Letters for numbers
Each letter stands for a number, and you work out a sum written in letters. Here’s how to get the right letter every time.
Rush doesn’t practise letters for numbers yet. These games practise related skills: Order & Algebra, Multiply & Divide and Add & Subtract.
Chapter 1
What are letters for numbers questions?
You’re given a key that gives each letter a number, like A = 2 and B = 3. Then you’re given a sum written in letters, like B + C × A.
You swap the letters for their numbers, work out the sum, then find the answer’s letter in the same key.
The key. Each letter has its own number. A is 1 only if the key says so.
The sum. Written in letters. Once the numbers are in, it’s ordinary maths.
The answers. Five letters, A to E. Your answer is a letter, never a number.
It sits in the verbal reasoning paper, but what it really tests is doing a sum in the right order.
If A = 2, B = 3, C = 7, D = 17 and E = 20, what is B + C × A? Give your answer as a letter.
- A
- B
- C
- D
- E
Chapter 2
Solve it in 3 steps.
Here’s the question, one step at a time.
The question
If A = 2, B = 3, C = 7, D = 17 and E = 20, what is B + C × A? Give your answer as a letter.
- A
- B
- C
- D
- E
-
Step 1
Swap every letter for its number.
Write the sum out again in numbers before you work anything out. B is 3, C is 7 and A is 2, so B + C × A becomes 3 + 7 × 2. Holding letters and numbers in your head at once is how slips happen.
The signs stay where they are.
-
Step 2
Times and divide come first.
Do × and ÷ before + and −, wherever they sit in the sum. Here that’s 7 × 2 = 14 first, then 3 + 14 = 17. If the order is new to you, the algebra page teaches it.
- 3 + 7 × 2
- = 3 + 14
- = 17
The × goes first, though it comes second.
-
Step 3
Turn it back into a letter.
Find 17 among the key’s numbers. D = 17, so the answer is D. If your number isn’t in the key, something has slipped: check the order and each sum again.
17 is D in this key.
Three mistakes this question sets
| The mistake | Do this instead |
|---|---|
| Working left to right. 3 + 7 = 10, then 10 × 2 = 20, and E is 20. Landing on a letter makes it look right. | Do 7 × 2 first, then add the 3. Being in the key doesn’t prove a number is right. |
| Picking a number from the sum. B, C and A are 3, 7 and 2, the numbers you started with. They’re in the options to catch a guess. | Only the number at the very end gets looked up. |
| Answering with the number. 17 is the answer to the sum, but the question wants a letter. Q, the 17th letter of the alphabet, isn’t it either. | Look 17 up in the key. It’s D. |
So it’s D. 3 + 7 × 2 is 3 + 14 = 17, and the key says D = 17.
Remember: Numbers in, order right, letter out.
Chapter 3
The catches.
The harder questions use these. Each card is one catch: the rule, an example and the trap.
-
12 − 4 + 2 = 10
8 ÷ 2 × 4 = 16
Same kind of sign? Left to right
× and ÷ are partners, and so are + and −. When partners meet, work along from the left.
12 − 4 + 2: 12 − 4 = 8, then 8 + 2 = 10. Trap: 6, from adding 4 + 2 first. In the same way, 8 ÷ 2 × 4 is 16, not 1.
-
- (9 − 3) × 2
- = 6 × 2
- = 12
Brackets go first
Work out anything in brackets before anything else, even × and ÷.
(9 − 3) × 2: 9 − 3 = 6, then 6 × 2 = 12. Trap: 3, from ignoring the brackets and doing 3 × 2 first.
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12 is only halfway. The answer is 7.
A halfway number can be in the key
Look a number up only when the whole sum is done.
With this key, A × B − C is 3 × 4 − 5. Halfway you have 12, and E is 12. Keep going: 12 − 5 = 7, which is D. Trap: stopping at E.
-
D is the 4th letter, but here D is 9.
Use the key, not the alphabet
A letter’s number comes from the key, and every question has its own.
With this key, C + D − E is 7 + 9 − 12 = 4, and A is 4. Trap: D, because D is the 4th letter of the alphabet. Here D is 9.
Learn the order by heart
Brackets, then squares, then × and ÷, then + and −. Its name is BIDMAS. When two signs are partners, go left to right. The algebra page has more on it.
- ( )1st
- A²2nd
- × ÷3rd
- + −4th
Chapter 4: In the exam
More ways the exam asks it.
Papers dress the same sum up in different ways. The hardest is worked through first, then the rest.
-
Step 1
Work out the full side.
A² means A × A, so it’s 3 × 3 = 9, not 3 × 2. Then 9 + 5 = 14.
- A² + B
- = 3 × 3 + 5
- = 14
Read A² as A × 2 and you get 11, which points to D: the trap.
-
Step 2
Make the other side match.
E is 20, so 20 take away the missing number must make 14. 20 − 14 = 6, and C is 6.
20 − 6 = 14, so it’s C.
Other ways the exam asks it
Some papers print one key for several sums. These four all use this one.
-
- BC − A
- = 3 × 4 − 2
- = 10
Letters side by side mean times
BC means B × C, and 3B means 3 × B.
BC − A is 3 × 4 − 2 = 10, which is E. Trap: D, from reading BC as B + C. In the same way, 3B is 3 × 3 = 9, not 33.
-
- = 12 ÷ 3
- = 4
A line means divide
Work out the top and the bottom, then divide.
The top is 2 + 10 = 12, and 12 ÷ 3 = 4, which is C. Trap: dividing only the E. 10 ÷ 3 isn’t whole, a sure sign of a slip: in the papers every divide comes out whole.
-
- 4 + 2 × 3 − 10 ÷ 2
- = 4 + 6 − 5
- = 5
Long sums: find the groups
Each × or ÷ joins its neighbours into a group. Work out the groups, then add and take away from the left.
C + A × B − E ÷ A is 4 + 6 − 5 = 5, which is D. A letter can come up twice. Trap: C, from working straight along.
-
C × B − E = 2
Answer: A
No options? Write the letter
Some papers give no options. Check whether they want a letter or a number.
C × B − E is 12 − 10 = 2. Asked for a letter, write A. Asked for a number, write 2. Trap: B, the 2nd letter of the alphabet.
Chapter 5
Try it yourself.
Easier ones first. Try each one before you open its answer.
-
Question 1Level 1
If A = 3, B = 5, C = 8, D = 10 and E = 13, what is E − C + B? Give your answer as a letter.
- A
- B
- C
- D
- E
Show answer
Answer: D.
How to answer it
- 13 − 8 + 5. Left to right: 13 − 8 = 5, then 5 + 5 = 10.
- D = 10. The answer is D.
Why the other answers are wrong
- BB is 5, which is only 13 − 8. B still has to be added.
- EE is 13, the number the sum starts from.
- CC is 8, the number taken away.
- AA is 3, and no step of this sum gives 3.
-
Question 2Level 1
If A = 2, B = 5, C = 6, D = 10 and E = 12, what is A × C − D? Give your answer as a letter.
- A
- B
- C
- D
- E
Show answer
Answer: A.
How to answer it
- 2 × 6 − 10. Multiply first: 2 × 6 = 12. Then 12 − 10 = 2.
- A = 2. The answer is A.
Why the other answers are wrong
- EE is 12, the number after A × C. D still has to be taken away.
- BB is the second letter of the alphabet, but in this key B stands for 5.
- DD is 10, the number taken away.
- CC is 6, one of the numbers multiplied.
-
Question 3Level 1
If A = 6, B = 7, C = 9, D = 13 and E = 16, what is E − B + A − C? Give your answer as a letter.
- A
- B
- C
- D
- E
Show answer
Answer: A.
How to answer it
- 16 − 7 + 6 − 9. Left to right: 16 − 7 = 9, 9 + 6 = 15, then 15 − 9 = 6.
- A = 6. The answer is A.
Why the other answers are wrong
- CC is 9, the number after the first step, 16 − 7. There are two more steps.
- BB is 7, the first number taken away.
- DD is 13, and no step of the sum gives 13.
- EE is 16, the number the sum starts from.
-
Question 4Level 2
If A = 3, B = 5, C = 6, D = 9 and E = 15, what is C + D ÷ A? Give your answer as a letter.
- A
- B
- C
- D
- E
Show answer
Answer: D.
How to answer it
- 6 + 9 ÷ 3. Divide first: 9 ÷ 3 = 3. Then 6 + 3 = 9.
- D = 9. The answer is D.
Why the other answers are wrong
- BB is 5, what you get working left to right: 6 + 9 = 15, then 15 ÷ 3 = 5.
- AA is 3, the answer to 9 ÷ 3 before C is added.
- EE is 15, which is 6 + 9 with the dividing left out.
- CC is 6, the number added, not the answer.
-
Question 5Level 2
If A = 2, B = 3, C = 11, D = 12 and E = 18, what is E − D ÷ A? Give your answer as a letter.
- A
- B
- C
- D
- E
Show answer
Answer: D.
How to answer it
- 18 − 12 ÷ 2. Divide first: 12 ÷ 2 = 6. Then 18 − 6 = 12.
- D = 12. The answer is D.
Why the other answers are wrong
- BB is 3, what you get working left to right: 18 − 12 = 6, then 6 ÷ 2 = 3.
- EE is 18, the number the sum starts from.
- CC is 11, one less than the answer.
- AA is 2, the number D is divided by.
-
Question 6Level 2
If A = 2, B = 3, C = 6, D = 8 and E = 18, what is (D − A) × B? Give your answer as a letter.
- A
- B
- C
- D
- E
Show answer
Answer: E.
How to answer it
- (8 − 2) × 3. Brackets first: 8 − 2 = 6. Then 6 × 3 = 18.
- E = 18. The answer is E.
Why the other answers are wrong
- CC is 6, the answer to the brackets alone. It still has to be multiplied by B.
- AA comes from ignoring the brackets: 2 × 3 = 6, then 8 − 6 = 2.
- DD is 8, the number inside the brackets before A is taken away.
- BB is 3, the number the brackets are multiplied by.
-
Question 7Level 3
If A = 2, B = 3, C = 6, D = 12 and E = 15, what is D ÷ (C − A) + B? Give your answer as a letter.
- A
- B
- C
- D
- E
Show answer
Answer: C.
How to answer it
- 12 ÷ (6 − 2) + 3. Brackets first: 6 − 2 = 4.
- Then divide: 12 ÷ 4 = 3. Then add: 3 + 3 = 6.
- C = 6. The answer is C.
Why the other answers are wrong
- BB is 3, the answer to 12 ÷ 4. B still has to be added.
- EE is 15, which is 12 + 3 with the dividing left out.
- DD is 12, the number the sum starts from.
- AA is 2, the number taken away inside the brackets.
-
Question 8Level 3
If A = 2, B = 4, C = 5, D = 6 and E = 10, which letter makes this true?
B² − D = A × ?
- A
- B
- C
- D
- E
Show answer
Answer: C.
How to answer it
- B² is 4 × 4 = 16. Then 16 − 6 = 10.
- The other side has to make 10: 2 times something is 10. 2 × 5 = 10.
- C = 5. The answer is C.
Why the other answers are wrong
- EE is 10, the number both sides make. The question asks for the letter that goes with A to make it.
- DD is 6, and 2 × 6 = 12, not 10.
- BB is 4, and 2 × 4 = 8.
- AA is 2, and 2 × 2 = 4.
For parents.
What parents ask us about letters for numbers.
Questions parents ask
What does letters for numbers mean in verbal reasoning?
Each letter in the question stands for a number given in a key. The child swaps the letters for numbers, works out the sum, then finds which letter the answer belongs to. So it's a sum, with a look-up in the key at the start and at the end.
Is A always 1?
No. Every question has its own key, and A can be any number. Only use alphabet places if the question tells you to. Writing the key down in a row helps keep the two apart.
What maths does my child need?
Times tables, adding and taking away, square numbers, and the order of operations. All of it is on the primary curriculum: square numbers come in Year 5 and the order of operations in Year 6. The order matters most, because a wrong order can give a number that is also in the key.
Some practice books work these sums left to right. Which is right?
The order of operations: × and ÷ before + and −. When every × and ÷ comes before any + or −, both ways give the same answer. When one comes later, as in C + D ÷ A, only the order of operations is correct maths, and it’s what your child is taught in school.
Practise the related skills in Rush.
- Order & AlgebraIt drills brackets first, then × and ÷, then + and −, the order every sum on this page depends on, and some of its questions ask for a missing letter in a formula.
- Multiply & DivideMost of these sums have a times or divide step, and a child who is quick with tables has more thought to spare for the order and the key.
- Add & SubtractThe adding and taking away steps need to be quick and right first time, which is what Add & Subtract trains.
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