Question

If the sum of two numbers is divided by 15, the quotient is 2 and the remainder is 10. If the difference of the same numbers is divided by 3 then the quotient is 4 and the remainder is 2. Find the numbers.
A. \[12,19\]
B. $29,11$
C. $27,13$
D. $9,21$

Answer 1

Hint: In this question, we will use algebraic methods of solving linear equations. Here we will express the given statements in the form of equations and by solving those equations simultaneously we will get the answer.

Complete step-by-step answer:
Let us suppose the numbers are $ x $ and $ y $.
We know that Dividend = Divisor $ \times $ Quotient + Remainder.
According to the question,“If the sum of two numbers is divided by 15, the quotient is 2 and the remainder is 10.” We can write this as :
$ \Rightarrow x + y = 15 \times 2 + 10 \Rightarrow x + y = 40$----(1)
Also, as the difference of the same numbers is divided by 3 then the quotient is 4 and the remainder is 2. We get,
$ \Rightarrow x - y = 3 \times 4 + 2 \Rightarrow x - y = 14$ - ---(2)
Now adding equations (1) and (2) ,we get
$\
   \Rightarrow \left( {x + y} \right) + \left( {x - y} \right) = 40 + 14 \Rightarrow 2x = 54 \\
   \Rightarrow x = 27 \\
\ $
We get $x = 27$,now substitute this value in either equation (1) or (2),
We are putting in equation (1), then we get,
$\
   \Rightarrow x + y = 40 \Rightarrow 27 + y = 40 \\
   \Rightarrow y = 40 - 27 \\
   \Rightarrow y = 13 \\
\ $
Hence, the numbers are $27,13$
The correct answer is : option (C)

Note: In these types of questions ,we have to make the linear equations according to the statements given in the question. After making equations we will solve them simultaneously by algebraic methods like substituting and through this we will get the answer.