S1G3 Math CH3 Notes (NBSS)

NAVAL BASE SECONDARY SCHOOL MATHEMATICS DEPARTMENT Sec 1 Express Mathematics Chapter 3 : Approximation and Estimation Notes Name: ________________________________ ( ) Class: _________ Date: ____________ 1 3.1 Introduction to Approximation Objectives: At the end of the lessons, students should be able to ● round off numbers to a required number of decimal places ● round off numbers to a required number of significant figures Approximation is the rounding of numbers to a required degree of accuracy. Prerequisite Let’s recap on the number placing positions. Fill in the blanks with a number from the following figure. 1894.237 (i) Thousands place: ______ (v) Thousandths place: ______ (ii) Hundreds place: ______ (vi) Hundredths place: ______ (iii) Tens place: ______ (vii) Tenths place: ______ (iv) Ones place: ______

2 Example 1 (Rounding off) Based on the following number, correct to: 16.49583 (a) the nearest whole number. Ans: 16 (b) two decimal places. Ans: 16.50 (c) the nearest integer Ans: 16 Practice 1 Round off the following correct to (i) two decimal places and (ii) the nearest whole number: (a) $12.3125 (b) 57.6283 (c) 1.99976 cm (d) 101.3333 kg (e) 3.23495 (f) $ 4.5671 When rounding off numbers: (1) Round down if the digit under consideration is 4 or less. Round up if the digit under consideration is 5 or more. (2) Check the value of approximation against the original value

3 Example 2 (Significant figures) Let us consider the following case. A grain of table salt weighs 0.0001208g. If the answer is rounded off to 3 decimal places, the weight of the salt is now: 0.0001208 g = 0.000 g (3 d.p.) This value we obtain is not useful for calculations! Instead of rounding off to 3 decimal places, we round off according to significant figures. 5 Rules for Determining Number of Significant Figures: All non-zero digits are significant. E.g. 34.58 has 4 significant figures. All zeros between non-zero digits are significant. E.g. 10237 has 5 significant figures. In a decimal, all zeros after decimal points and a non-zero number are significant. E.g. 1.7890 has 5 significant figures. In a decimal, all zeros after a non-zero digit are significant. E.g. 0.0056 has 2 significant figures. All zeros at the end of a whole number may or may not be significant depending on how the number is rounded off E.g. 10010 has 4 significant figures OR 5 significant figures.

4 Practice 2 Based on the following number, correct to: 21.03460970 (a) 2 significant figures. Ans: ___________ (b) 3 significant figures. Ans: ___________ (c) 4 significant figures. Ans: ___________ (d) 5 significant figures. Ans: ___________ (e) 7 significant figures. Ans: ___________ In a rounded off decimal, all digits, other than zeros preceding the first non-zero digit, are significant figures. E.g. 0.0503 has 3 significant figures ‘503’ 0.05030 has 4 significant figures ‘5030’ In a rounded off whole number, the ending zeros may or may not be significant. If it is the result of rounding off to the nearest 10, 100, 1000, …, then the last 1 zero, 2 zeros, 3 zeros, …, respectively, are not significant. All the other digits are significant figures. E.g. 23000 (to the nearest 10) has 4 significant figures 23000 (to the nearest 100) has 3 significant figures 23000 (to the nearest 1000) has 2 significant figures

5 Example 3 Express the following correct to 2 significant figures. (a) 0.04862 Solution: 0.049 (2SF) (b) 2.99 Solution: 3.0 (2SF) (c) 0.90945 Solution: 0.91 (2SF) (d) 54.0243 Solution: 54 (2SF) (e) 1.589 Solution: 1.6 (2SF) (f) 20.03 Solution: 20 (2SF) Practice 3(a) Write 1 354.154 correct to the number of significant figures indicated below. (a) 1 (b) 2 (c) 3 (d) 4 (e) 5

6 Practice 3(b) Round off (a) 29 470 to 3 significant figures (b) 98 836 to 2 significant figures (c) 851.02 to 4 significant figures (d) 750.645 to 4 significant figures (e) 0.079816 to 3 significant figures (f) 0.000 557 to 2 significant figures (g) 6407.37 to 5 significant figures (h) 3.0072 to 4 significant figures Practice 3(c) Express each of the following correct to (i) 2 decimal places, (ii) 2 significant figures. (a) 3.825 1 (b) 0.013 527 6 (c) 0.207 9 (d) 5.068 4 (e) 12.384 7 (f) 197.143 92

7 Practice 3(d) State the number of significant figures in each of the following. (a) 0.00063 (b) 7006.12 (c) 27.3752 (d) 392.6445 (e) 1.0780 Example 4 (a) Calculate 494.6 56.33 98.12 showing all the figures on your calculator display. (b) Give your answer correct to 1 decimal place. Solution: (a) 0.08948635556 (b) 0.1 Practice 4 (a) Calculate 384.8 47.23 91.43 showing all the figures on your calculator display. (b) Give your answer correct to 2 decimal place.

8 Example 5 The mass of a pebble is 25 grams correct to 2 significant figures. Find the minimum mass of the pebble. Solution: Thought process: To find the minimum mass of the pebble, we need to determine the range of values that round to 25 grams when rounded to 2 significant figures. Possible values include 24.5, 24.6, 24.7, … , 25.1, 25.2, 25.3, 25.4, and the smallest value is 24.5 grams. Practice 5 The length of a string is 340 cm correct to 3 significant figures. Find the maximum length of the string. Complete WS 1 by _________

9 3.2 Approximation & Approximation errors in Real-world contexts Learning Experience 1 Paste your receipt collected below and answer the questions that follow. (a) Express the total amount of the items correct to 3 significant figures. (b) Express the cash given correct to (i) 2 significant figures, (ii) 3 significant figures, (iii) 1 significant figure.

10 (c) Look at the below receipt issued by a restaurant in Singapore. Discuss with your partner and fill in the answers. (i) What was the amount charged on the customer? (ii) How was approximation used in this scenario? (iii) If you were the owner and did not want to lose out the few cents due to rounding off, what other mode of approximation can you do to solve this problem?