Recycling Drive Totals and a Suspicious Flyer
The environmental club at Riverside Middle School is preparing a report on its fall recycling drive. In October, students collected 4.8 × 10^5 grams of paper and 3.6 × 10^4 grams of aluminum. The club sponsor writes on a flyer: 'October total: 8.4 × 10^5 grams - a new record!' A student notices that the aluminum amount is less than one-tenth of the paper amount and questions the total. For comparison, September's drive collected 3.9 × 10^5 grams of paper and 2.7 × 10^4 grams of aluminum. The club's two-month goal is 1.0 × 10^6 grams of recyclables combined. The sponsor also wants to estimate next month's collection if 600 students each bring 2.5 × 10^3 grams. Before the report goes to the principal, the club needs to correct the flyer, decide whether the two-month goal was met, and judge whether the next-month estimate is reasonable.
1. Correct the October total and explain whether the flyer's 8.4 × 10^5 grams could be right. Identify the likely error.
Read the worked answer
Write both October amounts with the same power of 10: 4.8 × 10^5 + 3.6 × 10^4 = 4.8 × 10^5 + 0.36 × 10^5 = 5.16 × 10^5 grams, or 516,000 grams.
The flyer's 8.4 × 10^5 is not plausible: paper alone is 480,000 grams and aluminum is 36,000 grams, so the total must be between 480,000 and 520,000 grams. The likely error is adding the coefficients 4.8 + 3.6 = 8.4 while keeping 10^5, without first converting 3.6 × 10^4 to 0.36 × 10^5.
Rounded to two significant figures, the correct total is about 5.2 × 10^5 grams, so the 'more than half a million' part is true, but the coefficient is wrong.
2. Did the club meet its two-month goal of 1.0 × 10^6 grams? Show both monthly totals and the comparison.
Read the worked answer
September: 3.9 × 10^5 + 2.7 × 10^4 = 3.9 × 10^5 + 0.27 × 10^5 = 4.17 × 10^5 grams.
October: 5.16 × 10^5 grams.
Combined: 4.17 × 10^5 + 5.16 × 10^5 = 9.33 × 10^5 grams, or 933,000 grams.
The goal is 1.0 × 10^6 = 1,000,000 grams. Since 933,000 < 1,000,000, the club did not meet the goal. It is short by 67,000 grams, or 6.7 × 10^4 grams. The club reached 93.3% of its goal, so a statement such as 'we nearly reached one million grams' is accurate.
3. The sponsor estimates that 600 students each bringing 2.5 × 10^3 grams will meet a 1.5 × 10^6 gram target. Evaluate this estimate, including whether the result is exact or approximate and whether it leaves a margin.
Read the worked answer
600 × 2.5 × 10^3 = (6 × 10^2)(2.5 × 10^3) = 15 × 10^5 = 1.5 × 10^6 grams. So the estimate exactly equals the target on paper.
That does not leave a safety margin: if fewer than 600 students participate, or if each brings slightly less than 2.5 × 10^3 grams, the total falls below 1.5 × 10^6. Also, 2.5 × 10^3 grams is 2.5 kilograms, about 5.5 pounds per student, which is a large but possible amount for a drive.
The estimate is arithmetically correct but should be treated as a best-case scenario, not a guaranteed result.