Classwork Series and Exercises {Mathematics- SS2}: Approximation and Percentage Error

Introduction

Sometimes we do not need to measure of calculate things exactly. We may only wish to have a rough idea, or to calculate only to a certain degree of accuracy. It is in such cases we talk of decimal places, significant figures or rounding off to the nearest unit, tens, etc. When this happens, we say that we are approximating.

When we approximate we use such terms as

1. rounding off to the nearest unit, or ten, or hundred, or …..
2. decimal places
3. significant places etc.

ROUNDING OFF

In rounding off we recall the place-value of numbers

e.g. 743,285 means 7 hundreds of thousands

                         plus     4 tens of thousands

                           plus    3 units of thousands

                           plus    2 hundreds

                           plus    5 tens

                            and   8 units

which we can write in the abacus form as

H of Th	T of Th	Th	H	T	U
7	4	3	2	5	8

Now to round off this number to the nearest ten, for example, we look at the digit to the right of 5 tens. If this digit is equal to , or greater than 5 (³ 5) then we add 1 to the digit in the tens place. Like here the digit to the right of 5 tens is 8: and 8 > 5 then 5 tens becomes 6 tens i.e. to the nearest ten, 743,285 = 743,260.

But if the digit to the right of 5 tens is less than 5 (< 5) then we do not add 1 to the tens value

i.e. To the nearest ten, 743,253 = 743,250

Example

       488  =  490               to the nearest ten

       488  =  500               to the nearest hundred

119,347 =  119,300       to the nearest hundred

               =  120,000        to the nearest tens of thousands

               =  100,000        to the nearest hundreds of thousands

To round off a number to the nearest Unit usually involves numbers which are written in decimal form or mixed number form. If the fractional part is greater than ½ we add 1 to the unit digit of the number.

In the decimal form, if the digit after the decimal point is 5 or greater than 5, we add 1 to the unit digit of the number.

Decimal Places

In order to calculate to a required number of decimal place, we usually calculate to one place more than the required. If that last digit is less than 5, it will be discarded but if it is 5 or more than 5, then 1 is added to the digit just before it.

Examples

1. 45.475 to two decimal places= 45.48 since the last digit which is 8 is more than 5 we add 1 to 7
2. 122.184 to two decimal places = 122.18. We discard 4 which is less than 5

Significant Figures

The first non-zero digit in any number is its first significant figure. A zero between two significant figures is counted as a significant figure. We calculate to a required number of significant places by the same method a s we do for decimal places. Numbers can be rounded off to any required number of the significant figures.

Example

(i) To 3 significant figures, 0.02364 = 0.0236, Note we do not count the first digit, 0.

(ii) To 2 significant figures, 5286 = 5300, since 8 is more than 5, 1 is added to 2

(iii) Ti 1 significant figure, 9.103 = 9. (Since 1 is less than 5)

Percentage Error

When we make estimate or approximation we do not have the exact value of the result, but an approximation to it, i.e. an idea of the value.

Now the difference between the actual result and the estimated or approximated result can be calculated in percentage. This is known a s percentage error

Note The nearer the percentage error is to zero, the more accurate the result.

Example

The exact distance a boy walk to school every morning is 4km, when asked he said 4 ½ km while his parents said he walks 3 km to school, Find their percentage error.

Solution

Both the boy and the parents made errors. The boy over estimated by ½ km and the parents under estimated by 1 km.

½ km and 1 km are called the absolute errors.

 The percentage error of Boy = ½ /4 x 100% = 1/8 x 100 = 12 ½ %

The Percentage error of the Parents = ¼ x 100% = 25%

So the boy is more accurate than the parents.

The absolute error is difference between the estimate and the exact values. This percentage error is usually referred to as the Relative Error i.e. Relative error is often given as a percentage.

Percentage Error = absolute error/exact value x 100%

Approximation and Estimations in Everyday Life

1. A mother who wants to buy, say 3 loaves of bread, 5kg of garri. 1 packet of sugar and 1 bottle of oil will ensure that she has enough money for those items before leaving her house for the market. She may not know the exact costs of the items, but by estimation she will give some prices to the items, add the costs up, and then go shopping with approximately the amount needed for those items.
2. Cooks have to estimate the quantity of food that will satisfy a customer in a restaurant (or even at home), and multiply that quantity by the number of people to feed, then round off the quantities to take care of wastages etc. Life is full of activities involving estimations and approximations.

EXERCISES

Lets see how much you’ve learnt, attach the following answers to the comment below

1. Round off 1,478 to the 1000 A. 1,480 B. 1,479 C. 2000 D, 1,400
2. Round off 479 to the nearest 100 A. 480 B. 500 C. 789 D. 501

Find both the absolute errors and the Percentage errors of the following

3. Exact Value: ₦400, Approximate Measurement A. 27 and 6.25% B. 25 and 6.25% C. 25 and 7.35% 24 and 6.25%
4. Exact value : 25 m, Approximate Measurement: 5.5 m A. 0.65 and 12 B. 0.75 and 15  C. 0.75 and 12  D. 1.02 and 12
5. What is the first significant figure of 46.057? A. 46.10 B. 46.06 C. 46.5 D. 47

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