Miss Grady is buying multiple cakes to share with her friends. Mississauga comes in and takes 1/4 of the total cake after Miss Saga takes the cake Miss Molly, Aunt Ena and Miss Feldman come in and take 1/4 of the cake that were remaining. Mr. Alvarez comes in and takes half of the remaining cake if 20 cake remains at the end of the day how many cakes were there to begin with?



Answer :

Answer:

71

Step-by-step explanation:

Let's denote the initial number of cakes as \( x \).

1. **After Mississauga takes 1/4 of the cakes:**

- Mississauga takes \( \frac{1}{4} \) of \( x \) cakes, which is \( \frac{x}{4} \).

- Cakes remaining after Mississauga: \( x - \frac{x}{4} = \frac{3x}{4} \).

2. **After Miss Molly, Aunt Ena, and Miss Feldman take 1/4 of the remaining cakes:**

- They take \( \frac{1}{4} \) of \( \frac{3x}{4} \) cakes, which is \( \frac{3x}{16} \).

- Cakes remaining after them: \( \frac{3x}{4} - \frac{3x}{16} = \frac{9x}{16} \).

3. **After Mr. Alvarez takes half of the remaining cakes:**

- Mr. Alvarez takes \( \frac{1}{2} \) of \( \frac{9x}{16} \) cakes, which is \( \frac{9x}{32} \).

- Cakes remaining after Mr. Alvarez: \( \frac{9x}{16} - \frac{9x}{32} = \frac{18x}{32} - \frac{9x}{32} = \frac{9x}{32} \).

Given that 20 cakes remain at the end of the day, we have:

\[ \frac{9x}{32} = 20 \]

To find \( x \), multiply both sides by \( \frac{32}{9} \):

\[ x = 20 \times \frac{32}{9} \]

\[ x = \frac{640}{9} \]

\[ x = 71 \frac{1}{9} \]

Since \( x \) must be a whole number (you can't have a fraction of a cake), we round \( 71 \frac{1}{9} \) down to the nearest whole number.

Therefore, the initial number of cakes that Miss Grady had was \( \boxed{71} \).