Year 6: Missing numbers written with letters
Part of the Year 6 algebra chapter. Chapter overview · whole chapter on one page
Missing numbers written with letters
Sometimes a number is missing from a problem. We can use a letter for the missing number, and write the words as an equation. An equation has an equals sign, and the two sides are equal.
Read the words slowly. Each part of the story becomes a part of the equation. Then solve the equation by working backwards. Undo the last step first, using the opposite operation: add undoes subtract, and multiply undoes divide.
The bar model shows three bags with the same number of sweets, b, and 4 loose sweets. Together they make 25 sweets.
Key ideas
- A letter can stand for a missing number.
- Write the words as an equation with an equals sign.
- Undo the steps in reverse order, using the opposite operation.
- Always check your answer by putting it back into the equation.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.n + 9 = 20
- Undo the add 9: take 9 from both sides. Taking 9 away from both sides keeps the equation true.n = 11
- The letter is on its own. There is no number in front to undo.n = 11
- Check: put the answer back into the equation. If both sides match, the answer is right.11 + 9 = 20 ✓
- Write the answer. n = 11
Check. 11 + 9 = 20. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Write the bar model as an equation, then undo the add and the multiply.
- Write the equation from the bar model. 3 bags of b, plus 4 more, make 25.3b + 4 = 25
- Take 4 from both sides. This leaves the bags on their own.3b = 21
- Divide both sides by 3. This finds one bag.b = 21 ÷ 3 = 7
- Check. Put b back in.3 × 7 + 4 = 25 ✓
- Write the answer. b = 7
Check. 3 × 7 + 4 = 25. ✓
Watch out. Each bag holds the same number. Do not divide the total by 4.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.5p - 6 = 19
- Undo the take away 6: add 6 to both sides. Adding the same number to both sides keeps the equation true.5p = 19 + 6 = 25
- Undo the × 5: divide both sides by 5. Dividing both sides by 5 leaves the letter on its own.p = 25 ÷ 5 = 5
- Check: put the answer back into the equation. If both sides match, the answer is right.5 × 5 - 6 = 19 ✓
- Write the answer. p = 5
Check. 5 × 5 - 6 = 19. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Watch out
Do the opposite operation to both sides. In 2n + 5 = 13, take 5 from both sides first (to get 2n = 8), then divide by 2. Do not divide 5 by 2 first.
Exam tip
Estimate first. If the answer is bigger than the total, you have made a mistake. Then check: put the answer back in.
Exercise 8B Writing and solving equations
Non-calculatorKey words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.n + 8 = 17
- Undo the add 8: take 8 from both sides. Taking 8 away from both sides keeps the equation true.n = 9
- The letter is on its own. There is no number in front to undo.n = 9
- Check: put the answer back into the equation. If both sides match, the answer is right.9 + 8 = 17 ✓
- Write the answer. n = 9
Check. 9 + 8 = 17. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.k - 6 = 11
- Undo the take away 6: add 6 to both sides. Adding the same number to both sides keeps the equation true.k = 11 + 6 = 17
- The letter is on its own. There is no number in front to undo.k = 17
- Check: put the answer back into the equation. If both sides match, the answer is right.17 - 6 = 11 ✓
- Write the answer. k = 17
Check. 17 - 6 = 11. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.2x + 3 = 15
- Undo the add 3: take 3 from both sides. Taking 3 away from both sides keeps the equation true.2x = 12
- Undo the × 2: divide both sides by 2. Dividing both sides by 2 leaves the letter on its own.x = 12 ÷ 2 = 6
- Check: put the answer back into the equation. If both sides match, the answer is right.2 × 6 + 3 = 15 ✓
- Write the answer. x = 6
Check. 2 × 6 + 3 = 15. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.4y + 1 = 29
- Undo the add 1: take 1 from both sides. Taking 1 away from both sides keeps the equation true.4y = 28
- Undo the × 4: divide both sides by 4. Dividing both sides by 4 leaves the letter on its own.y = 28 ÷ 4 = 7
- Check: put the answer back into the equation. If both sides match, the answer is right.4 × 7 + 1 = 29 ✓
- Write the answer. y = 7
Check. 4 × 7 + 1 = 29. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.3z - 2 = 10
- Undo the take away 2: add 2 to both sides. Adding the same number to both sides keeps the equation true.3z = 10 + 2 = 12
- Undo the × 3: divide both sides by 3. Dividing both sides by 3 leaves the letter on its own.z = 12 ÷ 3 = 4
- Check: put the answer back into the equation. If both sides match, the answer is right.3 × 4 - 2 = 10 ✓
- Write the answer. z = 4
Check. 3 × 4 - 2 = 10. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.5w - 7 = 18
- Undo the take away 7: add 7 to both sides. Adding the same number to both sides keeps the equation true.5w = 18 + 7 = 25
- Undo the × 5: divide both sides by 5. Dividing both sides by 5 leaves the letter on its own.w = 25 ÷ 5 = 5
- Check: put the answer back into the equation. If both sides match, the answer is right.5 × 5 - 7 = 18 ✓
- Write the answer. w = 5
Check. 5 × 5 - 7 = 18. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Write the bar model as an equation, then undo the add and the multiply.
- Write the equation from the bar model. 3 bags of b, plus 4 more, make 25.3b + 4 = 25
- Take 4 from both sides. This leaves the bags on their own.3b = 21
- Divide both sides by 3. This finds one bag.b = 21 ÷ 3 = 7
- Check. Put b back in.3 × 7 + 4 = 25 ✓
- Write the answer. b = 7
Check. 3 × 7 + 4 = 25. ✓
Watch out. Each bag holds the same number. Do not divide the total by 4.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.6n - 4 = 26
- Undo the take away 4: add 4 to both sides. Adding the same number to both sides keeps the equation true.6n = 26 + 4 = 30
- Undo the × 6: divide both sides by 6. Dividing both sides by 6 leaves the letter on its own.n = 30 ÷ 6 = 5
- Check: put the answer back into the equation. If both sides match, the answer is right.6 × 5 - 4 = 26 ✓
- Write the answer. n = 5
Check. 6 × 5 - 4 = 26. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.2n + 9 = 31
- Undo the add 9: take 9 from both sides. Taking 9 away from both sides keeps the equation true.2n = 22
- Undo the × 2: divide both sides by 2. Dividing both sides by 2 leaves the letter on its own.n = 22 ÷ 2 = 11
- Check: put the answer back into the equation. If both sides match, the answer is right.2 × 11 + 9 = 31 ✓
- Write the answer. n = 11
Check. 2 × 11 + 9 = 31. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Put the number in place of the letter and see whether both sides are the same.
- Write the equation. We will put the number in and see if both sides are equal.3x + 2 = 16
- Put x = 4 into the left side. Work out the left side only.3 × 4 + 2 = 14
- Compare with the right side. If the answer is the same as the right side, it is a solution.14 ≠ 16
- Write the answer. No
Check. Solve it to be sure: 3x + 2 = 16 gives x = 14/3, which is not 4. ✓
Watch out. Both sides must be equal. A number that looks close is not a solution.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Put the number in place of the letter and see whether both sides are the same.
- Write the equation. We will put the number in and see if both sides are equal.2n + 5 = 13
- Put n = 9 into the left side. Work out the left side only.2 × 9 + 5 = 23
- Compare with the right side. If the answer is the same as the right side, it is a solution.23 ≠ 13
- Write the answer. No
Check. Solve it to be sure: 2n + 5 = 13 gives n = 4. ✓
Watch out. Both sides must be equal. A number that looks close is not a solution.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.2n + 8 = 26
- Undo the add 8: take 8 from both sides. Taking 8 away from both sides keeps the equation true.2n = 18
- Undo the × 2: divide both sides by 2. Dividing both sides by 2 leaves the letter on its own.n = 18 ÷ 2 = 9
- Check: put the answer back into the equation. If both sides match, the answer is right.2 × 9 + 8 = 26 ✓
- Write the answer. n = 9
Check. 2 × 9 + 8 = 26. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Undo the add or take away first, then undo the multiplication or division. Check at the end.
- Write the equation. Both sides are equal.5n = 60
- Undo the × 5: divide both sides by 5. Dividing both sides by 5 leaves the letter on its own.n = 60 ÷ 5 = 12
- Check: put the answer back into the equation. If both sides match, the answer is right.5 × 12 = 60 ✓
- Write the answer. n = 12
Check. 5 × 12 = 60. ✓
Watch out. Work backwards, doing the opposite operation. Do the last thing first.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Multiply both sides by 4 to undo the divide.
- Write the equation. The letter is being divided by 4.n ÷ 4 = 7
- Multiply both sides by 4. This undoes the divide by 4.n = 7 × 4 = 28
- Write the answer. 28
Check. 28 ÷ 4 = 7. ✓
Watch out. Multiply, not divide, to undo a division.
Key words. An equation has an equals sign, and both sides are equal. To solve an equation means to find the value of the letter that makes it true.
Method. Work backwards: undo the doubling, then the add 7, then the multiply by 3.
- Undo the doubling: divide 50 by 2. The last step was multiply by 2.50 ÷ 2 = 25
- Undo the add 7: take 7 from 25. The step before was add 7.25 - 7 = 18
- Undo the multiply by 3: divide 18 by 3. The first step was multiply by 3.18 ÷ 3 = 6
- Write the answer. 6
Check. 6 × 3 = 18, then 18 + 7 = 25, then 25 × 2 = 50. ✓
Watch out. Write the equation as 2(3n + 7) = 50 first. Then undo one step at a time.