Problem Solving

The prompt as posted has a typo.
It should read as follows:

As for your question:

(2.3 * 10¹⁴)/(3.9 * 10⁴) ≈ 2/3 * 10¹⁰ ≈ 0.66 * 10¹⁰ = 6.6 * 10⁹.

The closest answer choice is B.

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Student Review #1
Student Review #2
Student Review #3

We can solve the question using equivalent ratios.
We're comparing kilometers and inches, so the ratio is kilometers/inches

1 Kilometre is approximately 3.9 x 10^4 inches
So, we have the ratio: 1/(3.9 x 10^4)

We're told the number of inches to the sun (2.3 X 10^14) and we must find the number of equivalent kilometers.
Let d = the number of kilometers.
So, we have the ratio: 1/(2.3 X 10^14)

We can now write: 1/(3.9 x 10^4) = d/(2.3 X 10^14)
To solve for d, first cross multiply to get: d(3.9 x 10^4) = 1(2.3 X 10^14)
Solve for d to get: d = (2.3 X 10^14)/(3.9 x 10^4)
d = [2.3/3.9][(10^14)/(10^4)]
d = [0.59][10^10]
Check the answer choices...not there.
To get our response into the proper format, rewrite 0.59 as 5.9 x 10^-1 to get...
d = [5.9 x 10^-1][10^10]
d = 5.9 x 10^9
Answer: B

Cheers,
Brent

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Solution:

This problem is a unit conversion with an added twist of scientific notation. We need to convert 2.3 x 10^14 inches to KILOMETERS. We are given that 1 kilometer is approximately 3.9 X 10^4 inches. We also should recognize that we are being asked which of the following is CLOSEST to the average distance between the Sun and the planet, in kilometers. Because we are being asked for an approximation, we can use some estimation here.

To convert 2.3 x 10^14 inches to kilometers, we need to multiply 2.3 x 10^14 inches by the ratio of:

1 km/(3.9 x 10^4 inches)

However, before doing this multiplication, it will make things easier by “cleaning up” each scientific notation expression. Let’s start with 2.3 x 10^14 inches.

2.3 x 10^14 inches

is equivalent to

23 x 10^13 inches

Notice that because we turn 2.3 into 23, or move the decimal one place to the right, we have to then turn 10^14 into 10^13, or move the decimal one place to the LEFT to counterbalance the fact that we’ve moved the decimal one place to the right for 2.3.

Next we can adjust 3.9 x 10^4 inches. However, we can simply round this value up to 4 x 10^4 inches because the question only asks for the closest answer.

Since we’ve rounded 3.9 up to 4, let’s round 23 up to 24 also. That is, we are converting 24 x 10^13 inches into kilometers given that 1 km is approximately 4 x 10^4 inches:

(24 x 10^13 inches) x 1 km/(4 x 10^4 inches)

(24 x 10^13)/(4 x 10^4) km

We can break this work up into two separate calculations:

1) 24/4 = 6

2) 10^13/10^4 = 10^9

Thus, our answer is about 6 x 10^9 km.

We see that the closest answer is 5.9 x 10^9.

Answer: B

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