Question

1.2 measurements in chemistry
perform the following arithmetic and round off the answer to the correct number of significant figures. (47.346 - 47.312)/(7.42 × 1.56)
○ 0.0029
○ 0.003
● 0.00294
○ 0.002941

Explanation:

Step1: Calculate the numerator

First, we calculate the subtraction in the numerator: \(47.346 - 47.312 = 0.034\)

Step2: Calculate the denominator

Next, we calculate the multiplication in the denominator: \(7.42\times1.56 = 11.5752\)

Step3: Divide numerator by denominator

Now, we divide the result of the numerator by the result of the denominator: \(\frac{0.034}{11.5752}\approx0.002937\)

Step4: Determine significant figures

For subtraction \(47.346 - 47.312\), the number of decimal places in the least precise measurement is 3 (both have 3 decimal places), so the result \(0.034\) has 2 significant figures. For multiplication \(7.42\times1.56\), \(7.42\) has 3 significant figures and \(1.56\) has 3 significant figures, so the result \(11.5752\) should be considered as having 3 significant figures for the purpose of division (since when dividing, the result should have the same number of significant figures as the least precise measurement in the division, here the numerator has 2 significant figures? Wait, no, let's re - check. Wait, the numerator is \(47.346 - 47.312=0.034\) (the number of decimal places: \(47.346\) has 3 decimal places, \(47.312\) has 3 decimal places, so the result of subtraction has 3 decimal places, but the significant figures: \(0.034\) has 2 significant figures). The denominator \(7.42\times1.56\): \(7.42\) (3 sig figs), \(1.56\) (3 sig figs), so the product is \(7.42\times1.56 = 11.5752\) (we can consider it as having 3 sig figs for the multiplication step). Then when we do division \(\frac{0.034}{11.5752}\), the number of significant figures is determined by the least number of significant figures in the numerator and denominator. The numerator \(0.034\) has 2 significant figures, the denominator (from multiplication) has 3 significant figures. So the result of the division should have 2 significant figures? Wait, no, maybe I made a mistake. Wait, let's recalculate the numerator: \(47.346-47.312 = 0.034\) (the difference is \(0.034\), which is \(3.4\times 10^{- 2}\), so two significant figures). The denominator: \(7.42\times1.56\). Let's calculate \(7.42\times1.56\):

\(7.42\times1.56=(7 + 0.42)\times(1+0.56)=7\times1+7\times0.56 + 0.42\times1+0.42\times0.56=7+3.92+0.42 + 0.2352=11.5752\)

Now, when we divide \(0.034\) by \(11.5752\):

\(0.034\div11.5752\approx0.002937\)

Now, let's check the significant figures. The numerator \(47.346 - 47.312\): the rule for subtraction is that the result has the same number of decimal places as the least precise measurement. \(47.346\) has 3 decimal places, \(47.312\) has 3 decimal places, so the result \(0.034\) has 3 decimal places, but in terms of significant figures, leading zeros are not significant, so \(0.034\) has 2 significant figures. The denominator \(7.42\times1.56\): for multiplication, the result has the same number of significant figures as the factor with the least number of significant figures. Both \(7.42\) and \(1.56\) have 3 significant figures, so the product has 3 significant figures. When dividing, the result should have the same number of significant figures as the least precise measurement (the one with the least number of significant figures) in the division. So the numerator has 2 significant figures, so the result of the division should have 2 significant figures? Wait, but \(0.002937\) rounded to 2 significant figures is \(0.0029\) (because the first non - zero digit is 2, the next is 9, and the third is 3 which is less than 5? Wait, no, \(0.002937\): the significant figures are 2,9,3,7. If we take 2 significant figures, we look at the first two significant figures \(2\) and \(9\), the next digit is \(3\), which is less than 5, so we keep \(2\) and \(9\), so \(0.0029\). But wait, maybe I messed up the significant figures for the subtraction. Let's re - evaluate the subtraction:

\(47.346-47.312 = 0.034\). The values \(47.346\) and \(47.312\) have three decimal places. The difference is \(0.034\), which is \(3.4\times10^{-2}\). The number of significant figures in \(0.034\) is 2. The multiplication \(7.42\times1.56\): \(7.42\) (3 sig figs), \(1.56\) (3 sig figs), so the product is \(11.5752\) (3 sig figs). Then the division: \(\frac{0.034}{11.5752}\). The numerator has 2 sig figs, denominator has 3 sig figs. So the result should have 2 sig figs. \(0.002937\) rounded to 2 sig figs is \(0.0029\) (because the third digit after the leading zeros is 3, which is less than 5, so we don't round up the second significant figure).

Wait, but let's check the calculation again:

\(47.346 - 47.312=0.034\)

\(7.42\times1.56 = 11.5752\)

\(0.034\div11.5752=\frac{3.4\times 10^{-2}}{1.15752\times10^{1}}=\frac{3.4}{1.15752}\times10^{-3}\approx2.937\times10^{-3}=0.002937\)

Now, if we consider the significant figures for the subtraction: the precision of \(47.346\) and \(47.312\) is to the thousandth place (0.001). So the difference \(0.034\) is to the hundredth place? Wait, no, \(47.346-47.312 = 0.034\), which is \(0.034\) (the last digit is in the hundredth place? Wait, \(47.346\) is \(47\) units, \(3\) tenths, \(4\) hundredths, \(6\) thousandths. \(47.312\) is \(47\) units, \(3\) tenths, \(1\) hundredth, \(2\) thousandths. Subtracting the thousandths place: \(6 - 2=4\) thousandths. Subtracting the hundredths place: \(4 - 1 = 3\) hundredths. Tenths and units are the same. So the result is \(0.034\), which is \(3\) hundredths and \(4\) thousandths, so the number of decimal places is 3. So the significant figures: the non - zero digits start at the hundredth place? Wait, no, \(0.034\) can be written as \(3.4\times10^{-2}\), so it has 2 significant figures.

The multiplication \(7.42\times1.56\): \(7.42\) has 3 significant figures, \(1.56\) has 3 significant figures, so the product has 3 significant figures.

When dividing, the number of significant figures in the result is equal to the number of significant figures in the least precise measurement (the one with the fewest significant figures). So the numerator has 2 significant figures, so the result should have 2 significant figures. So \(0.002937\) rounded to 2 significant figures is \(0.0029\) (since the third significant figure is 3, which is less than 5, we don't round up the second significant figure).

Answer:

\(0.0029\)