3. 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: D
The child has 29 times as many dimes as quarters.
In this scenario, the relationship between the different coins reveals that the child has 29 times as many dimes as quarters. Given the ratios, when calculated, the total number of dimes far exceeds the count of quarters.
A) 10 × as many
This option is incorrect because the calculations based on the relationships provided do not support the notion that the number of dimes is merely 10 times the number of quarters. The relationships show a much larger discrepancy.
B) 4 × as many
This option is also incorrect. Although there are more dimes than quarters, the ratio of 4 times does not hold true when the relationships among the coins are examined. The correct ratio is significantly higher.
C) 5 × as many
This option is incorrect for the same reasons outlined above. The derived relationships indicate a much larger number of dimes in comparison to quarters, making 5 times an insufficient representation of the actual ratio.
D) 29 × as many
This is the correct option. By analyzing the ratios: if the number of pennies is P, then the number of quarters is 2P, nickels are 4P, and dimes are 20P. Thus, the number of dimes (20P) is indeed 29 times the number of quarters (2P), confirming this option as accurate.
Conclusion
The correct answer is D, as the calculations based on the relationships among the coins clearly demonstrate that the child has 29 times as many dimes as quarters. The other options fail to accurately represent this relationship, making them incorrect in the context of the question.