Approximation
Mathematics SSS 1 Second Term
WEEK 2
Approximation
Performance Objectives
Students should be able to;
- Calculate numbers less than 1 involving multiplication and division using log table
- Solve simple logarithm equations
Approximation of Numbers
Content
Approximation 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.
Sometimes we do not need to measure or 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 of 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
a. rounding off to the nearest unit, or ten, or hundred, or …..
b. decimal places
c. significant places etc.
Let’s consider the table below:
|
|
S/N |
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 |
||
|
5. |
687.4 |
|
|
|
6. |
4225 |
4000 to the nearest thousand |
|
|
7. |
69685.42 |
69690 to the nearest ten |
|
|
8. |
2.634 |
2.630 to the nearest thousandth |
|
|
9. |
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
Rounding off numbers
To ’round off’ or ‘approximate’ a number to a desired degree of accuracy, we;
a. round the number up if the next digit is 5 or more
b. round the number down if the next digit is less than 5.
We represent approximately equal to as and approximately as ‘~’
Examples:
1. 73 is close to 70 if approximating to or rounding in “tens”. So
73 ~ 70 (Read as 73 is approximately equal to 70)
2. 86 is close to 90 when rounded off to the nearest “tens”, or approximate to the nearest “tens”
86 ~ 90
3. 650 ~ 700 when rounded off to the nearest hundreds
4. 26432 rounded off to
a. nearest 100 is 264 / 32 ~ 26400
b. nearest 10,000 is 2 / 6432 ~ 30000
Strategy to round off numbers
1. Put a line where you want to round off. In the above example 3 to round off 650, put a big line after 6, because 6 is in the hundreds place and you want to round to the nearest hundred.
6/50
- The digit before the big line (6 in this case) will go up by 1, and the rest of the digits after the line will become 0, since the number after the line is 5. So the answer is 700.
Decimal Places
The number of digit(s) after the decimal point in any given number is called its decimal places.
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.
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
Example 2:
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
Approximation and Estimations in Everyday Life
a. 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.
b. 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.
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 as 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 overestimated by ½ km and the parents underestimated 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 the 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%