There are goats cows and sheep on a farm. 2/5 of the animals were goats. There were 3 times as many sheep as cows. If there were 45 more goats than cows, how many animals were there on the farm

Let the number of cows on the farm be \( C \), the number of sheep be \( S \), and the number of goats be \( G \).

From the problem, we know the following relationships:

1. \( \frac{2}{5} \) of the animals were goats, meaning:
   \[
   G = \frac{2}{5} \times \text{Total Animals}
   \]
   
2. There were 3 times as many sheep as cows:
   \[
   S = 3C
   \]
   
3. There were 45 more goats than cows:
   \[
   G = C + 45
   \]

Now, let the total number of animals be \( T \). The total number of animals is the sum of the cows, sheep, and goats:
\[
T = C + S + G
\]

Substitute the expression for \( S \) and \( G \) in terms of \( C \) from the equations above:
\[
T = C + 3C + (C + 45)
\]
Simplifying this expression:
\[
T = 5C + 45
\]

Now, since we know that \( G = \frac{2}{5}T \), we can also substitute \( G = C + 45 \) into this:
\[
C + 45 = \frac{2}{5}T
\]

Substitute \( T = 5C + 45 \) into the above equation:
\[
C + 45 = \frac{2}{5}(5C + 45)
\]
Simplify the right-hand side:
\[
C + 45 = 2C + 18
\]
Subtract \( C \) from both sides:
\[
45 = C + 18
\]
Solve for \( C \):
\[
C = 45 - 18 = 27
\]

Now that we know \( C = 27 \), we can find \( S \) and \( G \):
\[
S = 3C = 3 \times 27 = 81
\]
\[
G = C + 45 = 27 + 45 = 72
\]

Finally, the total number of animals is:
\[
T = C + S + G = 27 + 81 + 72 = 180
\]

Thus, the total number of animals on the farm is **180**.

Would you like further details on this solution, or have any other questions?

Here are 5 related questions to explore further:
1. What fraction of the animals were cows?
2. How many more sheep than goats were there on the farm?
3. If there were 10 fewer goats, how would the total number of animals change?
4. How many sheep would there be if the number of cows doubled?
5. What is the ratio of goats to sheep?

**Tip:** When dealing with word problems, break the information into smaller algebraic equations to simplify complex relationships.

There are goats, cows, and sheep on a farm. 2/5 of the animals were goats. There were 3 times as many sheep as cows. If there were 45 more goats than cows, how many animals were there on the farm?

This problem involves finding the total number of animals on a farm using algebraic equations and proportions. Learn to solve the system where 2/5 of the animals are goats, there are 3 times as many sheep as cows, and 45 more goats than cows. Suitable for students in Grades 6-8.

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