Question

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 \mathrm{~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?

Short answer

The maximum number of sets is 1,724, and the total mass is approximately 45,718.9 grams.

Step-by-step solution

Understanding Given Information

Each set consists of 3 quarters, 1 nickel, and 2 dimes. The given total masses are 33.871 kg for quarters, 10.432 kg for nickels, and 7.990 kg for dimes. Convert these masses to grams: 33,871 g for quarters, 10,432 g for nickels, and 7,990 g for dimes.

Calculate Number of Coins

Find out how many coins of each type can be obtained: for quarters, divide 33,871 g by 5.645 g (mass of one quarter), for nickels, divide 10,432 g by 4.967 g (mass of one nickel), and for dimes, divide 7,990 g by 2.316 g (mass of one dime). This gives approximately: - Quarters: 33,871 / 5.645 ≈ 6,000.17 - Nickels: 10,432 / 4.967 ≈ 2,100.65 - Dimes: 7,990 / 2.316 ≈ 3,449.14.

Determine Sets Based on Coins

Calculate the maximum possible sets based on the number of each coin: - For quarters, since each set needs 3 quarters, you can make 6,000 / 3 ≈ 2,000 sets. - For nickels, each set uses 1 nickel, so you can make 2,100 sets. - For dimes, each set requires 2 dimes, allowing for 3,449 / 2 ≈ 1,724.5 sets, round down to 1,724 sets.

Identify Limiting Factor for Sets

The limiting factor is the minimum number of sets possible from the available coins. From the above calculations, the limiting factor is the dimes, which allows for a maximum of 1,724 sets.

Calculate Total Mass of Assembled Sets

Calculate the total mass of 1,724 sets. Each set has a mass calculated as: \[ 3(5.645) + 4.967 + 2(2.316) = 16.923 + 4.967 + 4.632 = 26.522 \text{ g} \]The total mass is then 1,724 sets multiplied by 26.522 g, which equals approximately 45,718.9 g.

Key concepts

Mass Calculation

Mass calculation is the key to determining how many sets of coins can be assembled. In this exercise, we start by converting the total mass of coins from kilograms to grams. This conversion is essential because the mass of each individual coin is given in grams.
If we take 33.871 kg of quarters, we convert it to 33,871 grams. Similarly, for nickels and dimes, we convert 10.432 kg and 7.990 kg to 10,432 grams and 7,990 grams, respectively.
Next, determine how many coins of each type are available. For quarters, divide the total mass of quarters (33,871 g) by the mass of one quarter (5.645 g). This process helps us find that there are approximately 6,000 quarters on hand.
This method needs to be applied in the same way for the other coins to find 2,100 nickels and 3,449 dimes. This systematic approach of mass conversion followed by division allows us to

• find the count of available coins
• lay the groundwork for further stoichiometric calculations

Limiting Reactant

Understanding the concept of the limiting reactant is crucial in stoichiometry, even when dealing with a non-chemical example like assembling coin sets. Here, the limiting reactant determines the maximum number of complete sets that can be prepared based on available resources.
Each set uses a specific number of quarters, nickels, and dimes. Thus, to find the limiting reactant, calculate the maximum number of sets each type of coin can contribute.
For quarters, divide the total number (6,000) by 3, since each set needs three quarters. This means quarters could theoretically make 2,000 sets.
For nickels, only one is needed per set, permitting up to 2,100 sets from the available nickels.
For dimes, each set needs two, allowing approximately 1,724 sets from dimes.
Here, dimes become the limiting reactant as they restrict us from assembling more than 1,724 complete sets. This demonstrates how identifying limiting reactants helps allocate resources efficiently.

Coin Assembly

The process of coin assembly revolves around combining different coins in specific amounts to create complete sets. In this exercise, each set requires:

• 3 quarters
• 1 nickel
• 2 dimes

The goal is to construct as many complete sets as possible from a given mass of each coin. By calculating the potential sets each coin type can produce, we determine how all resources work together.
An interesting aspect of this task is that, even though we have sufficient quarters and nickels to make over 2,000 sets, the actual assembly is limited by the fewest resources available—which, in this case, are the dimes. In practical scenarios, such limitations require strategic planning to maximize output.

Dimensional Analysis

Dimensional analysis is a mathematical technique used to convert and compare different units of measure. It's an essential tool in stoichiometry, particularly when dealing with mass calculations and unit conversions.
For this exercise, dimensional analysis involves converting the weights of coins from kilograms to grams. We know 1 kilogram equals 1,000 grams, so this conversion helps us ensure all coin calculations are consistent.
After converting, we use dimensional analysis again in determining how many coins each total mass yields. By dividing the available total mass in grams by the mass per individual coin, we derive a count for each type of coin.
This technique is fundamental not only in chemical equations but across various fields requiring precise calculations and conversions. It ensures clarity and accuracy in results, showing why dimensional analysis is such a powerful concept.