Approximating means making a number simpler but keeping it close to the real value.It is a way of writing a number so that it is easy to work with while still showing almost the same size.

In mathematics, approximating is a way to simplify numbers so calculations are easier, while keeping results as close as possible to the true value.

1. It makes calculations easier to do mentally or quickly on paper.


2. It simplifies very long decimals or big numbers.


3. It helps when exact figures are not needed.


4. It is helpful when measuring tools are not very accurate.


5. It allows you to check if your exact answer is reasonable.

1. With large numbers or decimals where exact digits are less important.


2. In measurement, when instruments have limited precision.


3. In everyday life: money, distances, weights etc.


4. In checking results of exact calculation: you approximate and see whether your final result is near the approximate.

1. Rounding whole numbers – Round to the nearest ten, hundred or thousand. Example: 4,768 to the nearest hundred is 4,800.


2. Rounding decimals – Round decimals to the number of decimal places needed. Example: 3.746 to 2 decimal places is 3.75.


3. Significant figures – Keep only the first few important digits for very big or very small numbers. Example: 0.004563 to 2 significant figures is 0.0046.


4. Compatible numbers – Replace numbers with nearby easy numbers for quick mental maths. Example: 198 × 47 is about 200 × 50 = 10,000.


5. Truncation – Cut off extra digits without rounding. Example: 3.987 to 2 decimal places by truncation is 3.98.


6. Approximating fractions – Change fractions into simpler fractions or decimals. Example: 7/9 is about 0.78 or about 4/5.

Approximating the value of Addition and Subtraction

When adding or subtracting, we can first round the numbers to make mental calculation easier.

Examples

1. Add 347 + 289 (approximate to nearest ten)

   347 ≈ 350

   289 ≈ 290

   350 + 290 = 640 (approximate answer).

2. Add 5.82 + 3.47 (approximate to 1 decimal place)

   5.82 ≈ 5.8

   3.47 ≈ 3.5

   5.8 + 3.5 = 9.3 (approximate answer).

3. Subtract 1,248 – 627 (approximate to nearest hundred)

   1,248 ≈ 1,200

   627 ≈ 600

   1,200 – 600 = 600 (approximate answer).

4. Add 49 + 52 (nearest ten)

   49 ≈ 50

   52 ≈ 50

   50 + 50 = 100 (approximate answer).

5. Subtract 82.6 – 39.8 (nearest whole number)

   82.6 ≈ 83

   39.8 ≈ 40

   83 – 40 = 43 (approximate answer).

6. Add 7/9 + 4/5 (approximate each fraction to 1 decimal)

   7/9 ≈ 0.78 ≈ 0.8

   4/5 = 0.8

   0.8 + 0.8 = 1.6 (approximate answer).

2️⃣ Approximating the result of Multiplication and Division

Before multiplying or dividing, round the numbers first.

6 Worked Examples with Steps

1. 198 × 47 (nearest ten)

   198 ≈ 200

   47 ≈ 50

   200 × 50 = 10,000 (approximate answer).

2. 138 ÷ 4.6 (1 decimal place)

   138 ≈ 140

   4.6 ≈ 5

   140 ÷ 5 = 28 (approximate answer).

3. 305 × 19 (nearest ten)

   305 ≈ 300

   19 ≈ 20

   300 × 20 = 6,000 (approximate answer).

4. 3.78 ÷ 1.2 (1 decimal place)

   3.78 ≈ 3.8

   1.2 ≈ 1

   3.8 ÷ 1 = about 4 (approximate answer).

5. 62 × 47 (nearest ten)

   62 ≈ 60

   47 ≈ 50

   60 × 50 = 3,000 (approximate answer).

6. 279 ÷ 8.4 (1 decimal place)

   279 ≈ 280

   8.4 ≈ 8

   280 ÷ 8 = 35 (approximate answer).

HOW TO APPROXIMATE (General Steps)

1. Choose the place value or decimal place you want (nearest ten, hundred, 1 decimal place, etc.).


2. Look at the next digit.


3. If it is 5 or more, round up.


4. If it is 4 or less, round down.


5. Replace all digits to the right with zeros (for whole numbers) or cut off extra decimals.

TYPES OF NUMBERS WE CAN APPROXIMATE

1. Whole numbers (e.g. 347 → 350)


2. Decimals (e.g. 4.768 → 4.8)


3. Fractions (7/9 ≈ 0.8)


4. Very large numbers (5,123,000 → 5.1 million)


5. Very small numbers (0.00456 → 0.005)


6. Even irrational numbers like π (3.14159…) or √2 (1.414…) can be approximated to a chosen decimal place (e.g. π ≈ 3.14).