Question

Benjamin has a stick which is 80/7 inches long. He wishes to cut it into two pieces so that one piece is 32/5 inches longer than the other. How long should the longer piece be? (Estimate to the nearest tenths place).
1.5
2.5
8.9
3.2

Answer 1

Let the length of the shorter piece be \( x \) inches. Then, the longer piece will be \( x + \frac{32}{5} \) inches.

The total length of the stick is given to be \( \frac{80}{7} \) inches. Therefore, we can set up the following equation:

\[
x + \left( x + \frac{32}{5} \right) = \frac{80}{7}
\]

Simplifying this equation, we have:

\[
2x + \frac{32}{5} = \frac{80}{7}
\]

Next, let's isolate \( 2x \):

\[
2x = \frac{80}{7} - \frac{32}{5}
\]

To subtract these fractions, we need a common denominator. The least common multiple of 7 and 5 is 35. Rewrite each fraction with this common denominator:

\[
\frac{80}{7} = \frac{80 \times 5}{7 \times 5} = \frac{400}{35}
\]
\[
\frac{32}{5} = \frac{32 \times 7}{5 \times 7} = \frac{224}{35}
\]

Now substitute these fractions back into the equation:

\[
2x = \frac{400}{35} - \frac{224}{35} = \frac{176}{35}
\]

Next, we solve for \( x \):

\[
x = \frac{176}{70} = \frac{88}{35}
\]

Now we find the length of the longer piece:

\[
\text{Longer Piece} = x + \frac{32}{5} = \frac{88}{35} + \frac{32 \times 7}{5 \times 7} = \frac{88}{35} + \frac{224}{35} = \frac{312}{35}
\]

To convert \( \frac{312}{35} \) to a decimal:

\[
\frac{312}{35} \approx 8.914285714285714
\]

Rounding to the nearest tenths place gives:

\[
8.9
\]

Thus, the length of the longer piece is \(\boxed{8.9}\) inches.