Joanie is organizing her closet. She has 3 times as many black shirts as gray shirts, plus 3 new black shirts she just bought. She has 5 times as many white shirts as gray shirts, minus the 1 white shirt she donated. Joanie has 2 times more gray shirts than brown shirts. If she has the same number of black and white shirts, how many gray shirts does Joanie have?

Question

A.
4
B.
1
C.
2
D.
Joanie doesn't have any gray shirts.

Answer 1

Let's denote the number of gray shirts Joanie has as \( g \).

According to the information given:

1. The number of black shirts \( b \) is \( 3g + 3 \) (3 times as many black shirts as gray shirts, plus 3 new black shirts).
2. The number of white shirts \( w \) is \( 5g - 1 \) (5 times as many white shirts as gray shirts, minus 1 white shirt donated).
3. The number of gray shirts \( g \) is \( 2b \) (2 times more gray shirts than brown shirts).

From the problem, we also know that the number of black shirts \( b \) is equal to the number of white shirts \( w \). Therefore, we can set up the equation: \[ b = w \]

Substituting from our expressions for \( b \) and \( w \): \[ 3g + 3 = 5g - 1 \]

Now, let's solve for \( g \):

1. Rearrange the equation: \[ 3g + 3 + 1 = 5g \] \[ 4 = 5g - 3g \] \[ 4 = 2g \] \[ g = 2 \]

So Joanie has \( g = 2 \) gray shirts.

Let's find out the numbers of black and white shirts based on \( g = 2 \):

• The number of black shirts: \[ b = 3g + 3 = 3(2) + 3 = 6 + 3 = 9 \]
• The number of white shirts: \[ w = 5g - 1 = 5(2) - 1 = 10 - 1 = 9 \]

Thus, the number of black shirts (9) equals the number of white shirts (9).

Finally, the number of gray shirts Joanie has is \( \boxed{2} \).