During a shopping spree, Betty spent 1/5 of her money on a pair of skirt and 1/3 of the remaining money of 4 blouses and a belt. She spent the rest of her money on a bottle of perfume. The belt cost 1/4 as much as the bottle perfume. If she spent $21.50 on each blouse, how much did she pay for the bottle of perfume?

Let's break down the problem step by step.

### Step 1: Define variables
Let $ x $ represent the total amount of money Betty initially had.

### Step 2: Money spent on the skirt
She spent $ \frac{1}{5} $ of her money on the skirt, which leaves her with $ \frac{4}{5} $ of her total money.

### Step 3: Money spent on blouses and the belt
After spending on the skirt, she spent $ \frac{1}{3} $ of the remaining money on 4 blouses and a belt. The remaining money is $ \frac{4}{5}x $, so the amount spent on the blouses and the belt is:

\[
\frac{1}{3} \times \frac{4}{5}x = \frac{4}{15}x
\]

Thus, she has $ \frac{2}{3} \times \frac{4}{5}x = \frac{8}{15}x $ left after buying the blouses and the belt.

### Step 4: Cost of blouses and belt
The cost of the 4 blouses is:

\[
4 \times 21.50 = 86 \, \text{dollars}
\]

Let the cost of the belt be $ b $, so the total cost of the blouses and the belt is $ 86 + b $. From Step 3, we know the total spent on the blouses and belt is $ \frac{4}{15}x $, so we have the equation:

\[
86 + b = \frac{4}{15}x \quad \text{(Equation 1)}
\]

### Step 5: Money spent on the perfume
The rest of her money, $ \frac{8}{15}x $, was spent on a bottle of perfume. The belt cost $ \frac{1}{4} $ as much as the perfume, so if the cost of the perfume is $ p $, then the cost of the belt is:

\[
b = \frac{1}{4}p
\]

Substitute $ b = \frac{1}{4}p $ into Equation 1:

\[
86 + \frac{1}{4}p = \frac{4}{15}x \quad \text{(Equation 2)}
\]

### Step 6: Relationship between the remaining money and perfume
We also know that the remaining money $ \frac{8}{15}x $ was spent entirely on the perfume, so:

\[
p = \frac{8}{15}x \quad \text{(Equation 3)}
\]

### Step 7: Solve the system of equations
Now, we solve Equation 2 and Equation 3. Substitute $ p = \frac{8}{15}x $ from Equation 3 into Equation 2:

\[
86 + \frac{1}{4} \times \frac{8}{15}x = \frac{4}{15}x
\]

Simplify the equation:

\[
86 + \frac{2}{15}x = \frac{4}{15}x
\]

Now, subtract $ \frac{2}{15}x $ from both sides:

\[
86 = \frac{2}{15}x
\]

Multiply both sides by $ \frac{15}{2} $:

\[
x = 86 \times \frac{15}{2} = 645
\]

### Step 8: Calculate the cost of the perfume
Now that we know $ x = 645 $, substitute this value into Equation 3 to find $ p $:

\[
p = \frac{8}{15} \times 645 = 344 \, \text{dollars}
\]

### Final Answer:
The cost of the bottle of perfume is **$344**.

Would you like further details or have any other questions? Here are some related questions to expand on this topic:

1. How would the calculations change if the belt cost half as much as the perfume?
2. What if she had spent $ \frac{1}{4} $ of her remaining money on blouses and belt instead of $ \frac{1}{3} $?
3. How much would Betty have left if the perfume was cheaper, say $200?
4. What if Betty also bought a pair of shoes for $100 after purchasing the perfume?
5. How much total money did Betty spend on everything combined?

**Tip**: Always label unknown quantities clearly with variables when solving word problems to keep track of each part of the equation easily!

During a shopping spree, Betty spent 1/5 of her money on a pair of skirt and 1/3 of the remaining money on 4 blouses and a belt. She spent the rest of her money on a bottle of perfume. The belt cost 1/4 as much as the bottle of perfume. If she spent $21.50 on each blouse, how much did she pay for the bottle of perfume?

Solve this algebraic word problem where Betty spends part of her money on a skirt, blouses, a belt, and perfume. Learn how to use proportions, linear equations, and problem-solving strategies to determine the cost of the perfume. Includes step-by-step solution for students in Grades 8-10.

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