A bank teller is asked to assemble 1 sets of coins for his...

A bank teller is asked to assemble $\$ 1$ sets of coins for his clients. Each set is made up of three quarters, one nickel, and two dimes. The masses of the coins are quarter, $5.645 \mathrm{g} ;$ nickel, $4.967 \mathrm{g} ;$ and 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 grams) of the assembled sets of coins?

Key Concepts

Unit Conversion

Unit conversion involves changing quantities from one measurement unit to another to maintain consistency in calculations. In problems like this, converting kilograms to grams ensures that the masses of coins and the total mass available are in the same unit for accurate arithmetic operations.

Limiting Resource Analysis

In assembly or allocation problems, the limiting resource (or limiting reagent in chemistry terms) is the one that runs out first and thereby limits the number of complete sets or products that can be assembled. Here, by calculating how many sets can be made from each type of coin and then choosing the smallest number, one identifies which coin type restricts the total count of complete sets.

Integer Division and the Floor Function

When resources are divided to form complete sets, each set must use whole coins, meaning any fractional result must be rounded down to the nearest integer. This use of integer division (or the floor function) is essential to determine the maximum number of complete sets that can be assembled without having to split any coin.

Aggregate Mass Calculation

Once the number of complete sets is determined, computing the total mass involves summing the masses of individual coins per set and then multiplying by the number of sets. This step reinforces the idea of linear scaling in measurements where each set contributes a fixed mass to the overall total.

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