Question

Sum of two numbers is 97. If the larger number is divided by the smaller, the quotient is 7 and the remainder is 1. Find the numbers.
A) 75, 14
B) 90, 14
C) 75, 18
D) 85, 12

Answer 1

Hint: Here we will assume the two numbers be $x$ and $y$ in which $x$ be the larger. Then we will form the linear equations in two variables using the given information and then solve for the values of $x$ and $y$ to get the desired solution.

Complete step by step answer:
Let the two numbers be $x$ and $y$ in which $x$ be the larger number.
Now it is given that the sum of two numbers is 97.
Hence,
\[\Rightarrow x + y = 97\]………………… (1)
Also it is given that when the larger number is divided by the smaller, the quotient is 7 and the remainder is 1.
Hence, forming the linear equation using the remainder theorem which states that:-
Dividend $=$ Quotient $\times$ Divisor $+$ Remainder
We get:-
\[\Rightarrow x = 7y + 1\]……………………….(2)
Putting the value of $x$ from equation 2 in equation 1 we get:-
\[\Rightarrow 7y + 1 + y = 97\]
Simplifying it we get:-
\[\Rightarrow 8y = 96\]
Solving for the value of y we get:-
\[\Rightarrow y = 12\]
Now substituting this value of y in equation 2 we get:-
\[\Rightarrow x = 7\left( {12} \right) + 1\]
Simplifying it further we get:-
\[\Rightarrow x = 84 + 1\]
Solving for the value of $x$ we get:-
\[\Rightarrow x = 85\]
Hence, the larger number is 85 and the smaller number is 12.

Hence, we get 85 and 12 as the required numbers.

Note:
Students should note that the linear equation in two variables is the equation which contains two variables and each variable has power as 1.
Also, the dividend in remainder theorem is the number which gets divided and the divisor is the number from which the dividend gets divided.