Have you ever thought about the planet Mars? Take a look at this dilemma.
On certain dates, Mars is about \begin{align*}4.9 \times 10^7\end{align*} miles from Earth. If a spacecraft is headed toward Mars and has traveled 30% of the distance, how many miles has it gone?
To solve this problem, you will need to know how to use percents and scientific notation. You will learn all that you need to know in this Concept.
Guidance
Scientific notation is another useful mathematical tool that allows us to work with very large or very small numbers.
What is scientific notation?
Scientific notation is when a number is written as a factor and a power of 10. This means that we are using exponents to represent the power of 10.
Remember that any rational number can be written in scientific notation.
It follows the form:
@$\begin{align*}a \times 10^b\end{align*}@$
where @$\begin{align*}a\end{align*}@$ is a number greater than or equal to 1 but less than 10 and @$\begin{align*}b\end{align*}@$ is an exponent of 10
When we conduct operations involving percent with numbers in scientific notation, we can use any operation with the @$\begin{align*}a\end{align*}@$ value as we have seen in these sections. Then we will be sure to write our answer in scientific notation. We may need to adjust the @$\begin{align*}a\end{align*}@$ and @$\begin{align*}b\end{align*}@$ values. Let’s look at how this works.
Find 25% of @$\begin{align*}3 \times 10^{12}\end{align*}@$ .
@$\begin{align*}.25 \times 3 \times 10^{12} = .75 \times 10^{12}\end{align*}@$
Our @$\begin{align*}a\end{align*}@$ value is .75 which is not greater than or equal to 1.
We move the decimal point 1 place to the right on .75 to get 7.5.
Then adjust the exponent 1 integer less—if we make the @$\begin{align*}a\end{align*}@$ value bigger by a factor of 10, then we make the exponent 1 less.
@$\begin{align*}7.5 \times 10^{11}\end{align*}@$
This is our answer.
Very large and very small numbers are not always written in scientific notation. Writing numbers “normally” is called standard notation. We can still work with numbers that are very large but again must be most careful of decimal places. Making an error of 1 decimal place is like multiplying or dividing a number by 10. You would probably agree that there is a big difference between $50 and $500 even though the decimal place is only different by 1 place or we could say by multiplying by 10.
@$\begin{align*}50 \times 10 = 500\end{align*}@$
Take a look at this situation.
In the year 2000, the United States had a population of about 280,000,000 people. By 2010, the population is expected to be 308,000,000. What will the percent increase have been in those 10 years?
@$$\begin{align*}\text{Difference}: 308,000,000 - 280,000,000 &= 28,000,000\\ \text{Divide difference by original}: \frac{28,000,000}{280,000,000} &= .10 = 10\%\end{align*}@$$
The population will have grown by 10% in those 10 years.
Let’s review the steps we did here.
- We identified that we are looking for a percent.
- We found the difference between the original population and the new population.
- Then we divided the difference by the original population.
- Finally, we converted this decimal into our percent.
Write these steps down in your notebook.
Solve each problem.
Example A
Find 30% of .000567
Solution: @$\begin{align*}1.701 \times 10^{-4}\end{align*}@$
Example B
Find 10% of 123,000
Solution: @$\begin{align*}12,300\end{align*}@$
Example C
Find 25% of .0000987. You may round if needed.
Solution: @$\begin{align*}2.47 \times 10^{-5}\end{align*}@$
Now let's go back to the dilemma from the beginning of the Concept.
To solve this, we need to find 30% of @$\begin{align*}4.9 \times 10^7\end{align*}@$ . The @$\begin{align*}a\end{align*}@$ value is our factor that is 4.9 so we will find 30% of that and the power @$\begin{align*}10^7\end{align*}@$ is included in the product once we have multiplied the factor with the percent.
30% of @$\begin{align*}a\end{align*}@$ value: @$\begin{align*}.30 \times 4.9 \times 10^7 = 1.47 \times 10^7\end{align*}@$
The spacecraft has traveled @$\begin{align*}1.47 \times 10^7 \ miles\end{align*}@$ . We can leave our answer in the form of scientific notation
Our @$\begin{align*}a\end{align*}@$ value is now 1.47 which is greater than or equal to 1 and less than 10. There is no need to adjust it.
Guided Practice
Here is one for you to try on your own.
Find 32% of .00000054.
Solution
To work on this problem, we have to change the percent into a decimal first.
32% = .32
Then we notice the key word “of” which means multiply. We are going to multiply the percent times that decimal, which represents a very, very small number.
@$\begin{align*}.32 \times .00000054 = .0000001728\end{align*}@$
This is our answer.
Video Review
Explore More
Directions: Round the @$\begin{align*}a\end{align*}@$ values to the nearest hundredth and place your answers in scientific notation.
- Find 62% of @$\begin{align*}3.5 \times 10^9\end{align*}@$ .
- Find 5% of @$\begin{align*}9.1 \times 10^{13}\end{align*}@$ .
- Find 180% of @$\begin{align*}6.3 \times 10^{-17}\end{align*}@$ .
- Find 12% of @$\begin{align*}.18 \times 10^3\end{align*}@$ .
- Find 22% of @$\begin{align*}56.4 \times 10^{-2}\end{align*}@$ .
- Find 14% of @$\begin{align*}1.8 \times 10^{-5}\end{align*}@$ .
- Find 30% of @$\begin{align*}.999 \times 10^{12}\end{align*}@$ .
Directions: Answer each question and leave your answers in standard form.
- What percent of 8,570,000 is 152?
- Find 230% of .00000488
- The number .00036 is 45% of what number?
- Find 23% of 98.78
- Find 150% of .0000866
- Find 210% of .002368
- Find 30% of .000009
- A light year is about 5,880,000,000,000 miles. In one month, it travels about 8.2% of that distance. About how far does it travel in one month?