Question

If three times the larger of two numbers is divided by the smaller, we get $4$ as the quotient and $8$ as the remainder. If five times the smaller number is divided by the larger, we get $3$ as the quotient and $5$ as the remainder. Find the numbers.
A) $20,13$
B) $23,16$
C) $20,11$
D) $22,17$

Answer 1

Hint:
We will use a division algorithm to form the two equations with two variables. The variables will be the required numbers. We will solve the two equations simultaneously to obtain the result. The values for remainder and quotient are already given so we can use it to form the equation.

Complete step by step solution:
Let us assume that the larger number is $x$ and the smaller number is $y$.
It is given that, if three times the larger number is divided by the smaller, we get $4$ as the quotient and \[8\] as the remainder.
Similarly, it is also given that if five times the smaller number is divided by the larger, we get $3$ as the quotient and $5$ as the remainder.
For any two numbers $a$ and $b$, division algorithm states that if we divide $b$ by $a$ then there exists two numbers $p$ and $q$ such that :
$a = bq + r;0 \leqslant r < b$
Here $q$ is called quotient and $r$ is the remainder.
We will use this to form an equation from the given condition.
The first condition can be rewritten as follows by using the division algorithm.
$3x = 4y + 8$ … (1)
Similarly, the second condition can be rewritten as follows:
$5y = 3x + 5$ … (2)
Note that $x > y$ .
We will substitute ion (1) in the equation (2) and rewrite it as follows:
 $5y = \left( {4y + 8} \right) + 5$
Therefore, on simplifying we get,
$y = 13$
Substituting in the equation (1) we get,
$3x = 4\left( {13} \right) + 8$
Simplifying for $x$ we get,
$x = 20$
Therefore, the larger number is $20$ and the smaller number is $13$.

Therefore, the correct option is A.

Note:
Here the important point was to form the two equations by using the division algorithm. After that it is very easy to solve the two equations simultaneously and obtain the solutions. Be careful about which number is larger and which number is smaller. Also it is important to pay the attention towards the multiples.