 # Substitute Numbers for Letters in Complex Calculations (All Four Operations)

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 = 18, what is D ÷ C?

Can you calculate the answer as a number?

3

6

8

If a = 4, b = 36, c = 3 and d = 9, what is b ÷ c + a?

Can you calculate the answer as a number?

12

16

13

If A = 4, B = 36, 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 = 21, C = 3, D = 18.

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

## Column B

B ÷ C
7
C x A + D
30
A + B + C
28

You have been given the following information: a = 3, b = 36, c = 7, d = 12

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

## Column B

b ÷ a
12
d x a - c
29
b - c - d
17

Now you are going to be the examiner!

A student has been given the questions below using this information: A = 3, B = 36, C = 7, 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 = 36, C = 7, D = 12.

The student in the previous question answered that B ÷ A + C = 24.

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

Used the wrong numbers to complete the sum

Added C to A then multiplied by B

Another student has been given the questions below using this information: a = 8, b = 40, c = 5, d = 18.

Identify whether they have answered each question correctly or not.

Following on from the last question, you need to use this information again: a = 8, b = 40, c = 5, d = 18

The student in the previous question answered that b ÷ a + c = 37.

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

Used the wrong numbers to complete the sum

Confused the symbol and subtracted instead of divided

Completed the operations in the wrong order

Here is your final question as the examiner!

Another student has been given the questions below using this information: A = 5, B = 32, C = 8, D = 17.

Identify whether they have answered each question correctly or not.

• Question 1

If A = 4, B = 12, C = 3 and D = 18, what is D ÷ C?

Can you calculate the answer as a number?

6
EDDIE SAYS
The question tells us that D = 18 and C = 3.
If we swap these numbers in for the letters, then D ÷ C becomes 18 ÷ 3.
When we divide 18 by 3, we reach the answer of 6.
• Question 2

If a = 4, b = 36, c = 3 and d = 9, what is b ÷ c + a?

Can you calculate the answer as a number?

16
EDDIE SAYS
The question tells us that b = 36, c = 3 and a = 4.
If we swap these numbers in for the letters, then b ÷ c + a becomes 36 ÷ 3 + 4.
When we divide 36 by 3 then add 4, we reach the answer of 7.
We must calculate each step in the order it has been written to reach the correct answer. Did you keep the steps in order?
• Question 3

If A = 4, B = 36, C = 3 and D = 9, what is B ÷ C x A?

Can you calculate the answer as a number?

48
EDDIE SAYS
The question tells us that B = 36, C = 3 and A = 4, which is exactly the same as the last question.
If we swap these numbers in for the letters, then B ÷ C x A becomes 36 ÷ 3 x 4.
Did you spot the small change here from an addition sign to a multiplication sign? Often small changes exist in exam questions to try and check whether you are looking at all the details.
When we divide 36 by 3 then times it by 4, we reach the answer of 48.
• Question 4

You have been given the following information: A = 4, B = 21, C = 3, D = 18.

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

## Column B

B ÷ C
7
C x A + D
30
A + B + C
28
EDDIE SAYS
The question tells us that B = 21, C = 3, A = 4 and D = 18.
If we swap these numbers in for the letters in each of the equations, we get the following calculations: B ÷ C becomes 21 ÷ 3 = 7 C x A + D becomes 3 x 4 + 18 = 30 A + B + C becomes 4 + 21 + 3 = 28
How many of the equations did you calculate correctly? Were you able to keep the numbers matched to the correct letters?
• Question 5

You have been given the following information: a = 3, b = 36, c = 7, d = 12

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

## Column B

b ÷ a
12
d x a - c
29
b - c - d
17
EDDIE SAYS
The question tells us that b = 36, a = 3, d = 12 and c = 7.
If we swap these numbers in for the letters in each of the equations, we get the following calculations: b ÷ a becomes 36 ÷ 3 = 12 d x a - c becomes 12 x 3 - 7 = 29 b - c - d becomes 36 - 7 - 12 = 17
Well done if you got all three correct!
• Question 6

Now you are going to be the examiner!

A student has been given the questions below using this information: A = 3, B = 36, C = 7, D = 12.

Identify whether they have answered each question correctly or not.

EDDIE SAYS
The question tells us that B = 36, A = 3, D = 12 and C = 7.
If we swap these numbers in for the letters in each of the equations, we get the following calculations: B - A becomes 36 - 3 = 33, which the student got right! D x A becomes 12 x 3 = 36, which the student also got right! B ÷ A + C becomes 36 ÷ 3 - 7 = 19. The student answered 24 so this is incorrect.
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 = 36, C = 7, D = 12.

The student in the previous question answered that B ÷ A + C = 24.

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

Used the wrong numbers to complete the sum
EDDIE SAYS
The student got to an answer of 24 using the number options provided in the question, so they could have done only one of these things wrong: Option 1: we cannot be sure that they used the wrong numbers, but as the other two options are not true then this is the only possible explanation. Option 2: 36 ÷ 3 - 7 = 5 not 24, so this is not the error they made. Option 3: 7 + 3 ÷ 36 is not 24, so this is not the error they made.
• Question 8

Another student has been given the questions below using this information: a = 8, b = 40, c = 5, d = 18.

Identify whether they have answered each question correctly or not.

EDDIE SAYS
The question tells us that b = 40, d = 18, c = 5 and a = 8.
If we swap these numbers in for the letters in each of the equations, we get the following calculations: b - d becomes 40 - 18 = 22, which the student got right! d x c becomes 18 x 5 = 90, which the student also got right! d ÷ a + c becomes 40 ÷ 8 + 5 = 10. The student answered 37 so this is incorrect.
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 = 8, b = 40, c = 5, d = 18

The student in the previous question answered that b ÷ a + c = 37.

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

Confused the symbol and subtracted instead of divided
EDDIE SAYS
The student got to an answer of 37 using the number options provided in the question, so they could have only done one these things wrong: Option 1: There is no obvious combination which uses the numbers provided and reaches an answer of 37, so they could not have made this error. Option 2: 40 - 8 + 5 = 37 which is what the student answered, so it is likely that this is the error they made. Option 3: 8 + 5 ÷ 40 is not 37, so they could not have made this error. It is important to remember that you cannot switch the order of operations in a sum if you want to 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 = 5, B = 32, C = 8, D = 17.

Identify whether they have answered each question correctly or not.

EDDIE SAYS
The question tells us that B = 32, D = 17, A = 5 and C = 8.
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 5 = 160. The student answered 150, so they got this one wrong. B ÷ C + A becomes 32 ÷ 8 + 5 = 9, which the student also got right!
Well done for being the examiner and helping the student spot their errors. Hopefully it helped you to understand how you can answer these questions correctly every time!
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