Choosing a Reasonable Estimate for a School Store Purchase
A school store sells notebooks for $2.75 each and pencils for $0.50 each. A class buys 24 notebooks and 36 pencils. Student A estimates the total by rounding $2.75 to $3 and $0.50 to $1, getting 24 × $3 + 36 × $1 = $108. Student B rounds $2.75 to $3 and keeps $0.50, getting 24 × $3 + 36 × $0.50 = $90. The actual total is $84. The teacher asks which estimate is more reasonable and why. The class has been working on rounding money to the nearest dollar and on using estimation to check reasonableness. Several students say Student A’s estimate is wrong only because the answer is not exact. The teacher wants students to distinguish between an estimate that is reasonable and one that is not, and to explain how rounding choices affect the estimate.
1. Compute the exact total and verify that it is $84.
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Notebooks: 24 × $2.75. Since 24 × 2 = 48 and 24 × 0.75 = 18, the total is $48 + $18 = $66. Pencils: 36 × $0.50 = $18. Exact total: $66 + $18 = $84. Another way is 24 × 275 cents = 6,600 cents and 36 × 50 cents = 1,800 cents; 6,600 + 1,800 = 8,400 cents, which is $84.
2. Which estimate is more reasonable, Student A’s $108 or Student B’s $90? Explain using the rounding choices.
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Student B’s estimate of $90 is more reasonable. Student A rounded $0.50 up to $1, which doubles the price of every pencil. That creates an error of 36 × $0.50 = $18, so the estimate is $18 too high. Student B rounded only the notebook price, from $2.75 to $3, which overestimates by 24 × $0.25 = $6, giving $90. An estimate of $90 is close to $84 and is high, which is appropriate for a purchase check. Student A’s $108 is $24 too high and is not a useful estimate for deciding whether $84 is reasonable.
3. How could a student decide quickly whether $84 is reasonable without computing the exact total?
Read the worked answer
Use friendly numbers that stay close to the actual prices. Round $2.75 to $3 and $0.50 to $0.50. Then 24 × $3 = $72 and 36 × $0.50 = $18, so the estimate is $90. Since both rounded prices are at or above the actual prices, the exact total should be at or below $90. Also, 24 × $2.75 is close to 24 × $3 = $72, and 36 × $0.50 = $18, so $84 is plausible. A reasonable estimate should be close to the exact value and should not change one quantity by a large factor, as Student A did by doubling the pencil price.