ESTIMATION AND APPROXIMATION OF NUMBERS MATHEMATICS JSS 1

Approximation of Numbers

To approximate a number means to write a number near the original number that is a number not exactly the original number. It may be a bit more or less than the original. Whole numbers can be approximated to the nearest ten, hundred, thousand, million, etc.

Let’s consider the table below:

	NUMBER	APPROXIMATED
	186	190 to the nearest ten
	1586	1600 to the nearest hundred
	346	300 to the nearest hundred
	1481	1480 to the nearest ten
	687.4	687 to the nearest unit
	4225	4000 to the nearest thousand
	69685.42	69690 to the nearest ten
	2.634	2.630 to the nearest thousandth
	0.214	0.214 to the nearest thousandth

From the above table, we can see that when numbers are approximated, they do not give the exact result expected. In approximation, we only consider the next figure we are approximating. If it is up to 5 and above, we take it as one (1) and add the (1) to the figure we are approximating to. If it is less than 5, we make it zero (0) and add zero to the figure.

Examples:

1. Sum 48, 226 and 592 and approximate your answer to the nearest hundred.

Solution:

48 + 226 + 592 = 866.

To the nearest hundred 866 = 900

2. Simply 1984

Solution:                                    

To the nearest hundred = 700

3. The heights of three educators are 1.45m, 3m and 2.11m. Calculate the total height and approximate to the nearest hundredth.

Solution:

1.45m + 3.00m + 2.11m = 6.56m.

To the nearest hundredth, 6.56m is the answer.

Note that in the example above, the answer is exactly the as the number approximated, which is 6.56m.

NOTE: Correcting numbers to the nearest ten (10) means leaving your answer with only one zero at the back of your answer (the unit position), while to the nearest hundred (100) and thousand (1000) means leaving two zeros and three zeros at the back respectively

Example:

Write 7822 to the nearest

(a) ten

(b) hundred

(c) thousand

Solutions:

(a) 7822 = 7820 to the nearest ten because 7822 is nearer to 7820 than 7830

(b) 7822 = 7800 to the nearest ten because 7822 is nearer to 7800 than 7900

(c) 7822 = 8000 to the nearest ten because 7822 is nearer to 8000 than 7000

Significant Figures

Numbers could be rounded up to a given significant figure. Significant figures are obtained by counting the number of digits in a given number.

When rounding up or down a number to given significant figures we count the digits of the number from left hand side to the required significant figures, then we consider the next digit to do the rounding up or down. If the digit is less than 5, we round down the number to zero and if it is equal to or greater than 5, we round it to one (1)

Example:

1. Round off:

(a) 456.36 (b) 0.03278 to:

(i) 2 significant figures

(ii) 3 significant figures

Solution:

(a) (i) 456.36 = 460 (2 s.f)

(ii) 456.36 = 456 (3 s.f)

(b) (i) 0.03278 = 0.03 (2 s.f)

(ii) 0.03278 = 0.033 (3 s.f)

 Theory of Numbers Prime Factors, LCM and HCF and squares of numbers

Class Activity:

1. Approximate (a) 2.517 (b) 0.497 to the nearest whole, tenth and hundredth
2. In 2011 Nigeria general election, there were 22.8 million registered male and 25.12 million registered female voters. How many voters were there altogether correct to 3 significant figures?
3. Round off:

(a) 468.3907 (b) 0.009896 to:

(i) 3 significant figures

(ii) 4 significant figures and state in each case the type of rounding off.

 

Decimal Places

The number of digit(s) after the decimal point in any given number is called its decimal places.

Example:

Correct the following to

(i) 1 decimal place

(ii) 2 decimal places

(iii) 3 decimal places

(a) 0.10775

(b) 0.08017

(c) 2.1359

Solution

(a) 0.10775

0.10775 = 0.1 to 1d.p

0.10775 = 0.11 to 2d.p

0.10775 = 0.108 to 3d.p

(b) 08017

0.08017 = 0.1 to 1dp

0.08017 = 0.08 to 2dp

0.08017 = 0.080 to 3dp

(c) 2.1359

2.1359 = 2.1 to 1dp

2.1359 = 2.14 to 2dp

2.1359 = 2.136 to 3dp

Class Activity

Write the following numbers correct to 1, 2 and 3 decimal places

(a) 0.2176

(b) 0.7487

(c) 16.074

(d) 22.3262

(e) 0.007749

1. Estimation is an ________ way of finding an approximate answer. a. exact b. accurate c. approximate d. precise
2. When we round a number to the nearest ten, we look at the ________ digit. a. ones b. tens c. hundreds d. thousands
3. Approximation helps us get a ________ idea of a number. a. specific b. random c. general d. unique
4. If we estimate 48 to the nearest ten, the result is ________. a. 40 b. 45 c. 50 d. 55
5. Rounding 672 to the nearest hundred gives us ________. a. 600 b. 670 c. 700 d. 750
6. When estimating, we often use ________ numbers to make calculations easier. a. odd b. even c. whole d. decimal
7. Estimation is useful in making quick and ________ calculations. a. precise b. accurate c. exact d. rough
8. ________ is finding an answer that is close to the correct one but not necessarily exact. a. Rounding b. Approximation c. Precision d. Calculation
9. If we round 3.89 to the nearest tenth, the result is ________. a. 3.8 b. 3.9 c. 3.88 d. 3.9
10. The purpose of ________ is to simplify numbers for easier understanding. a. rounding b. precision c. accuracy d. estimation
11. When estimating, we often round to the nearest ________. a. hundred b. thousand c. ten d. hundredth
12. Estimation involves using ________ values to represent more complex ones. a. specific b. random c. close d. precise
13. If we round 56 to the nearest ten, the result is ________. a. 50 b. 55 c. 60 d. 65
14. The process of approximation helps us get a quick ________ of a numerical value. a. analysis b. calculation c. evaluation d. understanding
15. In estimation, we focus on the ________ digits of a number. a. smallest b. largest c. middle d. rightmost

 

PRACTICE QUESTIONS

1. Write each of the following numbers correct to 1, 2 and 3 significant figures

(a) 66.57

(b) 2.0079

(c) 0.0187

(d) 4 766

2. Write each of the following numbers correct to 1, 2 and 3 decimal places

(a) 0.05144

(b) 2.1359

(c) 159.82

3. Round off 2453.738 to:

(i) 2 decimal places

(ii) 3 significant figures

4. Approximate 0.0025349 to 4 significant figures
5. A room is 18.8m by 9.8m. Calculate correct to 2 decimal places the perimeter of the room.

ASSIGNMENT

1. Round off each of the following to: i) 2 decimal places ii) 3 significant figures

(a) 0.90907

(b) 0.9997

(c) 15.09624

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 correct to 5 decimal places?
3. A woman’s pace is about 70cm long. She takes 2858 paces to walk from her home to the market. Find the distance from her home to the market correct to 4 significant figures.
4. The perimeter of a school compound is 1 616m. In the perimeter of the fence, there are 203 fence posts equally spaced. Find, approximating to 3 decimal places, the distance between two posts?
5. A nautical mile is 1.853km long. Calculate to:

(i) 2 dp

(ii) 2 s.f.

(iii) How many kilometres are there in 243 nautical miles?