The problems based on this topic are nothing but puzzles in which letters take the place of digits in arithmetic operations and the task is to find the digits represented by the letters.

There are three conditions to be satisfied to solve each puzzle, i.e.

(i) Each letter represents only one digit.

(ii) A number cannot begin with zero, e.g. forty-five is written as 45 but not as 045.

(iii) A puzzle must have only one answer.


 ADDITION OF SINGLE-DIGIT NUMBERS

Example 1

Find the value of B and C in the following addition:


Solution:

We observe that a certain single-digit number is added three times in which:

(i)  the sum is a two-digit number; and

(ii) the given numbers are B, B, B and the unit digit of the sum is also B.

      It is something interesting. Right ?

 

So, let’s think like this:

1 + 1 + 1 = 3 

2 + 2 + 2 = 6

3 + 3 + 3 = 9

These three options are not acceptable.

Reason: The sum is not a two-digit number.

 

Now, think about the following:

4 + 4 + 4 = 12

It has a two-digit sum but it is also not acceptable.

Reason: The unit digit of the sum is 2.

               (Actually, 4 is expected. Right ?)

 

The next option is: 5 + 5 + 5 = 15

It has a two-digit sum and it is ACCEPTABLE.

Reason: The unit digit of the sum is 5 and this is expected too. Hurrah !

 

Now, what about the following options ?

6 + 6 + 6 = 18

7 + 7 + 7 = 21

8 + 8 + 8 = 24

9 + 9 + 9 = 27

In each of the above, the sum is a two-digit number but the unit digit of the sum doesn’t satisfy the condition. So, these four options are also not acceptable.

Thus, the correct option is: 5 + 5 + 5 = 15

Hence, C = 1 and B = 5.


Example 2

Find the value of A and B in the following addition:


Solution:

We observe that a certain single-digit number is added three times in which:

(i)  the sum is a two-digit number; and

(ii) the given numbers are B, B, B and the unit digit of the sum is 4.

 

So, let’s consider the following:

1 + 1 + 1 = 3 

2 + 2 + 2 = 6

3 + 3 + 3 = 9

4 + 4 + 4 = 12

5 + 5 + 5 = 15

6 + 6 + 6 = 18

7 + 7 + 7 = 21

8 + 8 + 8 = 24

9 + 9 + 9 = 27

We observe that 8 + 8 + 8 = 24 satisfies both the conditions, i.e.

-         The sum is a two-digit number; and

-         The unit digit of the sum is 4.

 

Therefore, the correct option is 8 + 8 + 8 = 24.

Hence, A = 8, B = 2.


PROBLEMS FOR PRACTICE

Question 1

Find the value of the letters in each of the following. Give reason for the steps involved:











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