A bag contains pennies, nickels, dimes, and quarters. There are 50 coins in all. Of the coins, 16% are pennies and 34% are dimes. There are more nickels than pennies. How much money does the bag contain?
The Correct Answer and Explanation is:
To find how much money is in the bag, we start by determining the number of each type of coin using the given percentages and conditions.
Step 1: Total number of coins
There are 50 coins in total.
Step 2: Number of pennies
16% of 50 coins are pennies:0.16×50=8 pennies0.16 \times 50 = 8 \text{ pennies}0.16×50=8 pennies
Step 3: Number of dimes
34% of 50 coins are dimes:0.34×50=17 dimes0.34 \times 50 = 17 \text{ dimes}0.34×50=17 dimes
Step 4: Number of remaining coins
So far we have:
- Pennies: 8
- Dimes: 17
Total so far:
8+17=25 coins8 + 17 = 25 \text{ coins}8+17=25 coins
Remaining coins:50−25=25 coins50 – 25 = 25 \text{ coins}50−25=25 coins
These remaining coins are nickels and quarters.
Step 5: Number of nickels
We are told there are more nickels than pennies, and there are 8 pennies.
Let us suppose there are 9 nickels.
Then the number of quarters would be:25−9=16 quarters25 – 9 = 16 \text{ quarters}25−9=16 quarters
Let us now compute the total value of all coins.
Step 6: Calculate total value
- Pennies: 8 × $0.01 = $0.08
- Dimes: 17 × $0.10 = $1.70
- Nickels: 9 × $0.05 = $0.45
- Quarters: 16 × $0.25 = $4.00
Add them together:0.08+1.70+0.45+4.00=6.230.08 + 1.70 + 0.45 + 4.00 = \boxed{6.23}0.08+1.70+0.45+4.00=6.23
Final Answer:
$6.23
Explanation
To find the total value of coins in the bag, we begin with the total number of coins, which is 50. The problem states that 16 percent of these coins are pennies. By multiplying 50 by 0.16, we find that there are 8 pennies. Next, we are told that 34 percent are dimes. Multiplying 50 by 0.34 gives us 17 dimes. Together, these account for 25 of the 50 coins, leaving 25 coins that are either nickels or quarters.
An important clue is that there are more nickels than pennies. Since we have 8 pennies, we must have at least 9 nickels. Let us try 9 nickels. That would mean the remaining coins are quarters: 25 minus 9 equals 16 quarters.
We then calculate the value of each type of coin. Pennies are worth one cent each, so 8 pennies are worth 8 cents. Dimes are ten cents each, so 17 dimes equal $1.70. Nickels are five cents each, and 9 nickels total 45 cents. Quarters are worth twenty-five cents each, and 16 quarters amount to $4.00.
When we add these amounts—$0.08, $1.70, $0.45, and $4.00—we get a total of $6.23. All the conditions are satisfied: the total number of coins is 50, the percentages are correct, and there are more nickels than pennies. Therefore, the final amount of money in the bag is $6.23.
