total.
- Take away a. 15 - 6 = 9.9
- Write the answer. 9
Check. 6 + 9 = 15. ✓
Method. Take a from the total.
- Take away a. 12 - 7 = 5.5
- Write the answer. 5
Check. 7 + 5 = 12. ✓
Method. Take a from the total.
- Take away a. 20 - 4 = 16.16
- Write the answer. 16
Check. 4 + 16 = 20. ✓
Method. Take a from the total.
- Take away a. 16 - 9 = 7.7
- Write the answer. 7
Check. 9 + 7 = 16. ✓
Method. Take 7 from 19.
- Subtract. 19 - 7 = 12.12
- Write the answer. 12
Check. 12 + 7 = 19. ✓
Method. Divide 21 by 3.
- Divide. 21 ÷ 3 = 7.7
- Write the answer. 7
Check. 3 × 7 = 21. ✓
Method. Add the equations: 2x = 26.
- Add the equations. 2x = 26.2x = 26
- Divide by 2. x = 13.13
- Write the answer. 13
Check. 13 + 7 = 20 and 13 - 7 = 6. ✓
Method. Add the two equations.
- Add. 2a = 16, so a = 8.a = 8
- Find b. 8 + b = 14, so b = 6.b = 6
- Write the answer. a = 8 and b = 6
Check. 8 - 6 = 2. ✓
Method. Try pairs of factors of 96 that add to 20.
- Try pairs. 4 × 24 = 96, but 4 + 24 is not 20.4 + 24 = 28
- Try the right pair. 8 × 12 = 96 and 8 + 12 = 20.8 + 12 = 20
- Write the answer. 8 and 12
Check. 8 × 12 = 96. ✓
Method. List the factor pairs of 24.
- List the pairs. (1, 24), (2, 12), (3, 8), (4, 6).(1, 24), (2, 12), (3, 8), (4, 6)
- Count them. There are 4 pairs.4
- Write the answer. 4
Check. Each pair multiplies to 24. ✓
Method. Subtract the first equation from the second.
- Subtract. (2x + y) - (x + y) = 14 - 10, so x = 4.x = 4
- Find y. 4 + y = 10, so y = 6.y = 6
- Write the answer. x = 4 and y = 6
Check. 2 × 4 + 6 = 14. ✓
Method. Try each value of x, then find y.
- Try x = 1, 2, 3 and 4. x = 2 gives 6 + 2y = 12, so y = 3; other values do not give a whole number from 1 to 5.(2, 3)
- Write the answer. (2, 3)
Check. 3 × 2 + 2 × 3 = 12. ✓
Method. Try two different values of a.
- Try a = 3. 3 + 7 = 10.(3, 7)
- Try a = 4. 4 + 6 = 10.(4, 6)
- Write the answer. Because a can be any number from 0 to 10, and b is then 10 minus a, so there are many pairs such as (3, 7) and (4, 6).
Check. Both pairs add to 10. ✓