Question

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: \(5.645 \mathrm{~g}\); nickel: \(4.967 \mathrm{~g}\); 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 g) of this collection of coins?

Short answer

The maximum number of 'one-dollar' sets of coins that can be formed and the total mass of this collection can be found using the mentioned steps.

Step-by-step solution

Calculate the number of possible sets each type of coin can form

To begin, we will first calculate how many sets each type of coin can form based on their total mass, and the mass of one coin. We are given total mass of quarters, nickels, and dimes. We will divide the total mass of each type by the mass of one coin to get the number of sets each can form. The formulas are as follows: For quarters: \(\frac{33.871 \times 10^3}{5.645 \times 3}\) (as 3 quarters are needed for a set) For nickels: \(\frac{10.432 \times 10^3}{4.967}\) (as only one nickel is needed for a set) For dimes: \(\frac{7.990 \times 10^3}{2.316 \times 2}\) (as 2 dimes are needed for a set)

Find the maximum number of sets that can be assembled

After getting the number of possible coin sets that can be formed by each type, we need to find the maximum number of sets. This will be the smallest number among the sets of quarters, nickels, and dimes as we need all types of coins to form a set.

Calculate the total mass of the collection of coins

The total mass of the collection of coins can be calculated by multiplying the number of sets by the mass of one set. Mass of one set is calculated as follows: (3 quarters \(\times\) mass of one quarter) + (1 nickel \(\times\) mass of one nickel) + (2 dimes \(\times\) mass of one dime). So, total mass will be `number of sets \(\times\) mass of one set`

Key concepts

coin sets

Coin sets are combinations of different types of coins assembled to meet a specific value, like a dollar. In our exercise, each "one-dollar" set comprises a mixture of U.S. coins: three quarters, one nickel, and two dimes. Each type of coin has a specific role and contributes to the overall target sum. Understanding coin sets is crucial in various real-life scenarios, such as banking or collecting. A systematic approach is essential when creating multiple sets to ensure the correct number and type of coins are used.

• "One-dollar" set formation: three quarters, one nickel, and two dimes.
• Purpose: To meet or approximate a specific monetary value using available coin resources.

quarters

A quarter is a U.S. coin worth 25 cents, making it one-fourth of a dollar. In terms of the mass, each quarter weighs approximately 5.645 grams. When forming "one-dollar" sets, three quarters are used, contributing 75 cents to the combination. When tasked with assembling sets, it's important to calculate how many complete sets can be assembled based on the available quarters.
In this exercise, for instance, you start by calculating the number of sets you can make from the total mass of quarters available. Divide the total mass by the mass required for three quarters to find out just how many sets are possible. Only then can you determine the contribution of quarters to the overall number of possible "one-dollar" sets.

nickels

Nickels are U.S. coins valued at 5 cents, and they play a special role in rounding out coin sets to precise values. Each nickel weighs around 4.967 grams. In a typical "one-dollar" coin set, one nickel is added to the mix, which, though modest in value, is crucial for reaching the targeted dollar sum.

When determining how many sets can be created, you must consider the total mass of all the nickels on hand, divided by the mass of a single nickel. This calculation tells you the potential number of sets that can be formed just from your nickels, before considering quarters and dimes. The nickel's addition prevents the need for smaller denominations like pennies, completing the swift formation of a simple yet efficient currency set.

dimes

Dimes are tiny contributors, packing a value of 10 cents each, into the "one-dollar" set. They are relatively lightweight, with an individual mass of approximately 2.316 grams. Despite needing two dimes per set, their contribution is substantial, forming part of the easy addition necessary to achieve the dollar goal.

To calculate your dimes' potential number of sets, you must factor the mass of two dimes into your equations. As with the previous coins, take the total mass in hand and divide it by the combined mass of two dimes. This will show how many sets can be produced purely from dimes available. Dimes serve as a fine-tuning element, closing the gap towards the expected dollar value with ease.