A bakery sold some butter cakes, chocolate cakes and almond cakes for $786. 1/2 of the cakes sold were butter cakes. 3/8 of the cakes sold were chocolate cakes and the rest were almond cakes. The bakery sold each butter cake for $15. each chocolate cake for $17 and each almond cake for $20. How many cakes did the bakery sell altogether?

To solve the problem, let's first set up the equations based on the information given.

### Step 1: Define variables
Let $ x $ represent the total number of cakes sold.

### Step 2: Determine the number of each type of cake sold
- Butter cakes: $ \frac{1}{2} $ of the cakes sold were butter cakes, so the number of butter cakes is $ \frac{1}{2}x $.
- Chocolate cakes: $ \frac{3}{8} $ of the cakes sold were chocolate cakes, so the number of chocolate cakes is $ \frac{3}{8}x $.
- Almond cakes: The rest were almond cakes. Since $ \frac{1}{2}x $ and $ \frac{3}{8}x $ have already been accounted for, the remaining fraction for almond cakes is:
  \[
  1 - \left(\frac{1}{2} + \frac{3}{8}\right)
  \]

Let's calculate the fraction for almond cakes:
\[
\frac{1}{2} = \frac{4}{8}
\]
\[
\frac{4}{8} + \frac{3}{8} = \frac{7}{8}
\]
\[
\text{Fraction of almond cakes} = 1 - \frac{7}{8} = \frac{1}{8}
\]
So, the number of almond cakes sold is $ \frac{1}{8}x $.

### Step 3: Set up the revenue equation
We know the total revenue was $786, so we can set up an equation:
\[
15 \times \frac{1}{2}x + 17 \times \frac{3}{8}x + 20 \times \frac{1}{8}x = 786
\]

### Step 4: Simplify and solve the equation
First, multiply each term:
\[
\frac{15x}{2} + \frac{51x}{8} + \frac{20x}{8} = 786
\]
Next, convert $ \frac{15x}{2} $ to have a denominator of 8:
\[
\frac{60x}{8} + \frac{51x}{8} + \frac{20x}{8} = 786
\]
Combine the terms:
\[
\frac{60x + 51x + 20x}{8} = 786
\]
\[
\frac{131x}{8} = 786
\]
Multiply both sides by 8 to solve for $ x $:
\[
131x = 786 \times 8
\]
\[
131x = 6288
\]
Finally, divide by 131 to find $ x $:
\[
x = \frac{6288}{131} = 48
\]

### Answer
The bakery sold a total of **48 cakes** altogether.

Would you like more details on any part of the solution? Here are some related questions for further practice:

1. What would happen if the price of butter cakes increased by $2?
2. If the bakery wanted to increase its revenue by 10%, how many more cakes should it sell?
3. How much revenue comes specifically from the sale of almond cakes?
4. If the proportion of chocolate cakes sold increased, how would it affect total revenue?
5. What would be the impact on total revenue if the price of almond cakes dropped to $18?

**Tip:** Always ensure the sum of the fractions representing parts of a whole adds up to 1.

Learn how to solve a bakery's cake sales problem involving fractions and equations. Calculate the total number of cakes sold based on given prices and revenue. Suitable for students in Grades 5-7.

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