Question

Solve the following cryptarithm
      \[A\] \[B\]
 \[ + \] \[3\] \[7\]
---------
      \[9\] \[A\]

Answer 1

Hint: Any number can be written in expanded form. For instance, \[124\] can be expanded as \[100 + 20 + 4\] . We can say that the number \[124\] contains \[100\] hundreds, \[20\] tens, and \[4\] ones.
This is the method of splitting the number into parts based on their place value. Using this rule, we need to first expand the given sum and form an equation. Then we will try to find the values of \[A\] & \[B\] by solving those equations.

Complete step by step answer:
  It is given that, \[AB + 37 = 9A\] where \[AB\] is one number with two digits \[A\& B\] . On adding \[37\] with the number \[AB\] we are getting a number \[9A\] with two digits \[9\& A\] .
In addition, we always add numbers by their place value. That is if we are adding two numbers first, we add their digits in ones-place then tens place then hundreds place, and so on.
Here, the two numbers are \[AB\] and \[37\] on adding this we are getting the answer as \[9A\] .
This problem has two cases, let’s see one by one.
Case 1:
As per the rule, we will first add the numbers in the ones-place that is \[B + 7 = A\] . Then we will add the numbers in tens place that is \[A + 3 = 9\] . Now, we got two equations \[B + 7 = A\] and \[A + 3 = 9\]
On solving this we will get the values of A and B.
From the second equation, we get \[A = 6\] . By substituting it in the first equation we get, \[B + 7 = 6 \Rightarrow B = - 1\] which is impossible, because a single digit in a number cannot be a negative value.
Case 2:
Let us consider the equation \[B + 7 = A\] .
Here there is another possibility that the value of the sum \[B + 7\] can be a two-digit number. Let us take \[B + 7 = A + 10\] so now the digit in the tens place will get carried over.
Thus, the second equation will become \[1 + A + 3 = 9\] . Now we get \[A = 9 - 3 - 1 = 5\] .
Therefore \[B + 7 = 5 + 10 = 15\] . The digit one in the tens place of \[15\] is got carried over.
Now let us find the value of \[B\] . We have \[B + 7 = 15\] , from this we are getting \[B = 8\] .
      \[1\]
      \[5\] \[8\]
 \[ + \] \[3\] \[7\]
---------
      \[9\] \[5\]
Therefore, \[A = 5\] and \[B = 8\] .
We aim to find the value of \[A + B\] and we know that \[A = 5\] and \[B = 8\] .
Thus, \[A + B = 5 + 8 = 13\]
Therefore, the value of \[A + B\] is \[13\] .

Note: We need to know how to expand the number based on its place value. This cryptarithm can be solved only by using the method of expanding the number.
Then, we cannot have a negative value for a single digit in a number so we have gone for the next possible case (double-digit).