**Approximation of numbers :**

In our daily life we need to know approximate values or measurements.

David bought a Lap Top for $499.85. When he wants to convey this amount to others, he simply says that he has bought it for $500.

This is the approximate value which is given in hundreds only.

Daniel buys a pair of slippers for $19.99. This amount may be considered approximately as $20 for convenience.

A photo frame has the dimensions of 35.23 cm long and 25.91 cm wide. If we want to check the measurements with our ordinary scale, we cannot measure accurately because our ordinary scale is marked in tenths of centimeter only.

In such cases, we can check the length of the photo frame 35.2 cm to the nearest tenth or 35 cm to the nearest integer value.

In the above situations we have taken the approximate values for our convenience. This type of considering the nearest value is called ‘Rounding off’ the digits.

Thus the approximate value corrected to the required number of digits is known as ‘Rounding off’ the digits.

Sometimes it is possible only to give approximate value, because

(a) If we want to say the population of a city, we will be expressing only the approximate value say 30 hundred thousands or 25 hundred thousands and so on.

(b) When we say the distance between two cities, we express in round number 350 km not 352.15 kilometers.

While rounding off the numbers we adopt the following principles.

(i) If the number next to the desired place of correction is less than 5, give the answer up to the desired place as it is.

(ii) If the number next to the desired place of correction is 5 and greater than 5 add 1 to the number in the desired place of correction and give the answer.

The symbol for approximation is usually denoted by **≃**

**Approximation nearest to TEN :**

**Illustration :**

Consider multiples of 10 before and after 521. ( i.e. 520 and 530 )

We find that 521 is nearer to 520 than to 530.

It has been illustrated in the picture given below.

Therefore, the approximate value of 521 is 520 in this case.

**Approximation nearest to HUNDRED :**

**Illustration :**

(i) Consider multiples of 100 before and after 521. ( i.e. 500 and 600 )

We find that 521 is nearer to 500 than to 600. So, in this case, the approximate value of 521 is 500.

(ii) Consider the number 625

Suppose we take the number line, unit by unit.

In this case, we cannot say whether 625 is nearer to 624 or 626, because it is exactly midway between 624 and 626. However, by convention we say that it is nearer to 626 and hence its approximate value is taken to be 626.

Suppose we consider multiples of 100, then 625 will be approximated to 600 and not 700

For the number 47,618

(a) Approximate value correct to the nearest tens = 47,620

(b) Approximate value correct to the nearest hundred = 47,600

(c) Approximate value correct to the nearest thousand = 48,000

(d) Approximate value correct to the nearest ten thousand = 50,000

**Illustration 1 :**

Consider the decimal number 36.729

(a) It is 36.73 round to two decimal places. (Since the last digit 9 > 5, we add 1 to 2 and make it 3).

Therefore, 36.729 **≃ **36.73 (Round to two decimal place)

(b) Look at the second decimal in 36.729, Here it is 2 which is less than 5, so we leave 7 as it is.

Therefore, 36.729 **≃ **36.7 (Round to one decimal place)

**Illustration 2 :**

Consider the decimal number 36.745

(a) It’s approximation is 36.75, round to two decimal places. Since the last digit is 5, we add 1 to 4 and make it 5.

(b) It’s approximation is 36.7, round to one decimal place. Since the second decimal is 4, which is less than 5, we leave 7 as it is.

36.745 **≃ **36.7

**Illustration 3 :**

Consider the decimal number 2.14829

(i) Approximate value rounded to one decimal place is 2.1

(ii) Approximate value rounded to two decimal places is 2.15

(iii) Approximate value rounded to three decimal places is 2.148

(iv) Approximate value rounded to four decimal places is 2.1483

**Illustration 4 :**

Round off the following numbers to the nearest integer:

(a) 288.29 (b) 3998.37 (c) 4856.795 (d) 4999.96

**Solution :**

(a) 288.29 **≃** 288

(b) 3998.37 **≃** 3998

(c) 4856.795 **≃ **4857

(d) 4999.96 **≃ **5000

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