Scientific Notation Word Problems Worksheet - Fill and Sign ... - Free Printable

Let's solve each of the scientific notation word problems step by step.

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Problem 1:

According to scientists, the Earth’s mass is \(5.98 \times 10^{24}\) kilograms. The mass of the Sun is \(1.989 \times 10^{30}\) kilograms. How much greater is the mass of the Sun than the mass of the Earth?

#### Step 1: Understand what is being asked
We need to find the difference in mass between the Sun and the Earth:
\[
\text{Mass of Sun} - \text{Mass of Earth}
\]

#### Step 2: Write down the values
- Mass of Sun = \(1.989 \times 10^{30}\) kg
- Mass of Earth = \(5.98 \times 10^{24}\) kg

#### Step 3: Make the exponents the same for subtraction
We can rewrite the smaller number (Earth's mass) so that it has the same exponent as the Sun’s mass.

Convert Earth's mass to \(10^{30}\):
\[
5.98 \times 10^{24} = 5.98 \times 10^{24} = 0.00000598 \times 10^{30}
\]
(We moved the decimal 6 places left because \(10^{30} \div 10^{24} = 10^6\))

So,
\[
5.98 \times 10^{24} = 5.98 \times 10^{-6} \times 10^{30} = 5.98 \times 10^{-6} \times 10^{30}
\]
But easier way: just write both with \(10^{30}\):

\[
5.98 \times 10^{24} = 5.98 \times 10^{24} = 0.00000598 \times 10^{30}
\]

Now subtract:
\[
1.989 \times 10^{30} - 0.00000598 \times 10^{30} = (1.989 - 0.00000598) \times 10^{30}
\]
\[
= 1.98899402 \times 10^{30} \text{ kg}
\]

But since we're looking for "how much greater", we can approximate:

\[
\approx 1.989 \times 10^{30} - 5.98 \times 10^{24} \approx 1.989 \times 10^{30} \quad \text{(since } 5.98 \times 10^{24} \text{ is very small compared to } 10^{30})
\]

But let’s do it more precisely:

\[
\text{Difference} = 1.989 \times 10^{30} - 5.98 \times 10^{24}
\]

Factor out \(10^{24}\):
\[
= 10^{24} (1.989 \times 10^6 - 5.98)
\]
\[
= 10^{24} (1,989,000 - 5.98) = 10^{24} (1,988,994.02)
\]

Now write in scientific notation:
\[
= 1.98899402 \times 10^6 \times 10^{24} = 1.98899402 \times 10^{30}
\]

So the Sun's mass is approximately \(1.989 \times 10^{30}\) kg greater than Earth’s.

But a better way to express this is:

\[
\boxed{1.989 \times 10^{30} - 5.98 \times 10^{24} = 1.98899402 \times 10^{30}} \text{ kg}
\]

Or rounded to 3 significant figures:
\[
\boxed{1.99 \times 10^{30}} \text{ kg}
\]

✔ Answer: The mass of the Sun is about \(1.99 \times 10^{30}\) kg greater than the mass of the Earth.

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Problem 2:

Last year, we noticed that the population of Tewkesbury was \(5.5 \times 10^3\). The population of Liverpool was \(1.3 \times 10^6\). Which town had a larger population and by how much?

#### Step 1: Compare the two populations
- Tewkesbury: \(5.5 \times 10^3 = 5,500\)
- Liverpool: \(1.3 \times 10^6 = 1,300,000\)

Clearly, Liverpool has a much larger population.

#### Step 2: Find the difference
\[
1.3 \times 10^6 - 5.5 \times 10^3
\]

Convert both to same exponent (use \(10^3\)):
\[
1.3 \times 10^6 = 1,300 \times 10^3
\]
So:
\[
1,300 \times 10^3 - 5.5 \times 10^3 = (1,300 - 5.5) \times 10^3 = 1,294.5 \times 10^3
\]

Now convert to proper scientific notation:
\[
1,294.5 \times 10^3 = 1.2945 \times 10^6
\]

✔ Answer: Liverpool had a larger population by \(1.2945 \times 10^6\) people.

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Problem 3:

In Russia, the population was 295 million people in 2002. The total health care costs for the country at that time were $1.7 trillion. Calculate the average amount spent per person on health care.

#### Step 1: Convert numbers to scientific notation
- Population = 295 million = \(295 \times 10^6 = 2.95 \times 10^8\)
- Total cost = $1.7 trillion = \(1.7 \times 10^{12}\) dollars

#### Step 2: Divide total cost by population
\[
\text{Average per person} = \frac{1.7 \times 10^{12}}{2.95 \times 10^8}
\]

Divide coefficients and subtract exponents:
\[
= \frac{1.7}{2.95} \times 10^{12 - 8} = \frac{1.7}{2.95} \times 10^4
\]

Calculate \( \frac{1.7}{2.95} \):
\[
\frac{1.7}{2.95} \approx 0.5763
\]

So:
\[
\approx 0.5763 \times 10^4 = 5.763 \times 10^3
\]

✔ Answer: The average amount spent per person on health care was approximately $5,763.

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✔ Final Answers:

1. The Sun's mass is \(1.99 \times 10^{30}\) kg greater than Earth's.
2. Liverpool had a larger population by \(1.2945 \times 10^6\) people.
3. Average health care spending per person in Russia was \$5,763.

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