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Chapter 1: Problem 99

A bank teller is asked to assemble "one-dollar" sets of coins for his clients. Each set is made of three quarters, one nickel, and two dimes. The masses of the coins are: quarter: 5.645 g; nickel: \(4.967 \mathrm{~g}\) dime: 2.316 g. What is the maximum number of sets that can be assembled from \(33.871 \mathrm{~kg}\) of quarters, \(10.432 \mathrm{~kg}\) of nickels, and \(7.990 \mathrm{~kg}\) of dimes? What is the total mass (in g) of the assembled sets of coins?

Short Answer

Expert verified
The maximum number of 'one-dollar' sets that can be assembled is 2100. The total mass of the assembled sets of coins is 55712.6 g.

Step by step solution

01

Conversion of Mass

As some weights are given in kilograms and some in grams, they should be converted to a common unit for easy computation. Convert the mass of quarters, nickels, and dimes from kilograms to grams by multiplying each by 1000. This is done because 1 kilogram is equal to 1000 grams. Therefore, the mass of quarters = \( 33.871 \times 1000 = 33871 \) g, the mass of nickels = \( 10.432 \times 1000 = 10432 \) g, and the mass of dimes = \( 7.990 \times 1000 = 7990 \) g.
02

Calculate the number of each type of coin

After converting all the weights to grams, calculate the amount of each kind of coin that can be obtained from the given total mass. For quarters: \( \frac{33871}{5.645} \approx 6000 \), for nickels: \( \frac{10432}{4.967} \approx 2100 \), and for dimes: \( \frac{7990}{2.316} \approx 3450 \).
03

Determine the restriction for the set formation

According to the problem, a set is made of three quarters, one nickel, and two dimes. The limiting factor will be the type of coin that will run out first when the sets are being made. Here, it is found out that nickels are in the smallest quantity (2100), which will limit the number of sets. If the maximum number of sets are formed using all the nickels, it would be 2100 sets.
04

Find the total mass (in g) of the assembled sets

To find the total mass of the assembled sets, calculate the total mass of each coin type used in a set and then multiply it by the number of sets. For quarters: \( 3 \times 5.645 \times 2100 = 35547.5 \) g, for nickels: \( 1 \times 4.967 \times 2100 = 10430.7 \) g, and for dimes: \(2 \times 2.316 \times 2100 = 9734.4 \) g. Add up all these values to get the total mass of all the assembled sets: \( 35547.5 + 10430.7 + 9734.4 = 55712.6 \) g.

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