Illustrative Mathematics (2024)

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Task

The price of copper fluctuates. Between 2002 and 2011, there were times when its price was lower than \$1.00 per pound and other times when its priace was higher than \$4.00 per pound. Copper pennies minted between 1962 and 1982 are 95% copper and 5% zinc by weight, and each penny weighs 3.11 grams. At what price per pound of copper does such a penny contain exactly one cent worth of copper? (There are 454 grams in one pound.)

IM Commentary

Comment 1:

Pennies have a monetary face value of one cent, but they are made of material that has a market value that is usually different. It is the value of the materials that requires attention in this problem. While it is interesting to compare the face value with the value of the materials, this does not have any bearing on the calculations. Interference between these two notions of value is a possible area of difficulty for some students.

Comment 2:

The number of grams of copper in one penny is 95% of $3.11 = (0.95)(3.11) = 2.9545$. The number 2.9545 is the numerical part of a rate with dimension “grams of copper per penny.” This rate appears in calculations in the form$$\frac{2.9545 \quad \text{grams copper}}{1 \quad \text{penny}}. \tag{1}$$If I multiply this by some number of pennies, I get an amount of copper. There are several other rates in this problem. One is the price of copper in dollars per pound, a rate that varies with time. The number of grams per pound can also be viewed as a rate: multiply a number of pounds by this number to get the number of grams. (The rate of 454 grams per pound given in the problem statement is an approximation. A more exact number is 453.5924.)

Comment 3:

This comment and the next one concern the motivation for Solution 2. Since the price of copper is given as a rate with dimension “dollars per pound of copper,” it is natural to convert the rate in (1) to a rate that involves pounds:$$\frac{2.9545 \quad \text{grams copper}}{1 \quad \text{penny}} =\frac{2.9545 \quad \text{grams copper}}{1 \quad \text{penny}} \cdot \frac{1 \quad \text{pound}}{454 \quad \text{grams}} =\frac{2.9545 \quad \text{pounds copper}}{454 \quad \text{pennies}}.$$Since $2.9545/454 \approx 0.0065$, we conclude that 100 pennies contain (about) 0.65 pounds copper. Thus, at any time when 0.65 pounds of copper sells for a dollar, then a penny contains one cent worth of copper. This is not an answer to the original question. We could continue to develop this line of reasoning to get to an answer, but instead we will look for other ways to display what we know in order to see if there is a more direct route to a solution.

Comment 4:

We can invert expression (1) to get the rate of pennies per gram copper. Following the motivation in Comment 3, we convert to pennies per pound copper:$$\frac{1 \quad \text{penny}}{2.9545 \quad \text{grams copper}} =\frac{1 \quad \text{penny}}{2.9545 \quad \text{grams copper}} \cdot \frac{454 \quad \text{grams}}{1 \quad \text{pound}} \approx\frac{154 \quad \text{pennies}}{1 \quad \text{pound copper}}.$$There is one pound of copper in 154 pennies, so when the price of copper is at \$1.54 per pound a penny contains one cent worth of copper. This solution is arrived at by attempting to display the given information in new ways, and staying alert the the possible interpretations of the expressions we see when we do that. The solution comes not from following a procedure, but by using simple procedures to look around, and remaining sensitive to the meaning.

Solutions

Solution:Solution 1

Let us find the value of the copper in one penny as a function of $x$, where $x$ is the price of copper in dollars per pound. There are $(0.95)(3.11) \approx 2.95$ grams of copper in one penny. Thus,$$\begin{align}\text{value of copper per penny} &\approx \frac{x \quad \text{dollars}}{1 \quad \text{pound copper}} \cdot \frac{1 \quad \text{pound}}{454 \quad \text{grams}} \cdot \frac{2.95 \quad \text{grams copper}}{1 \quad \text{penny}} \\&= x \cdot \frac{2.95 \quad \text{dollars}}{454 \quad \text{pennies}}\end{align}$$

We are looking for the value of $x$ that makes the following true:

$$0.01 = x \cdot \frac{2.95}{454}.$$

This is $x = \displaystyle \frac{4.54}{2.95} \approx 1.54$. In other words, if copper sells for \$1.54 per pound, then a penny contains (very close to) one cent worth of copper.

Solution:Solution 2

The amount of copper in a penny is (95% of 3.11 grams) $\approx$ 2.95 grams. The amount of copper in a pound of copper is 454 grams. Therefore, the number of pennies in a pound of copper is$$\frac{\text{grams copper in a pound}}{\text{grams copper in a penny}} = \frac{454}{2.95} \approx 154.$$Accordingly, if the price of copper is $1.54 per pound, the value of the copper in a penny is equal (very nearly) to one cent.