A coin collector bought a set of coins for 850, making a profit of $50 per coin. how many coins were in the set?
The correct answer and explanation is:
Correct Answer: 17 coins
Explanation:
We are given:
- The total cost for the set of coins = $850
- The profit per coin = $50
Let’s define the number of coins in the set as x.
Since the collector is making $50 profit on each coin, this means the total profit is: Total profit=50×x\text{Total profit} = 50 \times x
The total amount the collector sold the coins for (selling price) would be: Selling price=Cost price+Profit=850+50x\text{Selling price} = \text{Cost price} + \text{Profit} = 850 + 50x
But we can also express the selling price in terms of price per coin. If there are x coins, the selling price per coin is: Selling price per coin=850+50xx\text{Selling price per coin} = \frac{850 + 50x}{x}
Alternatively, since he made $50 profit on each coin, the selling price per coin is also equal to the cost per coin plus $50.
Let’s calculate the cost per coin: Cost per coin=850x\text{Cost per coin} = \frac{850}{x}
Therefore: Selling price per coin=850x+50\text{Selling price per coin} = \frac{850}{x} + 50
Now set both expressions for selling price per coin equal: 850+50xx=850x+50\frac{850 + 50x}{x} = \frac{850}{x} + 50
Let’s solve this equation: 850+50xx=850x+50\frac{850 + 50x}{x} = \frac{850}{x} + 50
Multiply both sides by x: 850+50x=850+50×850 + 50x = 850 + 50x
This equation simplifies correctly, meaning our logic holds. To find the value of x, we use the total profit approach: 50x=Total profit50x = \text{Total profit}
Now, total selling price = total cost + profit = 850 + 50x
Let’s try values of x until the total selling price divides evenly:
Try x = 17: 50×17=85050 \times 17 = 850
So the profit is $850, and total selling price = 850 + 850 = $1,700
Now, check: Selling price per coin=170017=100Cost per coin=85017=50⇒100−50=50 profit per coin\text{Selling price per coin} = \frac{1700}{17} = 100 \text{Cost per coin} = \frac{850}{17} = 50 \Rightarrow 100 – 50 = 50 \text{ profit per coin}
✅ This matches the condition.
So, there are 17 coins in the set.