In 1983 the composition of pennies in the United States was changed due, in part, to the rising cost of copper. Pennies minted after 1983 weigh 2.50 grams while the earlier copper pennies weigh 3.11 grams. a. A roll of pennies contains 50 pennies. If a particular roll of pennies weighs 138.42 grams, how many of the earlier copper pennies and how many of the after 1983 pennies does this roll contain? b. Suppose you measure a roll of pennies on a scale which only reads the weight to the nearest gram. The scale says that the roll of pennies weighs 131 grams. What can you conclude about the pennies in this roll? What if the roll weights 134 grams? Explain.
The correct answer and explanation is:
Correct Answer:
a. Let the number of copper pennies (3.11 g each) be x, and the number of zinc pennies (2.50 g each) be y.
We know:
- x+y=50x + y = 50 (since a roll contains 50 pennies)
- 3.11x+2.50y=138.423.11x + 2.50y = 138.42 (total weight equation)
Substitute y=50−xy = 50 – x into the weight equation: 3.11x+2.50(50−x)=138.423.11x + 2.50(50 – x) = 138.42 3.11x+125−2.50x=138.423.11x + 125 – 2.50x = 138.42 0.61x=13.420.61x = 13.42 x=13.420.61=22x = \frac{13.42}{0.61} = 22
So, 22 copper pennies and 50−22=2850 – 22 = 28 zinc pennies.
b.
If a roll weighs 131 grams, let us calculate the possible combinations:
- If all 50 are zinc pennies:
50×2.50=12550 \times 2.50 = 125 grams - If all 50 are copper pennies:
50×3.11=155.550 \times 3.11 = 155.5 grams
So any roll of 50 pennies will weigh between 125 g and 155.5 g depending on the mix.
If a roll weighs 131 g, that is closer to the zinc-only weight. That suggests the majority of the pennies are post-1983 zinc pennies.
If a roll weighs 134 g, that is still between 125 and 155.5 g, but higher than 131 g, suggesting a larger proportion of copper pennies than in the 131 g case. You can estimate how many copper pennies are present using: 3.11x+2.50(50−x)=1313.11x + 2.50(50 – x) = 131 0.61x=6⇒x=60.61≈9.84≈100.61x = 6 \Rightarrow x = \frac{6}{0.61} \approx 9.84 \approx 10
So, about 10 copper pennies and 40 zinc pennies.
Thus, from the weight alone, you can estimate the mix of pennies.