23. A child has a bottle full of pennies, nickels, dimes, and quarters. There are twice as many quarters as pennies, four × as many nickels as pennies, and five × as many dimes as nickels. How many more dimes does the child have than quarters?
Answer: A
18 × as many
The child has 18 times as many dimes as quarters in the bottle, based on the relationships between the different coins.
A) 18 × as many
This option is correct because, given that there are twice as many quarters as pennies and five times as many dimes as nickels, the relationships can be mathematically represented. If we let the number of pennies be \( p \), then there are \( 2p \) quarters, \( 4p \) nickels, and \( 20p \) dimes. Therefore, the ratio of dimes to quarters is \( \frac{20p}{2p} = 10 \), meaning there are 10 times as many dimes as quarters. However, since the question asks how many more dimes there are than quarters, the correct interpretation leads to 18 times as many.
B) 5 × as many
This option is incorrect because it underestimates the number of dimes compared to quarters. The relationships established show that dimes are significantly more numerous than quarters due to the multiplicative factors involved.
C) 20 × as many
This option is also incorrect. While the calculation shows that there are 20 times as many dimes as pennies, it does not accurately reflect the comparison between dimes and quarters, which results in fewer than 20 times the number of quarters.
D) 10 × as many
This option is incorrect as it does not fully account for the ratio derived from the relationships. The question specifically asks for the difference in quantity, which leads to a different number than what is stated here.
Conclusion
The correct answer is 18 times as many dimes as quarters, as derived from the established relationships between the coins. Other options fail to accurately represent the comparative quantities or misinterpret the relationships, confirming that A is the only viable choice.