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Problem 4: Calculate the answer up to appropriate numbers of significant digits: (a) 246.24 + 238.278 + 98.3 (b) (1.4 × 2.639) + 117.25 (c) (2.66 × 104) – (1.03 × 103) (d) (112 × 0.456)/(3.2 × 120) (e) 168.99 × 9 (f) 1023 + 8.5489 Rules for fin

Solution

Rules for finding appropriate number of significant digits

  1. Addition: When two or more than two numbers are added, the result (or sum) has number of decimal places equal to the least decimal places in the numbers to be added.
  2. Subtraction: In subtraction, the result has number of decimal places equal to the number with least decimal places.
  3. Product: When two numbers are multiplied, the result has the same number of significant digits as the smallest in the two numbers which are multiplied.
  4. Quotient: In division, the result has the same number of significant digits as the smallest in the two numbers.

Solution

(a) Now, 246.24 + 238.278 + 98.3 = 582.818
Here, 246.24 has 2 decimal places, 238.278 has 3 and 98.3 has 1. According to rule for addition or subtraction, the result (i-e, 582.818) would have 1 decimal place. Therefore, the answer is 582.8.

(b) (1.4 × 2.639) + 117.24
Now, 1.4 × 2.639 = 3.6946
Use the product rule (3 above), the result would be rounded off to the least number of significant digits of the input numbers. 1.4 has 2 sig figs and 2.639 has 4. Therefore, the result would be rounded off to 2 sig figs. All the number after 6 would be dropped. As there is 9 following 6, therefore, the accepted result is 3.7.
In the second step, we find 3.7 + 117.24 = 120.94
Use rule for addition (rule 1 above) and the result should be rounded off to 1 decimal place. (Remember, 3.7 has 1 decimal place). Therefore, the answer would be rounded off to 120.9.

(c) (2.66 × 104) – (1.03 × 103)
Now, (2.66 × 104) – (1.03 × 103) = {(2.66 × 10) – (1.03)} × 103 = (26.6 – 1.03) × 103 = 25.57 × 103 = 2.557 × 104
According to the product rule, the final result would be limited to the number with least sig figs. In this case both the input numbers, 26.6 and 1.03 have same number of sig figs (i-e, 3). Therefore, the accepted answer is 2.56 × 104. (How did we rounded off this number?)

(d) Now 112 × 0.456 = 51.072 and 3.2 × 120 = 384, therefore,
(112 × 0.456)/(3.2 × 120) = 51.072\384 = 0.133
The result is confined to 3 significant figures as the least number of sig figs in the input data is 3.

(e) 168.99 × 9
Now 168.99 × 9 = 1520.91
Apply product rule 3 above. The result should be rounded off to one sig fig. Therefore,
168.99 × 9 = 1520.91 = 2000

(f) 1023 + 8.5489 = 1,031.5489.
Now the least precise number in the input data is 1023, therefore, the result should be rounded off in such a way that there is no number after the decimal point. Therefore, the digit 5 should be dropped. However, according to rules, 1 should be added to the result. Therefore, the accepted result is 1032.

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