Significant Figures Part II: Calculations

Objectives When you complete this presentation, you will be able to determine the number of significant figures in a calculated answer

Introduction We know how to determine the number of significant figures in a measurement. 3.442 g has 4 significant figures 0.0025 m has 2 significant figures 140 s has 2 significant figures 0.000420 mL has 3 significant figures Now, we need to learn how to use those measurements in calculations.

Rounding In general, a calculated answer cannot be more precise than the least precise measurement from which it was calculated. The calculated value must be rounded to make it consistent with the measurements.

Rounding To round, we need to determine how many significant figures the answer needs to have. This depends on the measurements and the math process used to determine the answer. We will cover the details of that later in the presentation. First, let’s practice rounding numbers to a proper number of significant figures.

Rounding How do we round a numerical value? We use the same rules we have always used. If the digit to the immediately to the right of the last significant digit is less than 5, we drop the rest of the digits and the value of the last significant digit remains the same. Round 45.244 to 3 sig figs ➠ 45.244 ➠ 45.2 Round 0.85321 to 2 sig figs ➠ 0.85321 ➠ 0.85

Rounding How do we round a numerical value? We use the same rules we have always used. If the digit to the immediately to the right of the last significant digit is 5 or greater we drop the rest of the digits and the value of the last significant digit is increased by 1. Round 62.557 to 3 sig figs ➠ 62.557 ➠ 62.6 Round 0.0545 to 2 sig figs ➠ 0.0545 ➠ 0.055

Rounding Example 1: Round each of the following numbers to the indicated number of significant figures: 54,525.99 m to 3 sig figs 0.00741554 kg to 4 sig figs 37.255 s to 1 sig fig 0.78245 cm to 2 sig figs 355,000 km to 2 sig figs 0.0382574925 L to 3 sig figs 54,500 m 0.007146 kg 40 s 0.78 cm 360,000 km 0.0383 L

Calculations – addition and subtraction The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places. For example: 12.52 m + 349.0 m The 349.0 m measurement has the least number of decimal places. + 8.24 m 369.76 m

Calculations – addition and subtraction The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places. For example: 12.52 m + 349.0 m That measurement will control the number of decimal places in the answer. + 8.24 m 369.76 m

Calculations – addition and subtraction The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places. For example: 12.52 m + 349.0 m The reported answer should be 369.8 m + 8.24 m 369.76 m

Calculations – addition and subtraction Example 2: Perform each operation. Express the answer in the correct number of sig figs. 61.2 m + 9.35 m + 8.6 m 14.2 g + 8.73 g + 0.912 g 35 s + 72.1 s + 1.876 s 9.44 kg – 2.11 kg 1.36 cm + 10.17 cm 34.61 mL – 17.3 mL 79.2 m 23.8 g 109 s 7.33 kg 11.53 cm 17.3 mL

Calculations – multiplication and division The answer to a multiplication or division calculation should be rounded to the same number of significant digits as the measurement with the least number of significant digits. For example: 7.55 m × 0.34 m The 0.34 m measurement has the least number of significant digits (2). 2.567 m2

Calculations – multiplication and division The answer to a multiplication or division calculation should be rounded to the same number of significant digits as the measurement with the least number of significant digits. For example: 7.55 m × 0.34 m That measurement will control the number of decimal places in the answer. 2.567 m2

Calculations – multiplication and division The answer to a multiplication or division calculation should be rounded to the same number of significant digits as the measurement with the least number of significant digits. For example: 7.55 m × 0.34 m The reported answer should be 2.6 m2 2.567 m2

Calculations – multiplication and division Example 3: Perform each operation. Express the answer in the correct number of sig figs. 2.10 m × 0.70 m 2.4526 kg ÷ 8.4 8.3 m × 2.22 m 8432 kg ÷ 12.5 32 s × 15.125 34.61 mL ÷ 17.3 1.5 m2 0.29 kg 18 m2 675 kg 480 s 2.00 mL

Summary A calculated answer cannot be more precise than the least precise measurement from which it was calculated. The calculated value must be rounded to make it consistent with the measurements.

Summary Rules for rounding: If the digit to the immediately to the right of the last significant digit is less than 5, we drop the rest of the digits and the value of the last significant digit remains the same. If the digit to the immediately to the right of the last significant digit is 5 or greater we drop the rest of the digits and the value of the last significant digit is increased by 1.

Summary Significant figures in calculations: The answer to an addition or subtraction calculation should be rounded to the same number of decimal places (not digits) as the measurement with the least number of decimal places. The answer to a multiplication or division calculation should be rounded to the same number of significant digits as the measurement with the least number of significant digits.