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

OA. 4

OB. 2

OC. 1

D. Joanie doesn't have any gray shirts.

Answer 1

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

According to the problem, we can derive the following relationships:

1. Black shirts: Joanie has \( 3g + 3 \) black shirts (3 times as many black shirts as gray shirts plus the 3 new black shirts).
2. White shirts: Joanie has \( 5g - 1 \) white shirts (5 times as many white shirts as gray shirts minus the 1 white shirt she donated).
3. Brown shirts: Joanie has \( \frac{g}{2} \) brown shirts (2 times more gray shirts than brown shirts implies there are half as many brown shirts as gray shirts).
4. Equality of black and white shirts: Since the number of black shirts equals the number of white shirts, we can set up the equation: \[ 3g + 3 = 5g - 1 \]

Now, we can solve for \( g \):

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

2. Dividing by 2: \[ g = 2 \]

Now, we substitute \( g = 2 \) back into the equations to check:

• Black shirts: \( 3(2) + 3 = 6 + 3 = 9 \)
• White shirts: \( 5(2) - 1 = 10 - 1 = 9 \)
• Brown shirts: \( \frac{2}{2} = 1 \)

All values check out:

• Joanie has 9 black shirts and 9 white shirts, which are equal.
• Joanie has 2 gray shirts.
• Joanie has 1 brown shirt.

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