**Operations with scientific notation word problems :**

In this section, we are going to see, how to solve word problems on operations with scientific notation.

**Example 1 : **

The table below shows the population of the three largest countries in North America in 2011. Find the total population of these countries.

**Solution : **

**Method 1 :**

**Step 1 : **

First, write each population with the same power of 10.

United States : 3.1 x 10⁸

Canada : 0.338 x 10⁸

Mexico : 1.1 x 10⁸

**Step 2 : **

Add the multipliers for each population.

3.1 + 0.338 + 1.1 = 4.538

**Step 3 : **

Write the final answer in scientific notation :

**4.538 x 10⁸**

**Method 2 :**

**Step 1 : **

First, write each number in standard notation.

United States : 310,000,000

Canada : 33,800,000

Mexico : 110,000,000

**Step 2 : **

Find the sum of the numbers in standard notation.

310,000,000 + 33,800,000 + 110,000,000 = 453,800,000

**Step 3 : **

Write the final answer in scientific notation :

453,800,000 = **4.538 x 10⁸**

Using the population table in the last example, how many more people live in Mexico than in Canada ? Write your answer in scientific notation.

7.62 x 10⁷ more people live in Mexico than in Canada.

**Example 2 :**

When the Sun makes an orbit around the center of the Milky Way, it travels 2.025 × 10¹⁴ kilometers. The orbit takes 225 million years. At what rate does the Sun travel? Write your answer in scientific notation.

**Solution : **

**Key points : **

The answer is the number of kilometers per year that the Sun travels around the Milky Way.

Set up a division problem using

**Rate = Distance / Time **

to represent the situation.

**Step 1 : **

Substitute the values from the problem into the Rate formula.

**Step 2 : **

Write the expression for rate with years in scientific notation.

That is, 225 million = 2.25 x 10⁸

Then, we have

**Step 3 :**

Find the quotient by dividing the decimals and using the laws of exponents.

Divide the multipliers.

2.025 ÷ 2.25 = 0.9

Divide the powers of 10.

10¹⁴ ÷ 10⁸ = 10¹⁴ ⁻ ⁸

10¹⁴ ÷ 10⁸ = 10⁶

**Step 4 :**

Combine the answers to write the rate in scientific notation.

0.9 x 10⁶ = **9.0 x 10****⁵**

**Justify and Evaluate : **

Use estimation to check the reasonableness of your answer.

9.0 x 10⁵ is close 10⁶, so the answer is reasonable.

1) Light travels at a speed of 1.86 × 10⁵ miles per second. It takes light from the Sun about 4.8 × 10³ seconds to reach Saturn. Find the approximate distance from the Sun to Saturn. Write your answer in scientific notation.

2) Light travels at the speed of 1.17 × 10⁷ miles per minute. Pluto’s average distance from the Sun is 3,670,000,000 miles. On average, how long does it take sunlight to reach Pluto? Write your answer in scientific notation.

**Answers for the above questions : **

1) 8.928 × 10⁸ miles

2) 3.14 × 10² minutes

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