To solve this problem, we'll start by setting up an equation based on the given ratio and the difference between the lengths of the two pieces of string.
1. Define the lengths in terms of a variable:
Let the length of the shorter piece be [tex]\( 2x \)[/tex]. Because the ratio is 2:9, the length of the longer piece will be [tex]\( 9x \)[/tex].
2. Set up an equation from the given difference:
We are told that the difference between the lengths of the two pieces is 42 inches. Therefore, we can write:
[tex]\[
9x - 2x = 42
\][/tex]
3. Simplify the equation:
Combine like terms:
[tex]\[
7x = 42
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
Divide both sides of the equation by 7:
[tex]\[
x = \frac{42}{7}
\][/tex]
[tex]\[
x = 6
\][/tex]
5. Find the length of the shorter piece:
The length of the shorter piece is [tex]\( 2x \)[/tex]. Therefore:
[tex]\[
2x = 2 \times 6 = 12
\][/tex]
So, the length of the shorter piece of string is 12 inches.
