Question

The difference between two numbers is 48. The ratio of the two numbers is 7:3. What are the two numbers ?
A. 63, 27
B. 56, 24
C. 36, 84
D. 40, 80

Answer 1

Hint: We will first let the numbers be $x$ and $y$, and $x$ being the greater of two numbers. Then, express $x$ in terms of $y$ by using the given conditions. Formulate the equations and then solve the equations by substitution to get the required answer.

Complete step by step answer:
Let us suppose that the two numbers that we have to find be $x$ and $y$.
Let us also suppose that the number $x$ is greater than or equal to $y$, that is $x$ being the greater of two numbers.
According to the question, the difference between these numbers is 48. Therefore we can say according to our assumption of the two numbers be $x$ and $y$ with $x$ being greater, $x - y = 48$.
On rearranging we can say that $x = 48 + y$
Also it is given in the question that the ratio of the two numbers is 7:3.Therefore we can say according to assumption of the two numbers be $x$ and $y$ with $x$ being greater, $\dfrac{x}{y} = \dfrac{7}{3}$
On multiplying the equation throughout with $3y$, we get
$3x = 7y$
Substituting the value $48 + y$ for $x$ in the equation , we get
\[3\left( {48 + y} \right) = 7y\]
We can solve for $y$ from the above equation.
$
\Rightarrow 144 + 3y = 7y \\
 \Rightarrow 4y = 144 \\
  \Rightarrow y = 36 \\
$
We can now solve for $x$ by substituting the value 36 for $y$ in the equation \[x = y + 48\]
$
\Rightarrow x = 36 + 48 \\
  \Rightarrow x = 84 \\
$
Therefore the two numbers are 84 and 36.
Thus the option C matches with the solution and therefore is correct.

Note: Many students make mistakes in formulating equations and substituting the values. Also, this question can alternatively be solved using elimination method. The order of the numbers in the given option does not matter, that is, 36, 84 is equivalent to 84, 36.