Approximation

Rounding Off Numbers

Sometimes, when we perform a mathematical calculations, the result may give rise to many more digits than is required. We may approximate the result by giving it a smaller number of significant figures or decimal places. This is called Rounding off. Rounding off numbers may involve rounding up or rounding down. The last digit to be ignored is usually the clue to rounding up or rounding down of the last digit that is kept. If the last digit to be be ignored is 5 or more, the last digit to kept is increased by 1; if it is less than 5, it remains unchanged. In doing this however, the number that is being rounded off must be close to the original as possible.

Examples:

1. Round off the following to  (i) 2 significant figures  (ii) 3 significant figures

(a) 9562774  (b) 768.45  (c) 29042

2. Round off:

(a) 16540 to nearest thousand  (b) 48057 to nearest hundred  (c) 0.965 to nearest tenth

Solutions:

1. To 2 significant figures

(i) 9562774 = 9600000 to 2 s.f

(ii) 768.48 = 770.00 to 2 s.f

(iii) 29042 = 29000

2. (a) 16540 = 16600 to nearest thousand

(b) 48057 = 48100 to nearest hundred

(a) 0.965 = 1.000 to nearest tenth

CLASS ACTIVITY

Round off the following to the  (a) 2 significant figures  (b) nearest thousand  (c) nearest tenth

(i) 1435.087  (ii) 165.420  (iii) 54.6585  (iv) 23000.567  (v) 5.76456

Approximating the Results of Addition and Subtraction

In approximating a value, one needs to have a rough idea of the answer to the calculation prior to the actual calculation.

Examples:

Estimate the values of each of the following:

(i) 43 + 62 + 59  (ii) 3.1 + 4.8 + 4.1  (iii) Add up 18 and 19  (iv) 53 – 38  (v) 841 – 392

Solutions: 

(i) Actual calculation is 43 + 62 + 59 = 164

Approximation is 40 + 60 + 60 = 160

(ii) Actual calculation is 3.1 + 4.8 + 4.1 = 12

Approximation is 3.0 + 5.0 + 4.0 = 12

(iii) Actual calculation is 18 + 19 = 37

Approximation is 20 + 20 = 40

(iv) Actual calculation is 53 – 38 = 15

Approximation is 50 – 40 = 20

(v) Actual calculation is 841 – 392 = 449

Approximation is 840 – 390 = 450

CLASS ACTIVITY

1. Round off each of the following to 2 significant figures

(i) 37 + 52  (ii) 653 – 287  (iii) 133g + 452g  (iv) 943m – 482m  (v) #470 + #610

2. The populations of five towns are 15600, 17300, 62800, 74000 and 34400 each to the nearest hundred. Find the total population of the five towns to the nearest thousand.

Approximating the Result of Multiplication and Division

Examples

(i) Multiply 63 by 47  (ii) 17.32 × 1.07  (iii) 703 divided by 21

Solutions:

(i) Multiply 63 by 47

Actual calculation is 63 × 47

+22×4596442637101

Approximation is 63 × 47 ≅ 60 × 50 = 3000

(ii) 17.32 × 1.07

Solution:

Actual calculation:

17.32×1.0712124000000+17320018.5324

Approximation of 17.32 × 1.07 ≅ 17.0 × 1.0 = 17.0

(ii) 703 ÷ 21 = 70321 = 33.47619

Approximation of 703 ÷ 21 ≅ 700 ÷ 20 = 35

Class Activity

1. Calculate and approximate the following

(i) 12.48  (ii) 0.333 × 0.667  (iii) 9.6 × 5.8  (iv) #6470  ÷ #24

2. A shop sells about 340 magazines each week. The selling price of a magazine is #180. Estimate the amount of money the shop makes each week from selling magazines.
3. There are 36 teams who met at Kaduna stadium. Each team has 12 players. First estimate, and then find accurately, the total number of players at the stadium.

PRACTICE QUESTIONS

1. Round off each of the following to:  (a) 1 significant figure  (b) 2 significant figures

(i) 16.7 + 1.09  (ii) 7.55 – 3.45  (iii) 12.48 ÷ 4.8  (iv) 16.7 × 1.09

2. A packet of biscuits has a mass of 205g. if there are 28 biscuits in the packet, what is the approximate mass of 1 biscuit?
3. A woman’s pace is about 70cm long. She takes 2858paces to walk from her home to the market. Find, roughly, distance from her home to the market.
4. Round off (6.265 + 3.77) × (12.3 + 8.7) to 2 significant figures.
5. Round off (9.774 × 16.7) ÷ (3.64 × 6.7)