A farmer cuts a 300 feet fence into two pieces of different sizes. The longer piece is four times as long as the shorter piece. How long are the two pieces?

Question

Homi · Accepted Answer

Understand the Question

Let the length of the shorter piece be $x\text{ feet}$ .
The longer piece is $4$ times the shorter piece, so its length is $4x\text{ feet}$ .
The sum of the lengths of both pieces equals the total length of the fence ( $300\text{ feet}$ ).
Set up the linear equation $x + 4x = 300$ and solve for $x$ to find the lengths of both pieces.

Step 1 · Define Variables and Form the Equation

Let the length of the shorter piece be $x\text{ feet}$ .

Then, the length of the longer piece $= 4x\text{ feet}$ . Diagram 1

Given total length $= 300\text{ feet}$ :

\begin{aligned} x + 4x &= 300 \\ 5x &= 300 \end{aligned}

Step 2 · Find the Length of the Shorter Piece

Solve for $x$ by dividing both sides by $5$ :

\begin{aligned} x &= \dfrac{300}{5} \\[0.6em] &= 60 \end{aligned}

Thus, the length of the shorter piece is $60\text{ feet}$ .

Step 3 · Find the Length of the Longer Piece

Substitute $x = 60$ into the expression for the longer piece: $\text{Length of longer piece} = 4x = 4 \times 60 = 240\text{ feet}$

Answer

Shorter piece $= 60\text{ feet}$ , Longer piece $= 240\text{ feet}$

Common Mistakes

Incorrect Equation Setup: Writing $x + 4 = 300$ instead of $x + 4x = 300$ , confusing "four times as long" with "four feet longer".
Incomplete Answer: Finding $x = 60\text{ feet}$ and stopping without calculating the length of the longer piece ( $4x = 240\text{ feet}$ ).

Question 6

More questions in Exercise 2.2