Reasonable Answers with Decimals: Homework

Estimate first, solve second, and then ask whether the answer makes sense. A claim that is ten times too big is not a rounding error.

Part A — Estimate, Solve, Check / 9

Every price, weight and bill here is made up for practice. Estimate first and write it down, then find the exact answer, then say whether the two agree.

1.
You have $60. You want 3 shirts at $14.95 each and 2 hats at $9.50 each. Estimate to see whether $60 is enough, then find the exact total.Answer
2.
A truck can carry 25 kg. 12 bags of rice weigh 1.9 kg each. Estimate, then find the exact weight. Does it fit?Answer
3.
A car holds 12.5 gallons and goes 28.4 miles on each gallon. A friend says it can go 355 miles on a full tank. Estimate to check, then find the exact distance.Answer
4.
A phone case costs $8.25. How many can a store buy with $100? Estimate, then find the number it can really buy.Answer
5.
Four friends share a $73.60 bill equally. Estimate each share, then find it.Answer
6.
Your calculator says 6.8 × 4.2 = 285.6. Estimate to decide whether that is reasonable. If not, find the real product.Answer
7.
Which skill? Skill A (U2-L3, estimating) finds a quick answer when you do not have one. Skill B (this lesson, reasonable answers) checks an answer somebody already gave you. A calculator says 5.9 × 3.1 = 182.9. Which skill are you using, and is the answer reasonable? What should it be?Answer
8.
You buy 7 adult tickets at $12.75 and 5 child tickets at $8.25, and you have $150. Estimate whether that is enough, then find the exact total.Answer
9.
A recipe uses 0.75 cup of sugar for 12 cookies. How much sugar for 36 cookies? Estimate first.Answer

Part B — Two Marathon Records / 5

Real times. A marathon is 42.195 km, which is 42,195 meters. Times are hours:minutes:seconds. Both were world records when they were set.

1.
The men's marathon world record, twice:
Runner Race Time
Eliud Kipchoge Berlin, 2022 2:01:09
Kelvin Kiptum Chicago, 2023 2:00:35
a. Change Kipchoge's time into seconds.Answer
b. Change Kiptum's time into seconds.Answer
c. How many seconds faster was Kiptum?Answer
d. Estimate Kiptum's average speed in meters per second with 42,000 ÷ 7,000. Then find it to the nearest tenth.Answer
e. A student says Kiptum averaged 58 meters per second. Use your estimate to say whether that is reasonable.Answer

Runner	Race	Time
Eliud Kipchoge	Berlin, 2022	2:01:09
Kelvin Kiptum	Chicago, 2023	2:00:35

Part C — Critical Thinking / 2

Think it through, then write your answer and one sentence about how you knew.

1.
You estimate a shopping trip at about $40. One day the register says $38.46. Another day it says $84.60. Which would you question, and how close does an answer need to be to its estimate before you trust it?Answer
2.
A friend says a 3.9 meter board cut into 0.6 meter shelves makes 65 shelves. Use one quick estimate to show it cannot be right, then give the real number of whole shelves.Answer

Part D — Make Up a Problem / 1

Write your own, then trade with someone or solve it yourself.

1.
Write a shopping problem that has a claimed total, where a quick estimate shows the claim cannot be right. Give your estimate and the real total.Answer

Total / 17

Graded by

Notes

Answer key

Part A — the exact answer and what the estimate said.

$63.85. It is not enough. The estimate 3 × 15 + 2 × 10 = 65 already said so.
22.8 kg, so yes, it fits. The estimate was 12 × 2 = 24.
355 miles. The estimate 12 × 30 = 360 agrees, so the claim is reasonable.
12 cases. 100 ÷ 8.25 is about 12.1, and 12 cases cost $99.00. A 13th would not be paid for.
$18.40. The estimate was 72 ÷ 4 = 18.
28.56. The estimate 7 × 4 = 28 shows 285.6 has the decimal point in the wrong place.
18.29 is the real answer. This is Skill B. 6 × 3 = 18, so 182.9 is not reasonable; its decimal point is in the wrong place.
$130.50. Yes, $150 is enough. The estimate was 7 × 13 + 5 × 8 = 131.
2.25 cups. 36 is 3 times 12, and 3 × about 1 cup is about 3.

Part B — the answers.

a. 7,269 seconds b. 7,235 seconds c. 34 seconds d. 5.8 meters per second. The estimate was 6. e. No. The estimate is about 6, so 58 is ten times too big. The decimal point is in the wrong place.

Part C — one correct answer, and what a good explanation says.

The $84.60. It is about double the estimate, and rounding cannot cause that. $38.46 is close to $40, which is what rounding to easy numbers would give.
About 4 ÷ 0.5 = 8, so 65 is ten times too big. The real answer is 6 whole shelves (3.9 ÷ 0.6 = 6.5, and you cannot make half a shelf).

Part D — what a good problem looks like.

Answers vary. A good one has a claim that is far off, and an estimate that shows it quickly, such as: three items at $9.95, $4.10 and $6.05 with a claimed total of $40.10. The estimate 10 + 4 + 6 = 20 shows it is too high. The real total is $20.10.