Sasa had some 20-cent and 50-cent coins. Let x be the total number of coins. Then, 7/8 x were 20-cent coins, and the rest, 1 - 7/8 = 1/8 x, were 50-cent coins. After Sasa spent $72.50 worth of 50-cent coins, which is equivalent to 72.50 / 0.50 = 145 50-cent coins, and 5/7 of the 20-cent coins, she had 2/7 x coins left. Find the total amount of money Sasa left.

To solve this problem, let's break it down step by step.

### Step 1: Define the variables
Let:
- $ x $ be the total number of coins.
- $ \frac{7}{8}x $ be the number of 20-cent coins.
- $ \frac{1}{8}x $ be the number of 50-cent coins.

### Step 2: Sasa's expenditures
Sasa spent:
- **145 50-cent coins** (as given in the problem).
- **$\frac{5}{7}$ of the 20-cent coins.**

### Step 3: Set up the equation for the 50-cent coins
Initially, the number of 50-cent coins was $ \frac{1}{8}x $. After spending 145 50-cent coins, the remaining number of 50-cent coins is:
\[
\frac{1}{8}x - 145
\]

### Step 4: Set up the equation for the 20-cent coins
Initially, Sasa had $ \frac{7}{8}x $ 20-cent coins. After spending $ \frac{5}{7} $ of them, she had $ \frac{2}{7} $ of her 20-cent coins left. So the number of remaining 20-cent coins is:
\[
\frac{2}{7} \times \frac{7}{8}x = \frac{2}{8}x = \frac{1}{4}x
\]

### Step 5: Remaining coins
According to the problem, after spending some of her coins, Sasa had $ \frac{2}{7}x $ coins left. These are the remaining 20-cent and 50-cent coins combined, so:
\[
\frac{1}{4}x + \left(\frac{1}{8}x - 145\right) = \frac{2}{7}x
\]

### Step 6: Solve for $ x $
Now, solve the equation $ \frac{1}{4}x + \frac{1}{8}x - 145 = \frac{2}{7}x $.

1. Find a common denominator for the fractions:
\[
\frac{1}{4}x = \frac{2}{8}x
\]
So the equation becomes:
\[
\frac{2}{8}x + \frac{1}{8}x - 145 = \frac{2}{7}x
\]
\[
\frac{3}{8}x - 145 = \frac{2}{7}x
\]

2. Eliminate the fractions by multiplying the entire equation by 56 (the least common denominator of 8 and 7):
\[
56 \times \left(\frac{3}{8}x\right) - 56 \times 145 = 56 \times \left(\frac{2}{7}x\right)
\]
\[
21x - 8120 = 16x
\]

3. Rearrange to isolate $ x $:
\[
21x - 16x = 8120
\]
\[
5x = 8120
\]
\[
x = \frac{8120}{5} = 1624
\]

So, the total number of coins $ x $ is 1624.

### Step 7: Calculate how much money Sasa has left
Now, calculate the remaining amount of money:
1. The remaining number of 20-cent coins is $ \frac{1}{4}x = \frac{1}{4} \times 1624 = 406 $.
   - The value of the remaining 20-cent coins is $ 406 \times 0.20 = 81.20 $ dollars.

2. The remaining number of 50-cent coins is $ \frac{1}{8}x - 145 = \frac{1}{8} \times 1624 - 145 = 203 - 145 = 58 $.
   - The value of the remaining 50-cent coins is $ 58 \times 0.50 = 29 $ dollars.

### Step 8: Total amount of money Sasa has left
The total amount of money Sasa has left is:
\[
81.20 + 29 = 110.20 \text{ dollars}.
\]

Thus, Sasa has **$110.20** left.

---

Would you like to explore further details on any specific step? Here are some related questions:

1. How to solve equations with multiple fractions?
2. How are fractions simplified in linear equations?
3. How to calculate the total number of coins from fractions of a total?
4. How to handle multiple variables in coin-related word problems?
5. What is the importance of setting up correct relationships in word problems?

**Tip:** Always define all variables clearly when solving a word problem—it helps to set up equations systematically.

Learn how to solve a real-world word problem involving 20-cent and 50-cent coins using linear equations and fractional operations. This step-by-step solution walks through solving for the total number of coins and finding how much money is left. Suitable for Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation