Estimation and Approximation

1. Estimation and Approximation PREP I Chapter 4 – Book 1

2. KEY TERMS Estimation Approximate Rounding off Significant Figures Degree of Accuracy

3. Estimation Sometimes, we may have to guess or estimate numbers as close as possible to their real values. This process of guessing the numbers sensibly is called estimation. Generally you can use estimation to work out whether your calculation or the answer you are working towards is right or wrong. Estimation often involves rounding off

4. Rounding Off • To ’round off’ or ‘approximate’ a number to a desired degree of accuracy, we • round the number up if the next digit is 5 or more • round the number down if the next digit is less than 5. • We represent approximately equal to using the sign ‘ ~ ’ • 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)

5. 2.) 86 is close to 90 when rounded off to the nearest “tens”, or approximate to the nearest “tens” • Hence, 86 ~ 90 • 3.) 650 ~ 700 when rounded off to the nearest hundreds  • 4.) 26432 rounded off to  • nearest 100 is ~  26400 • nearest 10,000 is  ~  30000

6. Let’s look at some more examples. )  234 + 57 Here 234 is close to 230, and 57 is closer to 60. So 234 + 57 ~ 230 + 60  =  290, which is very close to 291 (the correct answer). 2.) 272 x 33 272 ~ 300, and 33 ~ 30 So 272 x 33 ~ 300 x 30 = 9000 The correct answer of 272 x 33 is 8976.

7. Rounding off a Number to a Given Number of Decimal Places Sometimes it is impractical or meaningless to have too many decimal places in a decimal. For example, it is absurd to give the speed of a car as 48.23456km/h. In such situations we round off a number to a desired number of decimal places. First you need to know if you are rounding to tenths, or hundredths, etc. Or maybe to "so many decimal places".

8. Hence, simply drop the extra digit if it is less than 5. If it is 5 or more, add 1 to the previous digit before dropping the extra digit.

9. Rounding Whole Numbers You may want to round to tens, hundreds, etc. In this case you replace the removed digits with zero.

10. Question • Without doing an exact calculation, determine whether you can afford all the items below if you have only $30. • 1 two-kilogram bottle of corn oil for $6.95 • 5 cans of peach at $1.95 per can • 300g of beef at $1.02 per 100g • 24 packets of recombined milk at $2.85 for 6. • Solution: • Estimated prices • Corn oil ~ $ 7 • Peach can ~ $ 2 • Beef ~ $ 1 • Milk ~ $ 3

11. Working of Total cost Corn oil (1 x 7) 7 Peach cans (5 x 2) 10 Beef (3 x 1) 3 Milk (4 x 3) 12 Total 32 No, we cannot afford all the items as the estimated total exceeds $30.

12. Significant Figures To find the number of significant figures in a number we have the following rules: The following figures in a number are significant: All non-zero figures (e.g. 7.12 has 3 significant figures). b) All zeros between significant figures (e.g. 2003 has 4 significant figures). c) All zeros at the end of a decimal (e.g. 22.300 has 5 significant figures).

13. The following figures in a number are not significant: All zeros at the beginning of a decimal less than 1 (e.g. 0.0000325 has 3 significant figures) b) All zeros at the end of a number may or may not be significant. It depends on how the estimation is made (e.g. 20,000 has 1 significant figure if estimation is made correct to the nearest 10,000)

18. The two zeros here are not significant.

22. Answers

24. References: New Syllabus Mathematics Book 1, 6th edition (Course book). http://www.slideshare.net/sbishop2/estimation-approximation http://www.sunshinemaths.com/topics/numbers-and-pre-algebra/whole-numbers/rounding-off-estimation-approximation/ http://www.mathsisfun.com/rounding-numbers.html