The Lumenary · Grades 3–5

Estimation and rounding

Name: ______________________

Date: ______________________

You will learn to

Round whole numbers to the nearest ten, hundred or thousand, and decimals to the nearest whole number or tenth, explaining the rule for halfway cases
Estimate sums, differences and products before calculating, and use estimates to check answers
Give lower and upper bounds for a quantity, such as "between $300$ and $400$ ", and combine bounds
Explain why rounding to the nearest ten changes a number by at most $5$ , and how rounding errors can add up

1.
Round $4372$ to the nearest hundred.
2.
Round $2950$ to the nearest hundred.
3.
Round $3.46$ to the nearest tenth.
- A. $3.5$
- B. $3.4$
- C. $3$
- D. $4$
4.
Kwame's calculator says $39 \times 61 = 1379$ . Which is the best quick check?
- A. $39 \times 61 \approx 40 \times 60 = 2400$ , so $1379$ must be wrong
- B.Calculators are never wrong, so it must be right
- C. $39 + 61 = 100$ , so the answer should be about $100$
- D.It ends in $9$ , and $9 \times 1 = 9$ , so it is right
5.
You know that $30 \le a \le 40$ and $20 \le b \le 30$ . What is the largest possible value of $a - b$ ?
6.
Wanjiru has $1000$ shillings. The prices are $289$ , $412$ and $198$ shillings. Which reasoning shows for certain that she has enough money, without adding exactly?
- A.Each price is at most $300$ , $500$ and $200$ , and $300 + 500 + 200 = 1000$
- B.The prices round to $300$ , $400$ and $200$ , and $900$ is less than $1000$
- C.Each price is at least $200$ , $400$ and $100$ , and $700$ is less than $1000$
- D.Three prices that each have three digits always add up to less than $1000$
7.
Work out $14 + 14 + 14$ in two ways: (1) round each $14$ to the nearest ten and add; (2) add exactly, then round the total to the nearest ten. What happens?
- A.Way (1) gives $30$ and way (2) gives $40$ : they differ
- B.Both ways give $40$
- C.Both ways give $30$
- D.Way (1) gives $40$ and way (2) gives $30$
8.
Ama rounds $2.45$ to the nearest tenth and gets $2.5$ . Then she rounds $2.5$ to the nearest whole number and gets $3$ . What is $2.45$ rounded directly to the nearest whole number?
- A. $2$
- B. $3$
- C. $2.5$
9.
In Lean, roundTen n is $n$ rounded to the nearest ten. Gideon wants to state precisely that rounding to the nearest ten changes $n$ by at most $5$ . Which statement says that?
- A.roundTen n ≤ n + 5 ∧ n ≤ roundTen n + 5
- B.roundTen n = n
- C.roundTen n ≤ 5
- D.roundTen n ≤ n + 5
10.
Write round_to(n, place) that rounds a whole number n to the nearest place (10, 100, 1000, …) using the school rule, so that halfway rounds up. The starter uses Python's built-in round(), which rounds halves to even, so some tests fail. Use whole-number arithmetic instead: add half of place, then knock off the lower digits with //.
Computational — Practice

Answer key — Estimation and rounding

1.
$4372$ lies between $4300$ and $4400$ , above the halfway point $4350$ . So it rounds up: $4372 \approx 4400$ .
2.
$2950$ is exactly halfway between $2900$ and $3000$ , and halfway rounds up, so $2950 \approx 3000$ . The rounding carries into the thousands.
3.
$3.46$ lies between $3.40$ and $3.50$ , above the halfway point $3.45$ . So to the nearest tenth, $3.46 \approx 3.5$ .
4.
$39 \approx 40$ and $61 \approx 60$ , so $39 \times 61 \approx 2400$ . The calculator's $1379$ is about $1000$ too small, so a key must have been pressed wrongly. Exactly, $39 \times 61 = 2379$ .
5.
The difference is largest when $a$ is as big as possible and $b$ as small as possible: $40 - 20 = 20$ . For differences, bounds work crosswise.
6.
Upper bounds give certainty: $289 \le 300$ , $412 \le 500$ , $198 \le 200$ , so the total is at most $300 + 500 + 200 = 1000$ . (Exactly, it is $899$ .) Rounding to the nearest hundred gives the estimate $900$ , which is useful but promises nothing, because some prices were rounded down.
7.
Way (1): $10 + 10 + 10 = 30$ . Way (2): $42 \approx 40$ . Each rounding changed $14$ by $4$ , and the errors added up to $12$ . So rounding at every step can change the final answer.
8.
$2.45$ is between $2$ and $3$ , and below the halfway point $2.5$ . So it rounds to $2$ . The first rounding ( $2.45 \to 2.5$ ) moved the number up onto the halfway point, and the second rounding moved it up again. Always round the original number once.
9.
roundTen n ≤ n + 5 ∧ n ≤ roundTen n + 5: the rounded value is no more than $5$ above $n$ and no more than $5$ below it. Lean proves the slightly stronger roundTen n ≤ n + 5 ∧ n ≤ roundTen n + 4 as roundTen_error.
10.
```
def round_to(n, place):
    return (n + place // 2) // place * place
```
Adding half of place pushes every number in the upper half (including the halfway number) past the next multiple; then // place * place knocks off the lower digits. Python's round() uses "round half to even", so round(2.5) == 2 and round(4.5) == 4. That rule is also a sensible convention (it has no upward bias), but it is not the school rule.