Fred left school to raise 700 goats and sheep. If he sold 3/4 of his goats and also sold 400 sheep, he would have an equal number of goats and sheep left. How many more sheep than goats did he have in the begining?


Let's denote the number of goats Fred initially had as \( G \), and the number of sheep as \( S \).

### Step 1: Set up the equations based on the problem description.

- Fred sold \(\frac{3}{4}\) of his goats, so the number of goats left would be:
  \[
  G - \frac{3}{4}G = \frac{1}{4}G
  \]
- Fred sold 400 sheep, so the number of sheep left would be:
  \[
  S - 400
  \]
- According to the problem, the number of goats left is equal to the number of sheep left:
  \[
  \frac{1}{4}G = S - 400
  \]

### Step 2: Consider the total number of animals.

- The total number of goats and sheep Fred initially had is 700:
  \[
  G + S = 700
  \]

### Step 3: Solve the system of equations.

We have two equations:
1. \( \frac{1}{4}G = S - 400 \)
2. \( G + S = 700 \)

#### Substitute \( S = 700 - G \) into the first equation:
\[
\frac{1}{4}G = (700 - G) - 400
\]
Simplify the equation:
\[
\frac{1}{4}G = 300 - G
\]

#### Multiply the entire equation by 4 to eliminate the fraction:
\[
G = 1200 - 4G
\]

#### Combine like terms:
\[
5G = 1200
\]

#### Solve for \( G \):
\[
G = 240
\]

### Step 4: Find \( S \) using \( G + S = 700 \):
\[
240 + S = 700
\]
\[
S = 460
\]

### Step 5: Find how many more sheep than goats Fred had initially:
\[
S - G = 460 - 240 = 220
\]

Fred initially had **220** more sheep than goats.

Would you like more details or have any questions?

Here are some related questions:
1. How would the answer change if Fred sold half of his goats instead of \(\frac{3}{4}\)?
2. What if Fred had 800 animals instead of 700?
3. How many goats would be left if Fred decided to sell all of them?
4. What would happen if Fred sold only 300 sheep?
5. If Fred sold 200 goats instead of \(\frac{3}{4}\) of them, how would the solution change?

**Tip:** When dealing with problems involving fractions of a quantity, it’s often useful to express the remaining quantity as a fraction of the original, as seen in this problem.

Fred left school to raise 700 goats and sheep. If he sold 3/4 of his goats and also sold 400 sheep, he would have an equal number of goats and sheep left. How many more sheep than goats did he have in the beginning?

Learn how to solve a system of linear equations with this goats and sheep problem. Discover how Fred balanced his herd after selling some of his animals, and determine how many more sheep than goats he had initially. Perfect for students in Grades 7-8.

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