Question

Find the solution to a math problem where the difference between two numbers is 5 times the smaller number and the larger number is 3 more than twice the smaller number.

Answer 1

Let the smaller number be denoted as \( x \) and the larger number as \( y \).

According to the problem, we have two equations based on the statements given:

1. The difference between the two numbers is 5 times the smaller number:
\[
y - x = 5x
\]

2. The larger number is 3 more than twice the smaller number:
\[
y = 2x + 3
\]

Now, we can solve these equations step by step.

**Step 1: Simplify the first equation.**

From the first equation, we can express \( y \) in terms of \( x \):
\[
y - x = 5x \implies y = 6x
\]

**Step 2: Substitute into the second equation.**

We now have two expressions for \( y \):
1. \( y = 6x \)
2. \( y = 2x + 3 \)

Set them equal to each other:
\[
6x = 2x + 3
\]

**Step 3: Solve for \( x \).**

Subtract \( 2x \) from both sides:
\[
6x - 2x = 3 \implies 4x = 3
\]

Now, divide by 4:
\[
x = \frac{3}{4}
\]

**Step 4: Find \( y \).**

Substitute \( x \) back into the equation for \( y \):
\[
y = 6x = 6 \left( \frac{3}{4} \right) = \frac{18}{4} = \frac{9}{2}
\]

**Final solution:**

The smaller number \( x \) is \( \frac{3}{4} \), and the larger number \( y \) is \( \frac{9}{2} \).

Thus, the solution is:
- Smaller number: \( \frac{3}{4} \)
- Larger number: \( \frac{9}{2} \)