The difference between two positive integers is 63. The ratio of the two integers is . Find the two integers.

Question

Homi · Accepted Answer

Understand the Question

Let the two positive integers be $2x$ and $5x$ , where $x$ is a positive multiplier.
The difference between the two integers is given as $63$ , so we can set up the linear equation $5x - 2x = 63$ .
Solving for $x$ allows us to find the exact values of both integers.

Step 1 · Set up the Equation and Solve for $x$

Let the two positive integers be $2x$ and $5x$ . Diagram 1

Given that their difference is $63$ :

\begin{aligned} 5x - 2x &= 63 \\ 3x &= 63 \\[0.6em] x &= \dfrac{63}{3} \\ x &= 21 \end{aligned}

Step 2 · Calculate the Two Integers

Substitute $x = 21$ into the expressions for both integers:

\begin{aligned} 2x &= 2 \times 21 \\ &= 42 \end{aligned}

\begin{aligned} 5x &= 5 \times 21 \\ &= 105 \end{aligned}

Answer

The two integers are $42$ and $105$ .

Common Mistakes

Order of Subtraction: Writing $2x - 5x = 63$ resulting in a negative multiplier $x = -21$ . Always subtract the smaller part from the larger part ( $5x - 2x$ ).
Stopping at $x$ : Finding $x = 21$ and stopping without multiplying back by $2$ and $5$ to find the actual integers ( $42$ and $105$ ).

Question 4