Estimating with Decimals: Homework

Round to easy numbers, do the quick arithmetic, and use the estimate to catch an answer that cannot be right.

Part A — Estimate It / 11

Round each number to its greatest place value, then do the easy arithmetic. For a list that bunches near one number, cluster. For division, pick numbers that divide evenly.

1.
Plot 2.7 and 5.4 and circle the whole number each is nearest. Then estimate 2.7 + 5.4.
Answer
2.
6.8 + 3.2 + 9.1Answer
3.
24.7 + 38.2 + 11.6Answer
4.
83.6 - 41.9Answer
5.
3.9 × 6.1Answer
6.
58.3 ÷ 7.8Answer
7.
41.7 + 38.2 + 40.5 + 39.6 (cluster)Answer
8.
$9.89 + $5.15 + $2.95Answer
9.
0.48 × 52Answer
10.
19.7 ÷ 4.1Answer
11.
612.4 - 387.9Answer

Part B — Don't Get Fooled / 6

Each of these has a trap. Think before you round.

1.
Estimate 3.2 × 0.48 by rounding each number to its greatest place. A student rounds 0.48 down to 0 and says the answer is about 0. Fix it.Answer
2.
A calculator says 8.3 × 4.9 = 4067. Use an estimate to decide where the decimal point really belongs.Answer
3.
Estimate 5.4 + 0.3 + 0.2 + 0.4 by rounding each number to its greatest place. Do NOT turn the small ones into zero.Answer
4.
Without multiplying, will 7.8 × 0.9 be more or less than 7.8? Why?Answer
5.
A clerk rings up $3.98, $6.02, $5.95 and $4.05 and says the total is $24.00. Estimate to see if that can be right.Answer
6.
Which skill? A friend buys items for $4.95, $2.10 and $7.89 and says the total is $29.94. Estimate the total yourself (Estimating, U2-L3), then say whether his answer is reasonable (judging an answer, U2-L10).Answer

Part C — Weighing Atoms and Counting Coins / 7

Real numbers. Atomic weights come from IUPAC and the coin widths from the U.S. Mint. Round to the nearest whole number first, then estimate.

1.
Atomic weights (the mass of one atom, in atomic mass units):
Atom Weight
Hydrogen 1.008
Carbon 12.011
Oxygen 15.999
Sodium 22.990
Chlorine 35.45
a. Water has 2 hydrogen atoms and 1 oxygen atom. Estimate its mass.Answer
b. Carbon dioxide has 1 carbon atom and 2 oxygen atoms. Estimate its mass.Answer
c. Table salt has 1 sodium atom and 1 chlorine atom. Estimate its mass.Answer
d. About how many times heavier is a chlorine atom than a hydrogen atom?Answer
2.
How wide each U.S. coin is, in millimeters:
Coin Width
Penny 19.05
Nickel 21.21
Dime 17.91
Quarter 24.26
a. Line up a penny, nickel, dime and quarter edge to edge. Estimate the length of the row.Answer
b. About how many quarters in a row would fit in 100 mm?Answer
c. About how much wider is a quarter than a dime?Answer

Atom	Weight
Hydrogen	1.008
Carbon	12.011
Oxygen	15.999
Sodium	22.990
Chlorine	35.45

Coin	Width
Penny	19.05
Nickel	21.21
Dime	17.91
Quarter	24.26

Part D — Critical Thinking / 2

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

1.
Round each number to the nearest whole number to estimate 4.4 + 4.4 and 3.6 + 4.4. Both estimates are the same. Find both exact sums. Which estimate is closer, and why?Answer
2.
Estimate 3.7 × 5.8 by rounding each factor to the nearest whole number. Without finding the exact answer, will the estimate be more or less than the exact product? Then check.Answer

Part E — Make Up a Problem / 1

Write your own, then work it.

1.
Write an addition problem with three decimals whose estimate is exactly 30 but whose exact sum is not 30. Show the estimate and the exact sum.Answer

Total / 27

Graded by

Notes

Answer key

Part A — the estimates.

about 8 (3 + 5)
about 19
about 70
about 40
about 24
about 7
about 160
about $18
about 25
about 5
about 200

Part B — the answer.

about 1.5. The greatest place of 0.48 is tenths, so it rounds to 0.5, and 3 × 0.5 = 1.5.
about 40, so the answer is 40.67. The point goes after the 0.
about 5.9
Less. 0.9 is less than one whole, so you take less than one whole 7.8. (The exact answer is 7.02.)
about 20 (4 + 6 + 6 + 4), so $24.00 is too high. The exact total is $20.00.
about 15 (5 + 2 + 8). His $29.94 is not reasonable; it is about double. The exact total is $14.94.

Part C — the estimates.

a. about 18 b. about 44 c. about 58 d. about 35 times
a. about 82 mm b. about 4 c. about 6 mm

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

Both estimates are 8. The sums are 8.8 and 8.0. The second estimate is exact. In the first sum both numbers rounded DOWN, so the errors piled up. In the second, 3.6 rounded up by 0.4 and 4.4 rounded down by 0.4, so the errors cancelled.
The estimate is 4 × 6 = 24. It is more, because both factors rounded UP. The exact answer is 21.46.

Part E — what a good problem looks like.

Answers vary. A good one shows the rounding and both sums, such as 9.6 + 10.4 + 9.8: the estimate is 10 + 10 + 10 = 30, and the exact sum is 29.8.