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The ratio of the number of goats in Farm [tex]$X$[/tex] to the number of goats in Farm [tex]$Y$[/tex] was [tex]$9: 4$[/tex]. After 35 goats were transferred from Farm [tex]$X$[/tex] to Farm [tex]$Y$[/tex], there was an equal number of goats in each farm. How many goats were there in Farm [tex]$Y$[/tex] at first?

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GinnyAnsweridn
Let's denote the number of goats in Farm [tex]\( Y \)[/tex] initially as [tex]\( y \)[/tex]. According to the problem, the ratio of the number of goats in Farm [tex]\( X \)[/tex] to the number of goats in Farm [tex]\( Y \)[/tex] was [tex]\( 9:4 \)[/tex]. This means that Farm [tex]\( X \)[/tex] initially had [tex]\( \frac{9}{4} \)[/tex] times the number of goats in Farm [tex]\( Y \)[/tex].

So, if Farm [tex]\( Y \)[/tex] had [tex]\( y \)[/tex] goats:
[tex]\[ \text{Number of goats in Farm } X = \frac{9}{4}y \][/tex]

After transferring 35 goats from Farm [tex]\( X \)[/tex] to Farm [tex]\( Y \)[/tex]:
- The number of goats left in Farm [tex]\( X \)[/tex] is:
[tex]\[ \frac{9}{4}y - 35 \][/tex]
- The number of goats in Farm [tex]\( Y \)[/tex] is:
[tex]\[ y + 35 \][/tex]

According to the problem, after the transfer, both farms have an equal number of goats. Therefore:
[tex]\[ \frac{9}{4}y - 35 = y + 35 \][/tex]

We can solve this equation step-by-step:
1. Multiply the whole equation by 4 to get rid of the fraction:
[tex]\[ 9y - 140 = 4y + 140 \][/tex]
2. Subtract [tex]\( 4y \)[/tex] from both sides to simplify:
[tex]\[ 5y - 140 = 140 \][/tex]
3. Add 140 to both sides:
[tex]\[ 5y = 280 \][/tex]
4. Divide both sides by 5:
[tex]\[ y = 56 \][/tex]

So, the initial number of goats in Farm [tex]\( Y \)[/tex] was [tex]\(\boxed{56}\)[/tex].

To check the solution:
1. Initial goats in Farm [tex]\( Y \)[/tex] = 56
2. Initial goats in Farm [tex]\( X \)[/tex]:
[tex]\[ \frac{9}{4} \times 56 = 126 \][/tex]
3. Goats left in Farm [tex]\( X \)[/tex] after transfer:
[tex]\[ 126 - 35 = 91 \][/tex]
4. Goats in Farm [tex]\( Y \)[/tex] after transfer:
[tex]\[ 56 + 35 = 91 \][/tex]

Both farms indeed have 91 goats each after the transfer, confirming that the initial number of goats in Farm [tex]\( Y \)[/tex] was correctly determined.

Before the transfer:
[tex]\[ \text{Farm } Y: 56 \][/tex]

After the transfer:
[tex]\[ \text{Farm } Y: 91 \][/tex]
[tex]\[ \text{Farm } X: 91 \][/tex]

Hence, the initial number of goats in Farm [tex]\( Y \)[/tex] was [tex]\(\boxed{56}\)[/tex].
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