Question

Find the values of the letters in the following and give reasons for the steps involved.

Answer 1

Hint: First we will take the last column of the given sum \[{\text{B}} + 7 = {\text{A}}\]. Since B can be 0, 1, 2, …, 9, so we have \[{\text{B}} + 7\], which will be 7, 8, 9, …, 16. Thus we will have either \[{\text{A}} + 3 = 6\] or if there is a carry-over of 1 on the first column of the given sum, which is the tens value of the sum of the last column of the given sum, then \[1 + {\text{A}} + 3 = 6\]. We will consider both the cases separately to find the value of B, which satisfies the conditions.

Complete step by step solution: We are given that

We know that the sum is the result of adding two or more numbers.

First, we will take the last column of the given sum \[{\text{B}} + 7 = {\text{A}}\].
Here, B can be 0, 1, 2, …, 9.
Then when we have \[{\text{B}} + 7\], which will be 7, 8, 9, …, 16.

Thus we will have either \[{\text{A}} + 3 = 6\] or if there is a carry-over of 1 on the first column of the given sum, which is the tens value of the sum of the last column of the given sum, then \[1 + {\text{A}} + 3 = 6\].

Case 1

Consider \[{\text{A}} + 3 = 6\].

Subtracting the above equation by 3 on both sides, we get
\[
   \Rightarrow {\text{A}} + 3 - 3 = 6 - 3 \\
   \Rightarrow {\text{A}} = 3 \\
 \]

Now, we have the sum

First, we will take the last column of the given sum \[{\text{B}} + 7 = 3\].
Then we know that B has to be 6 because the unit digit of the sum has to be 3 as only when 6 is added to 7, we get 13.
This implies that \[{\text{B}} = 6\].

So, we will have a carry-over of 1 on the first column of the given sum, which is the tens value of the sum of the last column of the given sum.

Thus we will have to add 1 with the first column of the given sum, we get
\[ \Rightarrow 3 + 3 + 1 = 7\]

Since \[7 \ne 6\].
Hence it is not possible, so the value of A is not equal to 3.

Case 2

Consider \[1 + {\text{A}} + 3 = 6\].

Rewriting the above equation, we get
\[ \Rightarrow {\text{A}} + 4 = 6\]
Subtracting the above equation by 4 on both sides, we get
\[
   \Rightarrow {\text{A}} + 4 - 4 = 6 - 4 \\
   \Rightarrow {\text{A}} = 2 \\
 \]

Now, we have the sum

First, we will take the last column of the given sum \[{\text{B}} + 7 = 2\].
Then we know that B has to be 5 because the unit digit of the sum has to be 2 as only when 5 is added to 7, we get 12.
This implies that \[{\text{B}} = 5\].

So, we will have a carry-over of 1 on the first column of the given sum, which is the tens value of the sum of the last column of the given sum.

Thus we will have to add 1 with the first column of the given sum, we get
\[ \Rightarrow 2 + 3 + 1 = 6\]

Since \[6 = 6\].
Hence, the value of A is equal to 2.

Note: In solving these types of questions, you need to know that a sum is a result of adding two or more numbers, by bringing two or more numbers together to make a new total. One needs to know that the number, which is a carry-over will be added to the first column of the given sum or else the solution will be wrong. We can also find the sum by counting our fingers for better understanding.