Question

A string is cut in two so that one section is 5 inches longer than the other. Twice the length of the longer section is three times the length of the shorter section. Write an equation that allows you to solve for the length of the shorter section. How long is the shorter section?

Answer 1

Let's define the lengths of the two sections of the string. Let \( x \) be the length of the shorter section, in inches. Then, the length of the longer section can be expressed as \( x + 5 \) inches.

According to the problem, twice the length of the longer section is equal to three times the length of the shorter section. We can set up the following equation based on this relationship:

\[
2(x + 5) = 3x
\]

Now, we can solve for \( x \).

1. Distribute on the left side:
\[
2x + 10 = 3x
\]

2. Next, subtract \( 2x \) from both sides:
\[
10 = 3x - 2x
\]
\[
10 = x
\]

Therefore, the length of the shorter section is \( \boxed{10} \) inches.

For validation, the longer section would be:
\[
x + 5 = 10 + 5 = 15 \text{ inches}
\]

Now, let's check the initial condition:
Twice the length of the longer section is:
\[
2 \times 15 = 30
\]
And three times the length of the shorter section is:
\[
3 \times 10 = 30
\]

Both calculations confirm that our equation was set up correctly. Thus, the length of the shorter section is indeed \( \boxed{10} \) inches.