Question

add or subtract the following measurements. be sure each answer you enter contains the correct number of significant digits. 12.777 ml - 1.9 ml = \square ml 9.670 ml + 11.50 ml = \square ml 11.720 ml + 14.50 ml = \square ml

Explanation:

First Calculation: \( 12.777 \, \text{mL} - 1.9 \, \text{mL} \)

Step 1: Perform the subtraction

Subtract the two volumes: \( 12.777 - 1.9 = 10.877 \). Now, consider significant figures. The number \( 1.9 \) has two significant figures after the decimal? Wait, no, \( 1.9 \) has two significant figures (the 1 and 9). When subtracting, the result should have the same number of decimal places as the least precise measurement. \( 12.777 \) has three decimal places, \( 1.9 \) has one decimal place. So we round to one decimal place? Wait, no, significant figures for subtraction: the number of decimal places. \( 1.9 \) has one decimal place, so the result should have one decimal place. Wait, \( 12.777 - 1.9 = 10.877 \), rounded to one decimal place is \( 10.9 \)? Wait, no, maybe I messed up. Wait, \( 12.777 - 1.9 = 10.877 \). Let's check the significant figures. The rule for addition/subtraction is that the result has the same number of decimal places as the term with the least number of decimal places. \( 1.9 \) has 1 decimal place, \( 12.777 \) has 3. So we round to 1 decimal place. \( 10.877 \) rounded to 1 decimal place is \( 10.9 \)? Wait, no, \( 10.877 \) to one decimal place: look at the second decimal, which is 7, so round up the first decimal: 8 + 1 = 9. So \( 10.9 \) mL. Wait, but maybe the problem is about significant figures in terms of the number of significant figures. Wait, \( 1.9 \) has two significant figures, \( 12.777 \) has five. Wait, no, the rule for addition/subtraction is decimal places, not significant figures. So \( 12.777 - 1.9 = 10.877 \), and since \( 1.9 \) has one decimal place, the result should have one decimal place. So \( 10.9 \) mL.

Step 2: Wait, maybe I made a mistake. Let's recalculate. \( 12.777 - 1.9 = 10.877 \). Now, \( 1.9 \) has one decimal place, so the answer should have one decimal place. So \( 10.9 \) mL.

Second Calculation: \( 9.670 \, \text{mL} + 11.50 \, \text{mL} \)

Step 1: Perform the addition

Add the two volumes: \( 9.670 + 11.50 = 21.170 \). Now, check the decimal places. \( 9.670 \) has three decimal places, \( 11.50 \) has two. So the result should have two decimal places (since 11.50 has two). So \( 21.170 \) rounded to two decimal places is \( 21.17 \) mL? Wait, no, \( 21.170 \) to two decimal places is \( 21.17 \) (since the third decimal is 0, which doesn't change the second). Wait, but \( 9.670 \) has three decimal places, \( 11.50 \) has two. So the least number of decimal places is two, so the result should have two decimal places. So \( 21.17 \) mL. Alternatively, maybe the significant figures: \( 9.670 \) has four significant figures, \( 11.50 \) has four. So the sum is \( 21.170 \), which can be written as \( 21.17 \) (since the trailing zero after the decimal in 9.670 is significant, but when adding, the decimal places matter). Wait, \( 9.670 \) is 9.670 (three decimal places), \( 11.50 \) is 11.50 (two decimal places). So the sum is \( 21.170 \), and we round to two decimal places: \( 21.17 \) mL.

Third Calculation: \( 11.720 \, \text{mL} + 14.50 \, \text{mL} \)

Step 1: Perform the addition

Add the two volumes: \( 11.720 + 14.50 = 26.220 \). Now, check the decimal places. \( 11.720 \) has three decimal places, \( 14.50 \) has two. So the result should have two decimal places (since 14.50 has two). So \( 26.220 \) rounded to two decimal places is \( 26.22 \) mL. Alternatively, the trailing zero in 11.720 is significant, but when adding, the decimal places determine the precision. So \( 26.22 \) mL.

Wait, but maybe the problem is not about decimal places but about significant figures in the sense of the number of significant figures. Let's re-examine:

First problem: \( 12.777 - 1.9 \). \( 1.9 \) has two significant figures, \( 12.777 \) has five. The result should have two significant figures? No, that's for multiplication/division. For addition/subtraction, it's decimal places. So \( 1.9 \) has one decimal place, so result has one decimal place: \( 10.9 \) mL.

Second problem: \( 9.670 + 11.50 \). \( 9.670 \) has four significant figures, \( 11.50 \) has four. The sum is \( 21.170 \), which has five significant figures, but we need to consider decimal places. \( 9.670 \) has three decimal places, \( 11.50 \) has two. So the sum should have two decimal places: \( 21.17 \) mL (since \( 21.170 \) rounded to two decimal places is \( 21.17 \)).

Third problem: \( 11.720 + 14.50 \). \( 11.720 \) has three decimal places, \( 14.50 \) has two. So sum has two decimal places: \( 26.22 \) mL (since \( 26.220 \) rounded to two decimal places is \( 26.22 \)).

But maybe the problem is simpler, just adding/subtracting without worrying about significant figures, just the arithmetic. Let's check:

First: \( 12.777 - 1.9 = 10.877 \)

Second: \( 9.670 + 11.50 = 21.17 \) (since \( 9.670 + 11.50 = 21.170 \), which can be written as \( 21.17 \))

Third: \( 11.720 + 14.50 = 26.22 \) (since \( 11.720 + 14.50 = 26.220 \), written as \( 26.22 \))

Wait, maybe the problem is just about the arithmetic, not significant figures. Let's do the arithmetic:

1. \( 12.777 - 1.9 = 10.877 \)
2. \( 9.670 + 11.50 = 21.17 \) (because \( 9.670 + 11.50 = 21.170 \), and maybe we can write it as \( 21.17 \))
3. \( 11.720 + 14.50 = 26.22 \) (because \( 11.720 + 14.50 = 26.220 \), written as \( 26.22 \))

But the problem says "Be sure each answer you enter contains the correct number of significant digits." So let's apply the significant figure rules for addition/subtraction:

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

First calculation: \( 12.777 \, \text{mL} - 1.9 \, \text{mL} \)

• \( 12.777 \) has 3 decimal places.
• \( 1.9 \) has 1 decimal place.
• So the result should have 1 decimal place.
• \( 12.777 - 1.9 = 10.877 \). Rounded to 1 decimal place: \( 10.9 \, \text{mL} \).

Second calculation: \( 9.670 \, \text{mL} + 11.50 \, \text{mL} \)

• \( 9.670 \) has 3 decimal places.
• \( 11.50 \) has 2 decimal places.
• So the result should have 2 decimal places.
• \( 9.670 + 11.50 = 21.170 \). Rounded to 2 decimal places: \( 21.17 \, \text{mL} \).

Third calculation: \( 11.720 \, \text{mL} + 14.50 \, \text{mL} \)

• \( 11.720 \) has 3 decimal places.
• \( 14.50 \) has 2 decimal places.
• So the result should have 2 decimal places.
• \( 11.720 + 14.50 = 26.220 \). Rounded to 2 decimal places: \( 26.22 \, \text{mL} \).

So the answers are:

1. \( 10.9 \, \text{mL} \)
2. \( 21.17 \, \text{mL} \)
3. \( 26.22 \, \text{mL} \)

Wait, but let's check the second calculation again. \( 9.670 + 11.50 = 21.170 \). The term \( 11.50 \) has 2 decimal places, so…

Answer:

1. \( 10.9 \, \text{mL} \)
2. \( 21.17 \, \text{mL} \)
3. \( 26.22 \, \text{mL} \)