Question

If the sum of two numbers is divided by 13, the quotient is 4 and the remainder 8. If the difference of the same numbers is divided by 4, the quotient is 3 and the remainder is 2. Find the numbers.

Answer 1

Hint: In this question, we have to use the formula of the relation between the dividend, divisor, quotient, and remainder, which is given as
\[\text{Dividend}=\left( \text{Divisor }\times \text{ Quotient} \right)+\text{ Remainder}\]
Now we will use the conditions given in the question to make equations. And lastly, we will solve them to get the required result.

Complete step-by-step answer:
We know that the formula for the division is:
\[\text{Dividend}=\left( \text{Divisor }\times \text{ Quotient} \right)+\text{ Remainder}\]
Now, let the two numbers that we have to find be x and y. Now we are given in the question that when the sum of these two numbers is divided by 13, it gives 4 as quotient and 8 as remainder. Applying the condition given above we get,
\[\left( x+y \right)=\left( 13\times 4 \right)+8\]
On solving the above equation, we get,
\[\Rightarrow x+y=60....\left( i \right)\]
Similarly, we are given that when the difference of both the numbers is divided by 4, the quotient is 3 and the remainder is 2. Applying this we get,
\[\left( x-y \right)=\left( 4\times 3 \right)+2\]
On solving the above equation, we get,
\[\Rightarrow x-y=14....\left( ii \right)\]
Subtracting (ii) and (i), we get,
\[x+y-\left( x-y \right)=60-14\]
\[\Rightarrow 2y=46\]
\[y=23....\left( iii \right)\]
Substituting (iii) in (i), we get,
\[x+y=60\]
\[\Rightarrow x+23=60\]
\[\Rightarrow x=60-23=37\]
\[\therefore x=37\]
Hence, the two numbers are 37 and 23.

Note: An alternate way of solving this question with two conditions is to simplify these equations as much as possible and then write any one simplified condition as the value of one variable. Then substitute this equation in the other simplified condition to get the value of that one variable. Now, again substitute the value of this variable in any of the above equations to get the value of the other variable. In these types of questions, always remember to read the question carefully and form proper conditions. If conditions are wrong, the whole solution will be incorrect.