Question

question 6 of 10
perform the following calculations and report your answers to the correct number of significant figures.
(\frac{35.934}{(24.6 - 20.4)} =)
((95.494 - 89.880) \times 16.460 =)

Explanation:

First Calculation: $\boldsymbol{\frac{35.934}{(24.6 - 20.4)}}$

Step1: Calculate the denominator

First, we calculate the value inside the parentheses: $24.6 - 20.4 = 4.2$

Step2: Perform the division

Now, we divide 35.934 by 4.2: $\frac{35.934}{4.2} \approx 8.555714$

Step3: Determine significant figures

The denominator $24.6 - 20.4$ has a result with two significant figures (because the least number of decimal places in the subtraction is one, but when subtracting, the result's precision is to the tenths place, and the number of significant figures for the result of subtraction here: 24.6 has three, 20.4 has three, the difference is 4.2 which has two significant figures? Wait, no: when subtracting, the number of decimal places matters. 24.6 has one decimal place, 20.4 has one decimal place, so the result (4.2) has one decimal place, and the number of significant figures: 4.2 has two significant figures? Wait, 4.2: the 4 and 2 are significant, so two. The numerator 35.934 has five significant figures. When dividing, the result should have the same number of significant figures as the least precise measurement, which is two (from the denominator 4.2). Wait, but wait: 24.6 - 20.4 = 4.2. 24.6 has three significant figures, 20.4 has three. The subtraction: 24.6 - 20.4 = 4.2 (the decimal places: both have one decimal place, so the result has one decimal place). The number of significant figures in 4.2: two (the 4 and the 2). The numerator is 35.934 (five sig figs). So when dividing, the result should have two significant figures? Wait, no, maybe I made a mistake. Wait, 24.6 - 20.4: 24.6 is three sig figs, 20.4 is three sig figs. The difference is 4.2, which is two sig figs? Wait, 4.2: the 4 is significant, the 2 is significant, so two. So the denominator has two sig figs, numerator has five. So the result of the division should have two sig figs? Wait, but let's check again. Wait, 24.6 - 20.4: 24.6 - 20.4 = 4.2. The precision is to the tenths place (one decimal place). The number of significant figures: 4.2 has two significant figures. So when we divide 35.934 (five sig figs) by 4.2 (two sig figs), the result should have two significant figures? Wait, no, maybe I messed up the rule. The rule for addition/subtraction is that the result has the same number of decimal places as the least precise measurement. The rule for multiplication/division is that the result has the same number of significant figures as the least precise measurement. So in the denominator, we have a subtraction: 24.6 (one decimal place) - 20.4 (one decimal place) = 4.2 (one decimal place). The number of significant figures in 4.2: two (the 4 and 2). Then, the division: 35.934 (five sig figs) divided by 4.2 (two sig figs) should give a result with two sig figs? Wait, but 35.934 / 4.2 = 8.5557... So rounding to two significant figures would be 8.6? Wait, no, 8.5557 rounded to two significant figures: the first two are 8 and 5, the next digit is 5, so we round up the 5 to 6, so 8.6? Wait, but maybe I made a mistake in the number of significant figures for the denominator. Wait, 24.6: three sig figs, 20.4: three sig figs. The difference is 4.2, which is two sig figs? Wait, 4.2: the 4 is the first significant figure, the 2 is the second. So yes, two. So the division result should have two sig figs. Wait, but let's check with another approach. Alternatively, maybe the denominator is considered to have two decimal places? No, 24.6 and 20.4 have one decimal place each. So the subtraction result has one decimal place. So 4.2 is correct. Then, 35.934 / 4.2: let's do the calculation more accurately. 35.934 ÷ 4.2. 4.2 × 8 = 33.6, 35.934…

Step1: Calculate the subtraction

First, we calculate the value inside the parentheses: $95.494 - 89.880 = 5.614$

Step2: Perform the multiplication

Now, we multiply 5.614 by 16.460: $5.614 \times 16.460 \approx 92.40644$

Step3: Determine significant figures

The subtraction: 95.494 (five sig figs) - 89.880 (five sig figs) = 5.614 (four sig figs? Wait, 95.494 has five decimal places? No, 95.494 has three decimal places, 89.880 has three decimal places. So the subtraction result has three decimal places: 5.614 (which is 5.614, three decimal places). The number of significant figures in 5.614: four (5,6,1,4). Then, the multiplication: 5.614 (four sig figs) × 16.460 (five sig figs). When multiplying, the result should have the same number of significant figures as the least precise measurement, which is four (from 5.614). So we round 92.40644 to four significant figures. 92.40644 rounded to four significant figures: the first four are 9,2,4,0. The next digit is 6, which is more than 5, so we round up the 0 to 1? Wait, no: 92.40644. Let's write it as 92.40644. The first significant figure is 9, second 2, third 4, fourth 0. The next digit is 6, so we round the fourth digit (0) up by 1, making it 1. So 92.41? Wait, no, wait: 92.40644. Let's count the significant figures: 9 (1), 2 (2), 4 (3), 0 (4), 6 (5), etc. Wait, 5.614 has four significant figures (5,6,1,4). 16.460 has five (1,6,4,6,0). So the multiplication result should have four significant figures. So 5.614 × 16.460 = 92.40644. Rounding to four significant figures: look at the fifth digit. The number is 92.40644. The first four significant figures are 9,2,4,0. The next digit is 6, which is ≥5, so we round the fourth digit (0) up to 1. So 92.41? Wait, but 5.614 has four sig figs, 16.460 has five. So the result should have four sig figs. So 92.41? Wait, but let's check the subtraction again. 95.494 - 89.880 = 5.614. 95.494 has five sig figs, 89.880 has five sig figs. The subtraction: the number of decimal places is three (both have three decimal places), so the result has three decimal places, and the number of significant figures: 5.614 has four (since the first non-zero digit is 5, then 6,1,4). So yes, four sig figs. Then, multiplying by 16.460 (five sig figs) gives a result with four sig figs. So 92.41? Wait, but 5.614 × 16.460: let's calculate it more accurately. 5 × 16.460 = 82.3, 0.6 × 16.460 = 9.876, 0.01 × 16.460 = 0.1646, 0.004 × 16.460 = 0.06584. Adding them up: 82.3 + 9.876 = 92.176; 92.176 + 0.1646 = 92.3406; 92.3406 + 0.06584 = 92.40644. So 92.40644. Rounding to four significant figures: the fourth significant figure is 0 (the number is 9 2 4 0 6...). The next digit is 6, so we round the 0 up to 1, making it 92.41. Wait, but maybe I made a mistake. Alternatively, maybe the subtraction result is 5.614, which has four sig figs, and 16.460 has five, so the multiplication result should have four sig figs. So 92.41.

Answer:

First calculation: $\boxed{8.6}$ (wait, but earlier I thought 8.5557 rounded to two sig figs is 8.6? Wait, 8.5557: first two sig figs are 8 and 5. The next digit is 5, so we round up the 5 to 6, so 8.6. Yes.

Second calculation: $\boxed{92.41}$ (rounded to four significant figures). Wait, but let's check again. Wait, 5.614 has four sig figs, 16.460 has five. So the product should have four sig figs. 92.40644 rounded to four sig figs: 92.41 (since the fourth digit is 0, next is 6, so round up 0 to 1). Yes.

Wait, but maybe the first calculation's denominator has two sig figs, so the result should have two. Let's confirm the significant figures rules:

• For addition/subtraction: the result has the same number of decimal places as the term with the least number of decimal places.

• For multiplication/division: the result has the same number of significant figures as the term with the least number of significant figures.

First calculation:

• Denominator: 24.6 - 20.4 = 4.2. 24.6 has 1 decimal place, 20.4 has 1 decimal place, so the result has 1 decimal place. The number of significant figures in 4.2: 2 (since 4 and 2 are significant).

• Numerator: 35.934 has 5 significant figures.

• Division: so the result should have 2 significant figures (since 4.2 has 2, which is less than 5).

35.934 / 4.2 = 8.5557... Rounded to 2 significant figures: 8.6 (because the third digit is 5, so we round up the second digit from 5 to 6).

Second calculation:

• Subtraction: 95.494 - 89.880 = 5.614. 95.494 has 3 decimal places, 89.880 has 3 decimal places, so the result has 3 decimal places. The number of significant figures in 5.614: 4 (5, 6, 1, 4).

• Multiplication: 5.614 (4 sig figs) × 16.460 (5 sig figs). So the result should have 4 sig figs.

5.614 × 16.460 = 92.40644. Rounded to 4 sig figs: 92.41 (the fourth sig fig is 0, the next digit is 6, so we round up 0 to 1).

So the answers are:

First: $\boxed{8.6}$

Second: $\boxed{92.41}$