Find the values of the letters in the following and give reasons for the steps involved.
| 3 A |
| + 2 5 |
| -------- |
| B 2 |
Answer:
The addition of A and 5 is giving 2 i.e., a number whose ones digit is 2.
This is possible only when digit A is 7.
In that case, the addition of A (7) and 5 will give 12 and thus, 1 will be the carry for the next step.
In the next step,
1 + 3 + 2 = 6
1 + 3 + 2 = 6
Therefore, the addition is as follows.
Clearly, B is 6.
| 3 7 |
| + 2 5 |
| -------- |
| 6 2 |
Clearly, B is 6.
Hence, A and B are 7 and 6 respectively.
Question 2:
Find the values of the letters in the following and give reasons for the steps involved.
| 4 A |
| + 9 8 |
| ------------ |
| C B 3 |
Answer:
The addition of A and 8 is giving 3 i.e., a number whose ones digit is 3.
This is possible only when digit A is 5.
In that case, the addition of A and 8 will give 13 and thus, 1 will be the carry for the next step.
In the next step,
1 + 4 + 9 = 14
Therefore, the addition is as follows.
Clearly, B and C are 4 and 1 respectively.
Hence, A, B, and C are 5, 4, and 1 respectively.
Question 4:
Find the values of the letters in the following and give reasons for the steps involved:
Therefore, the addition is as follows.
| 4 5 |
| + 9 8 |
| ------------ |
| 1 4 3 |
Clearly, B and C are 4 and 1 respectively.
Hence, A, B, and C are 5, 4, and 1 respectively.
Question 4:
Find the values of the letters in the following and give reasons for the steps involved:
| A B |
| + 3 7 |
| ------------ |
| 6 A |
Answer:
The addition of A and 3 is giving 6. There can be two cases.
(1) First step is not producing a carry
In that case, A comes to be 3 as 3 + 3 = 6.
Considering the first step in which the addition of B and 7 is giving A (i.e., 3),
B should be a number such that the units digit of this addition comes to be 3.
It is possible only when B = 6.
In this case, A = 6 + 7 = 13.
However, A is a single digit number. Hence, it is not possible.
(2) First step is producing a carry
In that case, A comes to be 2 as 1 + 2 + 3 = 6. Considering the first step in which
the addition of B and 7 is giving A (i.e., 2), B should be a number such that the units
digit of this addition comes to be 2.
It is possible only when B = 5 and 5 + 7 = 12.
Hence, the values of A and B are 2 and 5 respectively.
Question 5:
Find the values of the letters in the following and give reasons for the steps involved.
Answer:
The multiplication of 3 and B gives a number whose ones digit is B again.
Hence, B must be 0 or 5.
Let B is 5.
Multiplication of first step = 3 × 5 = 15
1 will be a carry for the next step.
We have, 3 × A + 1 = CA
This is not possible for any value of A.
Hence, B must be 0 only. If B = 0, then there will be no carry for the next step.
We should obtain, 3 × A = CA
That is, the one’s digit of 3 × A should be A. This is possible when A = 5 or 0.
However, A cannot be 0 as AB is a two-digit number.
Therefore, A must be 5 only. The multiplication is as follows:
In this case, A = 6 + 7 = 13.
However, A is a single digit number. Hence, it is not possible.
(2) First step is producing a carry
In that case, A comes to be 2 as 1 + 2 + 3 = 6. Considering the first step in which
the addition of B and 7 is giving A (i.e., 2), B should be a number such that the units
digit of this addition comes to be 2.
It is possible only when B = 5 and 5 + 7 = 12.
| 2 5 |
| + 3 7 |
| ------------ |
| 6 2 |
Hence, the values of A and B are 2 and 5 respectively.
Question 5:
Find the values of the letters in the following and give reasons for the steps involved.
| A B |
| ✖️ 3 |
| ------------ |
| C A B |
Answer:
The multiplication of 3 and B gives a number whose ones digit is B again.
Hence, B must be 0 or 5.
Let B is 5.
Multiplication of first step = 3 × 5 = 15
1 will be a carry for the next step.
We have, 3 × A + 1 = CA
This is not possible for any value of A.
Hence, B must be 0 only. If B = 0, then there will be no carry for the next step.
We should obtain, 3 × A = CA
That is, the one’s digit of 3 × A should be A. This is possible when A = 5 or 0.
However, A cannot be 0 as AB is a two-digit number.
Therefore, A must be 5 only. The multiplication is as follows:
| 5 0 |
| ✖️ 3 |
| ------------ |
| 1 5 0 |
Hence, the values of A, B, and C are 5, 0, and 1 respectively.
Question 6:
Find the values of the letters in the following and give reasons for the steps involved.
| 5 0 |
| ✖️ 3 |
| ------------ |
| 1 5 0 |
Answer:
The multiplication of B and 5 is giving a number whose ones digit is B again. This is
possible when B = 5 or B = 0 only.
In case of B = 5, the product, B × 5 = 5 × 5 = 25
2 will be a carry for the next step.
We have, 5 × A + 2 = CA, which is possible for A = 2 or 7
The multiplication is as follows.
| 2 5 | 7 5 | |
| ✖️ 5 | ✖️ 5 | |
| ------------ | ---------- | |
| 1 2 5 | 3 7 5 |
If B = 0,
B × 5 = B ⇒ 0 × 5 = 0
There will not be any carry in this step.
In the next step, 5 × A = CA
It can happen only when A = 5 or A = 0
However, A cannot be 0 as AB is a two-digit number.
Hence, A can be 5 only. The multiplication is as follows:
| 5 0 | ||
| ✖️ 5 | ||
| ------------ | ||
| 2 5 0 |
Hence, there are 3 possible values of A, B, and C.
(i) 5, 0, and 2 respectively
(ii) 2, 5, and 1 respectively
(iii) 7, 5, and 3 respectively
Question 7:
Find the values of the letters in the following and give reasons for the steps involved:
| A B |
| ✖️ 6 |
| ------------ |
| B B B |
Answer:
The multiplication of 6 and B gives a number whose one’s digit is B again.
It is possible only when B = 0, 2, 4, 6, or 8
If B = 0, then the product will be 0. Therefore, this value of B is not possible.
If B = 2, then B × 6 = 12 and 1 will be a carry for the next step.
6A + 1 = BB = 22 ⇒ 6A = 21 and hence, any integer value of A is not possible.
If B = 6, then B × 6 = 36 and 3 will be a carry for the next step.
6A + 3 = BB = 66 ⇒ 6A = 63 and hence, any integer value of A is not possible.
If B = 8, then B × 6 = 48 and 4 will be a carry for the next step.
6A + 4 = BB = 88 ⇒ 6A = 84 and hence, A = 14. However, A is a single digit number.
Therefore, this value of A is not possible.
If B = 4, then B × 6 = 24 and 2 will be a carry for the next step.
If B = 4, then B × 6 = 24 and 2 will be a carry for the next step.
6A + 2 = BB = 44 ⇒ 6A = 42 and hence, A = 7
The multiplication is as follows:
| 7 4 |
| ✖️ 6 |
| ------------ |
| 4 4 4 |
Hence, the values of A and B are 7 and 4 respectively.
Question 8:
Find the values of the letters in the following and give reasons for the steps involved:
| A 1 |
| + 1 B |
| -------- |
| B 0 |
Answer:
The addition of 1 and B is giving 0 i.e., a number whose ones digits is 0.
This is possible only when digit B is 9.
In that case, the addition of 1 and B will give 10 and thus, 1 will be the carry for the next step.
In the next step,
1 + A + 1 = B
Clearly, A is 7 as 1 + 7 + 1 = 9 = B
Therefore, the addition is as follows.
1 + A + 1 = B
Clearly, A is 7 as 1 + 7 + 1 = 9 = B
Therefore, the addition is as follows.
| 7 1 |
| + 1 9 |
| -------- |
| 9 0 |
Hence, the values of A and B are 7 and 9 respectively.
Question 9:
Find the values of the letters in the following and give reasons for the steps involved.
| 2 A B |
| + A B 1 |
| -------- |
| B 1 8 |
Answer:
The addition of B and 1 is giving 8 i.e., a number whose ones digits is 8.
This is possible only when digit B is 7.
In that case, the addition of B and 1 will give 8.
In the next step,
A + B = 1
Clearly, A is 4.
4 + 7 = 11 and 1 will be a carry for the next step. In the next step,
1 + 2 + A = B
1 + 2 + 4 = 7
Therefore, the addition is as follows.
A + B = 1
Clearly, A is 4.
4 + 7 = 11 and 1 will be a carry for the next step. In the next step,
1 + 2 + A = B
1 + 2 + 4 = 7
Therefore, the addition is as follows.
| 2 4 7 |
| + 4 7 1 |
| -------- |
| 7 1 8 |
Hence, the values of A and B are 4 and 7 respectively.
Question 10:
Find the values of the letters in the following and give reasons for the steps involved.
| 1 2 A |
| + 6 A B |
| -------- |
| A 0 9 |
Answer:
The addition of A and B is giving 9 i.e., a number whose ones digits is 9.
The sum can be 9 only as the sum of two single digit numbers cannot be 19.
Therefore, there will not be any carry in this step.
In the next step, 2 + A = 0
It is possible only when A = 8
2 + 8 = 10 and 1 will be the carry for the next step.
1 + 1 + 6 = A
Clearly, A is 8. We know that the addition of A and B is giving 9.
In the next step, 2 + A = 0
It is possible only when A = 8
2 + 8 = 10 and 1 will be the carry for the next step.
1 + 1 + 6 = A
Clearly, A is 8. We know that the addition of A and B is giving 9.
As A is 8, therefore,
B is 1.
Therefore, the addition is as follows:
Therefore, the addition is as follows:
| 1 2 8 |
| + 6 8 1 |
| -------- |
| 8 0 9 |
Hence, the values of A and B are 8 and 1 respectively.
