Understanding Estimation & Rounding

Understanding Rounding

Rounding means making a number simpler but keeping its value close to what it was. The result is less accurate, but easier to use. Rounding is a key skill used in estimation.

General Rule for Rounding:

  1. Identify the digit in the place value you are rounding to.
  2. Look at the next digit to its right.
  3. If this next digit is 5 or more, increase the digit in the rounding place by one (round up).
  4. If this next digit is 4 or less, keep the digit in the rounding place the same (round down).
  5. Change all digits to the right of the rounding digit to zeros (for whole numbers). For decimals, you might drop these digits if they are after the decimal point.

Examples of Rounding:

What is Estimation?

Estimation is the process of finding an approximate value or calculation, rather than an exact one. It's a useful skill for quickly checking if an answer is reasonable or for making calculations when exact numbers are not necessary or available.

To estimate, we usually round numbers (as explained above) to a sensible degree of accuracy (e.g., to the nearest 10, 100, or to one significant figure) before performing the calculation.

The symbol ≈ is often used to mean "approximately equal to".

Estimation: Worked Examples

1. Estimating a Sum

Problem: Estimate the value of 487 + 312.

Workings:

  1. Round each number to the nearest hundred:
    • 487 rounds to 500 (because 87 is greater than or equal to 50).
    • 312 rounds to 300 (because 12 is less than 50).
  2. Add the rounded numbers: 500 + 300 = 800.

Estimated Answer: 487 + 312 ≈ 800

(Exact answer: 487 + 312 = 799)

2. Estimating a Product (Multiplication)

Problem: Estimate the value of 28 × 63.

Workings:

  1. Round each number to the nearest ten (or one significant figure):
    • 28 rounds to 30.
    • 63 rounds to 60.
  2. Multiply the rounded numbers: 30 × 60.
    • Think of it as (3 × 10) × (6 × 10) = 3 × 6 × 10 × 10 = 18 × 100 = 1800.
    • Or, 3 × 6 = 18, then add the two zeros: 1800.

Estimated Answer: 28 × 63 ≈ 1800

(Exact answer: 28 × 63 = 1764)

3. Estimating a Division

Problem: Estimate the value of 587 ÷ 22.

Workings:

  1. Round numbers to make the division easier. Often rounding to one significant figure is helpful.
    • 587 rounds to 600 (nearest hundred or 1 s.f.).
    • 22 rounds to 20 (nearest ten or 1 s.f.).
  2. Divide the rounded numbers: 600 ÷ 20.
    • You can cancel a zero from both: 60 ÷ 2.
    • 60 ÷ 2 = 30.

Estimated Answer: 587 ÷ 22 ≈ 30

(Exact answer: 587 ÷ 22 ≈ 26.68)

4. Estimating with Decimals

Problem: Estimate the value of 19.7 × 3.2.

Workings:

  1. Round each decimal to the nearest whole number (or one significant figure):
    • 19.7 rounds to 20.
    • 3.2 rounds to 3.
  2. Multiply the rounded numbers: 20 × 3 = 60.

Estimated Answer: 19.7 × 3.2 ≈ 60

(Exact answer: 19.7 × 3.2 = 63.04)

Test Yourself: Estimation & Rounding

1. Round 5,483 to the nearest thousand.

Answer: 5,000

Explanation: The thousands digit is 5. The next digit (hundreds) is 4. Since 4 is less than 5, we keep the thousands digit the same and change the rest to zeros.

2. Estimate the total cost of items priced at £18.99, £32.50, and £8.75.

Estimated Answer: ≈ £60

Explanation:

  1. Round each price:
    • £18.99 ≈ £20 (to nearest £ or £10)
    • £32.50 ≈ £30 (to nearest £10) or £33 (to nearest £)
    • £8.75 ≈ £10 (to nearest £ or £10)
  2. Add the rounded prices (using nearest £10 for simpler mental math): £20 + £30 + £10 = £60.
  3. (Exact answer: £18.99 + £32.50 + £8.75 = £60.24)

3. A car travels 295 miles on a full tank of petrol. The tank holds 11.2 gallons. Estimate the car's miles per gallon.

Estimated Answer: ≈ 30 miles per gallon

Explanation:

  1. Round the values to make division easy:
    • 295 miles ≈ 300 miles
    • 11.2 gallons ≈ 10 gallons
  2. Estimate the division: 300 miles ÷ 10 gallons = 30 miles per gallon.
  3. (Exact answer: 295 ÷ 11.2 ≈ 26.34 mpg)

4. Round 0.0762 to two significant figures.

Answer: 0.076

Explanation:

  1. The first significant figure is 7 (the first non-zero digit from the left).
  2. The second significant figure is 6.
  3. The next digit is 2 (which is 4 or less).
  4. So, keep the 6 the same. Result: 0.076.

Key Points for Estimation & Rounding