# Substitute Numbers for Letters in Complex Calculations

In this worksheet, students will replace letters with numbers and carry out questions using the four number operations, with some calculations in sets of brackets. They must give their final answers as numbers not letters.

Key stage:  KS 2

Curriculum topic:   Verbal Reasoning

Curriculum subtopic:   Maths Codes

Difficulty level:

### QUESTION 1 of 10

Make sure that you’ve got your thinking cap on as we are going to be number decoders!

In this activity, we are going to swap numbers for letters to solve maths problems.

Let’s take a look at this question first:

If a = 4, b = 12, c = 3 and d = 9, what is b ÷ a?

This is called algebra, which is where numbers are replaced with letters. We need to swap the letters back into numbers to solve this problem.

If we put the numbers back in, then b ÷ a becomes 12 ÷ 4. We know this is 3.

If any part of the question is written in brackets, this part has to be calculated first.

Let’s try this question next:

If a = 7, b = 9, c = 6 and d = 4, what is b + (a x d)?

Let’s swap the letters back to numbers. This would be written as: 9 + (7 x 4) = ___.

We must work out 7 x 4 first as it is in brackets. This is 28.

Our number sentence now looks like this: 9 + (28) = ___. The answer is 37.

Let’s try one more together:

If a = 8, b = 10, c = 4 and d = 12, what is b (a + c)?

Where there isn’t a -, +, x or ÷ between the first letter and the bracket, it means we have to multiply the outside number by what is inside the brackets. We must imagine an invisible times sign here.

This can be written as: b x (a + c) = ___.

In numbers this is: 10 x (8 + 4) = ____.

We must solve the brackets first: 8 + 4 = 12.

Our number sentence now becomes: 10 x (12) = 120. So our answer is 120.

Now it’s your turn to crack the codes. Good luck number detective!

If A = 4, B = 12, C = 3 and D = 9, what is D ÷ C?

Can you calculate the answer as a number?

3

4

8

If a = 4, b = 27, c = 3 and d = 9, what is b ÷ c + a?

Can you calculate the answer as a number?

12

14

13

If A = 4, B = 27, C = 3 and D = 9, what is B ÷ C x A?

Can you calculate the answer as a number?

You have been given the following information: A = 4, B = 24, C = 3, D = 12.

Can you match each equation to its correct answer (as a number)?

## Column B

B ÷ D
2
C x A + B
36
A + B + C
31

You have been given the following information: a = 3, b = 30, c = 6, d = 12.

Can you match each equation to its correct answer (as a number)?

## Column B

b ÷ c
6
d x a - b
5
b - c - d
12

Now you are going to be the examiner!

A student has been given the questions below using this information: A = 3, B = 30, C = 6, D = 12.

Identify whether they have answered each question correctly or not.

Following on from the last question, you need to use this information again: A = 3, B = 30, C = 6, D = 12

The student in the previous question answered that B - A = 33.

Which of the mistakes below could they have made to reach their answer?

Used the wrong numbers to complete the sum

Subtracted the numbers the wrong way around

Another student has been given the questions below using this information: a = 7, b = 35, c = 5, d = 16.

Identify whether they have answered each question correctly or not.

Following on from the last question, you need to use this information again: a = 7, b = 35, c = 5, d = 16

The student in the previous question answered that d x c = 21.

Which of the mistakes below could they have made to reach their answer?

Used the wrong numbers to complete the sum

Multiplied the numbers the wrong way around

Here is your final question as the examiner!

Another student has been given the questions below using this information: A = 4, B = 32, C = 7, D=17.

Identify whether they have answered each question correctly or not.

• Question 1

If A = 4, B = 12, C = 3 and D = 9, what is D ÷ C?

Can you calculate the answer as a number?

3
EDDIE SAYS
The question tells us that D = 9 and C = 3.
If we swap these numbers in for the letters, then D ÷ C becomes 9 ÷ 3.
When we divide 9 by 3, we reach the answer of 3.
If you were asked to write the answer as a letter, which letter would it be?
• Question 2

If a = 4, b = 27, c = 3 and d = 9, what is b ÷ c + a?

Can you calculate the answer as a number?

13
EDDIE SAYS
The question tells us that b = 27, c = 3 and a = 4.
If we swap these numbers in for the letters, then b ÷ c + a becomes 27 ÷ 3 + 4.
When we divide 27 by 3 then add 4, we reach the answer of 13.
• Question 3

If A = 4, B = 27, C = 3 and D = 9, what is B ÷ C x A?

Can you calculate the answer as a number?

36
EDDIE SAYS
The question tells us that B = 27, C = 3 and A = 4, which is the same as the previous question.
If we swap these numbers in for the letters, then B ÷ C x A becomes 27 ÷ 3 x 4.
Did you spot the small change here from the previous question? The addition sign changed to a multiplication sign. Often small changes exist in exam questions to try and check whether you are looking at all the detail.
When we divide 27 by 3 then multiply our answer by 4, we reach the total of 36.
• Question 4

You have been given the following information: A = 4, B = 24, C = 3, D = 12.

Can you match each equation to its correct answer (as a number)?

## Column B

B ÷ D
2
C x A + B
36
A + B + C
31
EDDIE SAYS
The question tells us that B = 24, D = 12, C = 3 and A = 4.
If we swap these numbers in for the letters in each of the equations, we get the following calculations: B ÷ D becomes 24 ÷ 12 = 2 C x A + B becomes 3 x 4 + 24 = 36 A + B + C becomes 4 + 24 + 3 = 31
How many of the equations did you calculate correctly? Well done if you got all three correct! If not then don't worry, as we are going to practice some more now.
• Question 5

You have been given the following information: a = 3, b = 30, c = 6, d = 12.

Can you match each equation to its correct answer (as a number)?

## Column B

b ÷ c
5
d x a - b
6
b - c - d
12
EDDIE SAYS
The question tells us that b = 30, c = 6, d = 12 and a = 3.
If we swap these numbers in for the letters in each of the equations, we get the following calculations: b ÷ c becomes 30 ÷ 6 = 5 d x a - b becomes 12 x 3 - 30 = 6 b - c - d becomes 30 - 6 - 12 = 12
Were you able to get all three answers matched?
• Question 6

Now you are going to be the examiner!

A student has been given the questions below using this information: A = 3, B = 30, C = 6, D = 12.

Identify whether they have answered each question correctly or not.

EDDIE SAYS
The question tells us that B = 30, A = 3, C = 6 and D = 12.
If we swap these numbers in for the letters in each of the equations, we get the following calculations: B - A becomes 30 - 3 = 27. The student answered 33 so this is incorrect. D x A becomes 12 x 3 = 36, which the student got right! B ÷ A + C becomes 30 ÷ 3 + 6 = 16, which the student also got right!
How did you find being the examiner? Was it easy or hard to spot the student's errors?
• Question 7

Following on from the last question, you need to use this information again: A = 3, B = 30, C = 6, D = 12

The student in the previous question answered that B - A = 33.

Which of the mistakes below could they have made to reach their answer?

EDDIE SAYS
The student got to an answer of 33 using the number options provided in the question, so they could have only done one these things wrong: Option 1: They used 30 and 3 which were the correct numbers, so they could not have made this error. Option 2: 30 + 3 is 33 which was the answer the student gave. So it is likely that they confused the symbol and added the numbers instead of subtracting them. Option 3: 3 - 30 is not 33, so the student could not have made this error.
• Question 8

Another student has been given the questions below using this information: a = 7, b = 35, c = 5, d = 16.

Identify whether they have answered each question correctly or not.

EDDIE SAYS
The question tells us that b = 35, a = 7, d = 16 and c = 5.
If we swap these numbers in for the letters in each of the equations, we get the following calculations: b - d becomes 35 - 16 = 19, which the student got right! d x c becomes 16 x 5 = 80. The student answered 21 so this is incorrect. b ÷ a + c becomes 35 ÷ 7 + 5 = 10, which the student also got right!
Are you getting better at spotting the incorrect answers? Examiners need to make sure they don't mark any questions right that are wrong and vice versa!
• Question 9

Following on from the last question, you need to use this information again: a = 7, b = 35, c = 5, d = 16

The student in the previous question answered that d x c = 21.

Which of the mistakes below could they have made to reach their answer?

EDDIE SAYS
The student got to an answer of 21 using the number options provided in the question, so they could have only done one these things wrong: Option 1: Using any combination of the available numbers for d and c cannot give a total of 21, so they could not have made this error. Option 2: 16 + 5 gives 21 which was the student's answer, so this is likely to be the mistake they made. Option 3: 16 x 5 and 5 x 16 give the same answer, so they could not have made this error. It is important to remember that you can switch the order of numbers in a multiplication sum and still reach the correct answer.
• Question 10

Here is your final question as the examiner!

Another student has been given the questions below using this information: A = 4, B = 32, C = 7, D=17.

Identify whether they have answered each question correctly or not.

EDDIE SAYS
The question tells us that B = 32, A = 4, D = 17 and C = 7.
If we swap these numbers in for the letters in each of the equations, we get the following calculations: B - D becomes 32 - 17 = 15, which the student got right! B x A becomes 32 x 4 = 128, which the student got right! B ÷ A + C becomes 32 ÷ 4 + 7 = 15, which the student also got right!
The student answered them all correctly! Hopefully you were as successful as an examiner too.
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