ICSE Class 8 Maths Numbers Chapter 05 Approximation Rounding Off

Approximation (Rounding Off)

Suppose, 86375 people attended a football match. To get a rough idea of the number of spectators, we may remember the first two digits from the left and put zeros in places of the other digits. Thus, it can be reported that approximately 86000 people attended the match. This is called rounding off. Here, we have rounded off 86375 to the nearest thousand. Similarly, we can round off numbers to the nearest ten, hundred, ten thousand, lakh, million, etc.

Rounding Off Whole Numbers

Consider the number 3264. Suppose you want to round it off to the nearest ten. You find that 3264 is nearer to 3260 than to 3270. So, 3264 rounded off to the nearest ten is 3260.

Now suppose you want to round off 3264 to the nearest hundred. 3264 is nearer to 3300 than to 3200. So, you round it off to 3300.

The general rule for rounding off a whole number to the nearest hundred is as follows.

Rule - If the digit in the tens place is less than 5, write 0 in the tens place as well as in the units place and keep the digits in the other places as they are. If the digit in the tens place is 5 or more than 5, write 0 in the tens place as well as in the units place and add 1 to the digit in the hundreds place.

Examples (i) 3826 rounded off to the nearest hundred is 3800 as the digit in the tens place = 2 < 5.

(ii) 4283 rounded off to the nearest hundred is 4300 as the digit in the tens place = 8 > 5.

Similarly, a whole number can be rounded off to the nearest thousand, ten thousand, etc.

(i) 4763 is rounded off to 5000 (nearest thousand).

(ii) 23625 is rounded off to 20000 (nearest ten thousand).

Rounding Off Decimal Fractions

Decimal fractions can be rounded off to the nearest one, tenth, hundredth, thousandth, etc.

The general rule for rounding off decimals is as follows.

Rule - First, find the answer to one more place than you need. If the digit in the extra place is 5 or more, add 1 to the digit in the place just before it. If the digit in the extra place is less than 5, leave the digit in the place just before it as it is.

Examples (i) 43.7 rounded off to the nearest unit is 44 because the digit in the first decimal place is 7 > 5.

(ii) 43.78 rounded off to 1 decimal place is 43.8 because the digit in the second decimal place is 8 > 5.

(iii) 43.783 rounded off to 2 decimal places is 43.78 because the digit in the third decimal place is 3 < 5.

Significant Figures

The digits used to express a number to a specified degree of accuracy are the significant figures.

Finding Significant Figures Of A Number

Method - Ignore the decimal point and read the number from left to right. The first significant figure is the first nonzero digit. All digits after that are also significant.

Examples - 3.246 has 4 significant figures-3, 2, 4 and 6. The number 0.0004057 also has 4 significant figures-4, 0, 5 and 7. The zeros before the digit 4 are not significant but the zero after 4 is significant.

The number 243.57 has 5 significant figures-2, 4, 3, 5 and 7.

In the number 243.57, the digit 2 is the most significant figure as it has a place value of 200, while the digit 7 is the least significant figure as it has a place value of \(\frac{7}{100}\).

Rounding Off To Significant Figures

Method - First, find the answer to one more significant figure than you need. If the extra digit is 5 or more than 5, add 1 to the digit just before it. If the extra digit is less than 5, leave the digit just before it as it is.

Examples - 243.528 correct to 5 significant figures = 243.53. As the sixth significant figure = 8 > 5, we add 1 to the fifth significant figure, that is, 2.

243.528 correct to 4 significant figures = 243.5 as the fifth significant figure = 2 < 5.

Teacher's Note

Rounding off is used daily when reporting statistics, weather forecasts, or financial figures to make information easier to understand and communicate.

Solved Examples

Example 1 - Round off the following.

(i) 0.016297 to two significant figures and to two decimal places

(ii) 3.5792 to three significant figures and to three decimal places

Solution - (i) 0.016297 = 0.016, rounded off to two significant figures = 0.02, rounded off to two decimal places.

(ii) 3.5792 = 3.58, rounded off to three significant figures = 3.579, rounded off to three decimal places.

Example 2 - Round off the following.

(i) 7835284 to the nearest 10000

(ii) Rs 25.634 to the nearest paisa

(iii) 21.78 m to the nearest 10 cm

(iv) 38.523 cm to the nearest mm

(v) 3.5672 kg to the nearest g

Solution - (i) 7835284 is closer to 7840000 than to 7830000. Therefore, 7835284 = 7840000, to the nearest 10000.

(ii) Rs 25.634 = 2563.4 p, which is closer to 2563 p (that is, Rs 25.63) than to 2564 p. Therefore, Rs 25.634 = Rs 25.63, to the nearest paisa.

(iii) 21.78 m = 2178 cm, which is closer to 2180 cm (that is, 21.80 m) than to 2170 cm. Therefore, 21.78 m = 21.80 m, to the nearest 10 cm.

(iv) 38.523 cm = 385.23 mm, which is closer to 385 mm (that is, 38.5 cm) than to 386 mm. Therefore, 38.523 cm = 38.5 cm, to the nearest mm.

(v) 3.5672 kg = 3567.2 g, which is closer to 3567 g (that is, 3.567 kg) than to 3568 g. Therefore, 3.5672 kg = 3.567 kg, to the nearest g.

Example 3 - Express \(\frac{2}{171}\) as a decimal number correct to (i) three decimal places, (ii) three significant figures.

Solution - 2 ÷ 171.

171 ) 2.00 (0.01169

- 171

290

- 171

1190

- 1026

1640

- 1539

101

(i) \(\frac{2}{171}\) = 0.012, correct to three decimal places (as 6 is the fourth decimal place and 6 > 5).

(ii) \(\frac{2}{171}\) = 0.0117, correct to three significant figures (as 9 is the fourth significant digit and 9 > 5).

Teacher's Note

Understanding significant figures helps in scientific experiments and engineering projects where precision matters in calculations and measurements.

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