Worksheet - Estimation and Approximation Techniques

DIRECTIONS: Each question has five answer choices. Select the one best answer. Do not use a calculator.

Section A - Rounding and Basic Estimation - Questions 1 to 7

1. Which of the following is the best estimate of 487 × 52?

1. 2,500
2. 20,000
3. 25,000
4. 30,000
5. 50,000

2. The value of \(\frac{3,987}{198}\) is closest to which integer?

1. 15
2. 18
3. 20
4. 22
5. 25

3. Rounded to the nearest tenth, what is the value of 47.862?

1. 47.8
2. 47.86
3. 47.9
4. 48.0
5. 50.0

4. A reasonable estimate for 21% of 398 is

1. 40
2. 60
3. 80
4. 100
5. 120

5. Which expression gives the best estimate of \(\frac{5.97 \times 8.12}{3.89}\)?

1. \(\frac{5 \times 8}{3}\)
2. \(\frac{6 \times 8}{4}\)
3. \(\frac{6 \times 9}{4}\)
4. \(\frac{5 \times 9}{4}\)
5. \(\frac{6 \times 8}{3}\)

6. The sum 1,872 + 4,923 + 2,104 is approximately

1. 7,000
2. 8,000
3. 9,000
4. 10,000
5. 11,000

7. If a store sells 48 items at $9.95 each, approximately how much total revenue does the store collect?

1. $400
2. $450
3. $500
4. $550
5. $600

Section B - Multi-Step Estimation and Operations - Questions 8 to 14

8. What is a reasonable estimate for \(\sqrt{83}\)?

1. 7
2. 8
3. 9
4. 10
5. 11

9. The expression \(\frac{(19.8)^2}{4.1}\) is closest to

1. 50
2. 100
3. 150
4. 200
5. 250

10. A student estimates \(3.87 \times 10^5 \div 1.92 \times 10^2\) by computing \(\frac{4 \times 10^5}{2 \times 10^2}\). What is the result of the student's estimate?

1. 2,000
2. 20,000
3. 200,000
4. 2,000,000
5. 20,000,000

11. Which of the following is the best approximation of \(\frac{897 + 1,123}{52 - 11}\)?

1. 25
2. 40
3. 50
4. 75
5. 100

12. If \(x = 4.93\) and \(y = 2.08\), then \(\frac{x^2}{y}\) is approximately

1. 10
2. 12
3. 15
4. 20
5. 25

13. The product 0.487 × 0.0312 is closest to

1. 0.0015
2. 0.015
3. 0.15
4. 1.5
5. 15

14. A reasonable estimate for \(\sqrt{47} + \sqrt{83}\) is

1. 13
2. 16
3. 19
4. 22
5. 25

Section C - Advanced Application - Questions 15 to 20

15. A rectangular field is 287 meters long and 193 meters wide. Which of the following is the best estimate of the perimeter of the field in meters?

1. 500
2. 800
3. 960
4. 1,000
5. 1,200

16. A factory produces 1,892 units per day. If the factory operates for 21 days in a month, approximately how many units are produced that month?

1. 20,000
2. 30,000
3. 40,000
4. 50,000
5. 60,000

17. The circumference of a circle is given by \(C = 2\pi r\). If a circle has a radius of 48.7 cm, which of the following is the best estimate of its circumference in centimeters? (Use \(\pi \approx 3\))

1. 150
2. 250
3. 300
4. 350
5. 450

18. If \(n = \frac{987 \times 412}{197}\), which of the following is closest to \(n\)?

1. 1,500
2. 2,000
3. 2,500
4. 3,000
5. 3,500

19. A store marks up items by 18% above cost. If an item costs $47.90, approximately what is the selling price?

1. $50
2. $54
3. $57
4. $60
5. $65

20. The expression \(\frac{5.2 \times 10^6 \times 3.8 \times 10^{-3}}{9.7 \times 10^2}\) is approximately equal to

1. 20
2. 200
3. 2,000
4. 20,000
5. 200,000

Answer Key

Quick Reference

1C 2C 3C 4C 5B 6C 7C 8C 9B 10A

11C 12B 13B 14B 15D 16C 17C 18B 19C 20A

Detailed Explanations

Question 1 - Correct Answer: C

Round 487 to 500 and round 52 to 50.
Compute 500 × 50 = 25,000.
The best estimate is 25,000.

Choice B results from rounding 487 to 400 instead of 500, yielding 400 × 50 = 20,000.

Question 2 - Correct Answer: C

Round 3,987 to 4,000 and round 198 to 200.
Compute \(\frac{4,000}{200} = 20\).
The value is closest to 20.

Choice D results from estimating with 3,987 rounded to 4,400 or from computational error in the division.

Question 3 - Correct Answer: C

The digit in the tenths place is 8.
The digit in the hundredths place is 6, which is 5 or greater.
Round up: 47.862 becomes 47.9.

Choice A results from failing to round up and instead truncating the number at the tenths place.

Question 4 - Correct Answer: C

Round 398 to 400.
Round 21% to 20%.
Compute 20% of 400: \(0.20 \times 400 = 80\).
A reasonable estimate is 80.

Choice D results from rounding 21% to 25% and computing 25% of 400 = 100.

Question 5 - Correct Answer: B

Round 5.97 to 6.
Round 8.12 to 8.
Round 3.89 to 4.
The best estimate is \(\frac{6 \times 8}{4}\).

Choice A results from rounding 5.97 down to 5 and 3.89 down to 3, which yields less accurate estimates.

Question 6 - Correct Answer: C

Round 1,872 to 2,000.
Round 4,923 to 5,000.
Round 2,104 to 2,000.
Compute 2,000 + 5,000 + 2,000 = 9,000.
The sum is approximately 9,000.

Choice B results from rounding 4,923 to 4,000, yielding 2,000 + 4,000 + 2,000 = 8,000.

Question 7 - Correct Answer: C

Round 48 to 50.
Round $9.95 to $10.
Compute 50 × 10 = 500.
Approximately $500 in total revenue.

Choice B results from rounding 48 to 45 instead of 50, yielding 45 × 10 = 450.

Question 8 - Correct Answer: C

\(9^2 = 81\) and \(10^2 = 100\).
Since 83 is between 81 and 100, \(\sqrt{83}\) is between 9 and 10.
83 is closer to 81 than to 100, so \(\sqrt{83}\) is closer to 9.
A reasonable estimate is 9.

Choice D results from choosing the midpoint between 9 and 10 without considering proximity of 83 to 81.

Question 9 - Correct Answer: B

Round 19.8 to 20.
Round 4.1 to 4.
Compute \(\frac{(20)^2}{4} = \frac{400}{4} = 100\).
The expression is closest to 100.

Choice A results from incorrectly computing \(20^2\) as 200 instead of 400, yielding \(\frac{200}{4} = 50\).

Question 10 - Correct Answer: A

The student computes \(\frac{4 \times 10^5}{2 \times 10^2}\).
Simplify the coefficients: \(\frac{4}{2} = 2\).
Simplify the powers of 10: \(10^5 \div 10^2 = 10^3\).
Compute \(2 \times 10^3 = 2,000\).

Choice B results from incorrectly computing \(10^5 \div 10^2\) as \(10^4\) instead of \(10^3\), yielding 20,000.

Question 11 - Correct Answer: C

Round 897 to 900 and 1,123 to 1,100.
The numerator is approximately 900 + 1,100 = 2,000.
Round 52 to 50 and 11 to 10.
The denominator is approximately 50 - 10 = 40.
Compute \(\frac{2,000}{40} = 50\).

Choice D results from rounding the denominator to 52 - 10 = 42, then estimating \(\frac{2,000}{42}\) as approximately 48, and selecting the nearest value of 75 due to computational error.

Question 12 - Correct Answer: B

Round 4.93 to 5.
Round 2.08 to 2.
Compute \(\frac{5^2}{2} = \frac{25}{2} = 12.5\).
The value is approximately 12.

Choice A results from computing \(\frac{x \times y}{2}\) instead of \(\frac{x^2}{y}\).

Question 13 - Correct Answer: B

Round 0.487 to 0.5.
Round 0.0312 to 0.03.
Compute 0.5 × 0.03 = 0.015.
The product is closest to 0.015.

Choice C results from miscounting decimal places and computing 0.5 × 0.3 = 0.15 instead of recognizing 0.0312 as approximately 0.03.

Question 14 - Correct Answer: B

\(\sqrt{47}\) is slightly less than \(\sqrt{49} = 7\), approximately 6.9.
\(\sqrt{83}\) is slightly greater than \(\sqrt{81} = 9\), approximately 9.1.
Sum: 6.9 + 9.1 = 16.
A reasonable estimate is 16.

Choice C results from estimating \(\sqrt{47}\) as 7 and \(\sqrt{83}\) as 9.5, yielding 7 + 9.5 = 16.5, then rounding to 19.

Question 15 - Correct Answer: D

Round 287 to 300 and 193 to 200.
Perimeter = 2 × (length + width) = 2 × (300 + 200) = 2 × 500 = 1,000.
The best estimate is 1,000 meters.

Choice C results from computing 2 × 287 + 2 × 193 without proper rounding, yielding approximately 574 + 386 = 960.

Question 16 - Correct Answer: C

Round 1,892 to 2,000.
Round 21 to 20.
Compute 2,000 × 20 = 40,000.
Approximately 40,000 units are produced.

Choice B results from rounding 1,892 to 1,500 or 21 to 20, yielding 1,500 × 20 = 30,000.

Question 17 - Correct Answer: C

Round the radius 48.7 to 50.
Use \(\pi \approx 3\).
Compute \(C = 2 \times 3 \times 50 = 300\) cm.
The best estimate is 300 cm.

Choice B results from using \(\pi \approx 2.5\) or incorrectly multiplying, yielding approximately 250.

Question 18 - Correct Answer: B

Round 987 to 1,000.
Round 412 to 400.
Round 197 to 200.
Compute \(\frac{1,000 \times 400}{200} = \frac{400,000}{200} = 2,000\).
The value is closest to 2,000.

Choice A results from rounding 987 to 1,000 and 412 to 300, yielding \(\frac{1,000 \times 300}{200} = 1,500\).

Question 19 - Correct Answer: C

Round the cost $47.90 to $48.
Round 18% to 20%.
Compute 20% of 48: \(0.20 \times 48 = 9.60\), approximately $10.
Selling price = 48 + 10 = $58.
The closest estimate is $57.

Choice D results from computing 25% markup instead of 18%, yielding 48 + 12 = $60.

Question 20 - Correct Answer: A

Round 5.2 to 5, 3.8 to 4, and 9.7 to 10.
Compute \(\frac{5 \times 10^6 \times 4 \times 10^{-3}}{10 \times 10^2}\).
Multiply coefficients: \(5 \times 4 = 20\).
Combine powers of 10: \(10^6 \times 10^{-3} \div 10^2 = 10^{6-3-2} = 10^1 = 10\).
Compute \(\frac{20 \times 10}{10} = 20\).
The expression is approximately 20.

Choice B results from incorrectly computing the exponent as \(10^{6-3+2} = 10^5\), yielding \(\frac{20 \times 10^5}{10} = 200\).