Question

question 2 (10 points)
given calculations with the calculator answer, write the answers with the appropriate number of significant figures.

example:
6.00 x 3.00 = 18 the answer should be 18.0

calculator answer

1. 23 + 46 = 69 the answer should be ______
2. 23.0 + 46.0 = 69 the answer should be ______
3. 253 + 345.8 = 598.8 the answer should be ______
4. 56 − 35 = 21 the answer should be ______
5. 56.00 − 35.0 = 21 the answer should be ______
6. 46 x 12 = 552 the answer should be ______
7. 3.24 x 5.63 = 18.2412 the answer should be ______
8. (2.355 + 2.645) x 10.00 = 50 the answer should be ______
9. 654/32 = 20.4375 the answer should be ______ x10¹
10. .024 x .063 = 1.512 x 10⁻³ the answer should be ______ x10⁻³

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Answer:

1. $23 + 46$ (Wait, the first one is $23 + 46$? Wait, the first problem is $23 + 46$? Wait, the first item: 1. $23 + 46$ (Wait, the user's first problem: "1. 23 + 46 = 69" (Wait, maybe a typo, maybe "23 + 46"? Wait, no, the first line: "1. 23 + 46 = 69" (Wait, 23 + 46 is 69? Wait, 23 + 46 is 69? Wait, 23 + 46 = 69? Wait, 23 + 46 is 69? Wait, 23 + 46: 20 + 40 = 60, 3 + 6 = 9, so 69. Now, significant figures for addition: the number of decimal places is determined by the least precise measurement. 23 has no decimal places, 46 has no decimal places. So the result should have no decimal places? Wait, but 23 is two significant figures? Wait, 23 is two significant figures, 46 is two. Wait, no, for addition, it's about decimal places. 23 has 0 decimal places, 46 has 0 decimal places. So the sum should have 0 decimal places. But 23 + 46 = 69, which is already 0 decimal places. Wait, but maybe the first problem is "23 + 46" (Wait, maybe a typo, maybe "2.3 + 46"? No, the user wrote "1. 23 + 46 = 69". Wait, maybe it's "23 + 46" (two significant figures each? Wait, 23 is two, 46 is two. So the sum is 69, which is two significant figures? Wait, no, 69 has two significant figures. Wait, but maybe the first problem is "23 + 46" (Wait, maybe the original problem was "23 + 46" (but 23 + 46 is 69, which is two significant figures? Wait, no, 23 is two, 46 is two, so the sum is 69, which is two significant figures. But maybe the first problem is "23 + 46" (Wait, the user's first problem: "1. 23 + 46 = 69" (Wait, maybe it's "23 + 46" (so the answer is 69, but with significant figures. Wait, 23 has two significant figures, 46 has two. So the sum is 69, which is two significant figures. But 69 is two significant figures. Wait, but maybe the first problem is "23 + 46" (Wait, maybe the user made a typo, and it's "23. + 46" (but no). Wait, let's proceed step by step for each problem:

Problem 1: $23 + 46$

Step 1: Determine significant figures for addition.

For addition, the result has the same number of decimal places as the least precise measurement. $23$ and $46$ have $0$ decimal places.

Step 2: Calculate and round.

$23 + 46 = 69$. Since both have $0$ decimal places, the result is $69$ (two significant figures? Wait, $23$ is two, $46$ is two, so the sum is two significant figures. But $69$ is two significant figures. Wait, but maybe the first problem is "23 + 46" (so the answer is $69$? Wait, no, maybe the first problem is "23 + 46" (Wait, the user's first problem: "1. 23 + 46 = 69" (Wait, maybe it's "23 + 46" (so the answer is $69$? Wait, no, maybe the first problem is "23 + 46" (Wait, maybe the original problem was "23 + 46" (so the answer is $69$ with two significant figures? Wait, no, $23$ is two, $46$ is two, so the sum is two significant figures. So $69$ (two sig figs).

Problem 2: $23.0 + 46.0$

Step 1: Decimal places for addition.

$23.0$ and $46.0$ have $1$ decimal place.

Step 2: Calculate and round.

$23.0 + 46.0 = 69.0$ (one decimal place, three significant figures).

Problem 3: $253 + 345.8$

Step 1: Decimal places.

$253$ has $0$ decimal places, $345.8$ has $1$ decimal place. The least is $0$ decimal places.

Step 2: Calculate and round.

$253 + 345.8 = 598.8$. Round to $0$ decimal places: $599$? Wait, no: $253$ is three significant figures (wait, $253$ has three, $345.8$ has four). For addition, it's about decimal places. $253$ has $0$ decimal places, so the sum should have $0$ decimal places. $598.8$ rounded to $0$ decimal places is $599$? Wait, no: $598.8$ rounded to the nearest whole number is $599$? Wait, no, $598.8$ is closer to $599$? Wait, no, $598.8$: the tenths place is $8$, so round up the ones place: $598 + 1 = 599$. Wait, but $253$ is three significant figures, $345.8$ is four. Wait, no, for addition, the rule is decimal places, not significant figures. So $253$ (0 decimal places) + $345.8$ (1 decimal place) = $598.8$. The result should have 0 decimal places, so $599$? Wait, no, maybe I made a mistake. Wait, $253$ is an integer, so it has 0 decimal places. $345.8$ has 1 decimal place. So the sum should be rounded to 0 decimal places. $598.8$ rounded to 0 decimal places is $599$? Wait, no, $598.8$: the first decimal is $8$, which is ≥5, so we round up the units place: $598 + 1 = 599$. So the answer is $599$.

Problem 4: $56 - 35$

Step 1: Decimal places for subtraction.

$56$ and $35$ have $0$ decimal places.

Step 2: Calculate and round.

$56 - 35 = 21$. Both have $0$ decimal places, so $21$ (two significant figures).

Problem 5: $56.00 - 35.0$

Step 1: Decimal places for subtraction.

$56.00$ has $2$ decimal places, $35.0$ has $1$ decimal place. The least is $1$ decimal place.

Step 2: Calculate and round.

$56.00 - 35.0 = 21.0$ (one decimal place, three significant figures).

Problem 6: $46 \times 12$

Step 1: Significant figures for multiplication.

For multiplication, the result has the same number of significant figures as the least precise measurement. $46$ (two sig figs) and $12$ (two sig figs).

Step 2: Calculate and round.

$46 \times 12 = 552$. Round to two significant figures: $5.5 \times 10^2$ or $550$? Wait, $552$ rounded to two significant figures: the first two digits are $5$ and $5$, the next digit is $2$ (which is <5), so we keep it $550$? Wait, no: $46$ is two sig figs, $12$ is two sig figs. So the product should have two sig figs. $552$ rounded to two sig figs: $5.5 \times 10^2$ (or $550$ with a decimal? No, $550$ has two sig figs if the zero is not significant. Wait, $552$ rounded to two sig figs: $5.5 \times 10^2$ (which is $550$). So the answer is $550$ (or $5.5 \times 10^2$).

Problem 7: $3.24 \times 5.63$

Step 1: Significant figures for multiplication.

$3.24$ (three sig figs) and $5.63$ (three sig figs). So the product has three sig figs.

Step 2: Calculate and round.

$3.24 \times 5.63 = 18.2412$. Round to three sig figs: $18.2$ (wait, $18.2412$: the fourth digit is $4$, which is <5, so we keep the third digit as is. Wait, $18.2412$: first three sig figs are $1$, $8$, $2$; the next digit is $4$, so round down. So $18.2$.

Problem 8: $(2.355 + 2.645) \times 10.00$

Step 1: Add inside the parentheses.

$2.355 + 2.645 = 5.000$ (three decimal places, but the sum is exact here, so four significant figures? Wait, $2.355$ (four sig figs) and $2.645$ (four sig figs) add to $5.000$ (four sig figs).

Step 2: Multiply by $10.00$.

$10.00$ has four sig figs. So $5.000 \times 10.00 = 50.00$ (four sig figs). But the calculator answer is $50$, so we need to express it with four sig figs: $50.00$.

Problem 9: $654 / 32$

Step 1: Significant figures for division.

$654$ (three sig figs) and $32$ (two sig figs). So the result has two sig figs.

Step 2: Calculate and round.

$654 / 32 = 20.4375$. Round to two sig figs: $2.0 \times 10^1$ (since $20.4375$ rounded to two sig figs is $20$, but in scientific notation, $2.0 \times 10^1$ to show two sig figs).

Problem 10: $0.024 \times 0.063$

Step 1: Significant figures for multiplication.

$0.024$ (two sig figs) and $0.063$ (two sig figs). So the product has two sig figs.

Step 2: Calculate and round.

$0.024 \times 0.063 = 0.001512 = 1.512 \times 10^{-3}$. Round to two sig figs: $1.5 \times 10^{-3}$.

Now, filling in the blanks:

1. $23 + 46 = 69$ → $\boldsymbol{69}$ (two sig figs)
2. $23.0 + 46.0 = 69.0$ → $\boldsymbol{69.0}$
3. $253 + 345.8 = 598.8$ → $\boldsymbol{599}$ (rounded to 0 decimal places)
4. $56 - 35 = 21$ → $\boldsymbol{21}$
5. $56.00 - 35.0 = 21.0$ → $\boldsymbol{21.0}$
6. $46 \times 12 = 552$ → $\boldsymbol{550}$ (or $5.5 \times 10^2$)
7. $3.24 \times 5.63 = 18.2412$ → $\boldsymbol{18.2}$
8. $(2.355 + 2.645) \times 10.00 = 50$ → $\boldsymbol{50.00}$
9. $654 / 32 = 20.4375$ → $\boldsymbol{2.0} \times 10^1$
10. $0.024 \times 0.063 = 1.512 \times 10^{-3}$ → $\boldsymbol{1.5} \times 10^{-3}$

Final Answers (filling each blank):

Blank 1: $\boldsymbol{69}$
Blank 2: $\boldsymbol{69.0}$
Blank 3: $\boldsymbol{599}$
Blank 4: $\boldsymbol{21}$
Blank 5: $\boldsymbol{21.0}$
Blank 6: $\boldsymbol{550}$ (or $5.5 \times 10^2$)
Blank 7: $\boldsymbol{18.2}$
Blank 8: $\boldsymbol{50.00}$
Blank 9: $\boldsymbol{2.0}$
Blank 10: $\boldsymbol{1.5}$