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.

Answer 1

Hint:
Let x and y be the two required numbers.
Using the information provided in the question, write the numbers using the division algorithm.
Division algorithm is stated as: Dividend = (Divisor \[ \times \] Quotient) + Remainder.
Thus, form a pair of linear equations.
On solving the pair of linear equations, we get the two required numbers.

Complete step by step solution:
Let the numbers be x and y.
It is given that If the sum of two numbers i.e. x + y is divided by 15, the quotient is 2 and the remainder is 10.
Now, we will write the above statement in the form of a division algorithm, which can be stated as Dividend = (Divisor \[ \times \] Quotient) + Remainder.
So, the above given statement can be written as $\left( {x + y} \right) = \left( {15 \times 2} \right) + 10$ .
Now, we will simplify the above equation.
 $\Rightarrow x + y = 30 + 10$
 $\Rightarrow x + y = 40$ … (1)
Also, it is given that, If the difference of the same numbers i.e. x - y is divided by 3 then the quotient is 4 and the remainder is 2.
So, writing the above statement in the form of division algorithm as $\left( {x - y} \right) = \left( {3 \times 4} \right) + 2$
 $\Rightarrow x - y = 12 + 2$
 $\Rightarrow x - y = 14$ … (2)
Now, to find the values of x and y, we will solve the pair of equations (1) and (2).
 $\Rightarrow x + y = 40$ … (1)
 $\Rightarrow x - y = 14$ … (2)
 $
  \Rightarrow 2x = 54 \\
  \Rightarrow x = \dfrac{{54}}{2} \\
  \Rightarrow x = 27 \\
 $
Thus, we get \[x = 27\] . Substituting the value \[x = 27\] in equation (1) gives
 $
  27 + y = 40 \\
  \Rightarrow y = 40 - 27 \\
  \Rightarrow y = 13 \\
 $
Thus, we get \[y = 13\] .

So, the numbers are 27 and 13.

Note:
Division algorithm:
The division algorithm computes the values of quotients as well as the remainder. The division algorithm can be stated as “The dividend of any division is the sum of product of quotient and divisor and the remainder.”
Thus, division algorithm: Dividend = (Divisor \[ \times \] Quotient) + Remainder.
For example, the number 158 in form of division algorithm can be written as
 $158 = \left( {9 \times 7} \right) + 5$.