Question

SAT Question (Please help, thanks!)

"In 2010, the population of the United States was roughly 309.3 million. Which of the following is the best approximation of the fraction of the US population that developed a case of the mumps in 2010?

A) 8 x 10^3 %
B) 8 x 10^4 %
C) 8 x 10^5%
D) 8 x 10^6 %


Please explain. :D

Answer 1

`8.4 * 10^(-4)\%`

Are these choices suppose to have negative exponents?

Explanation:

So 2600 were present in 2010 while

there were 309.3 million people present in the US( roughly).

The fraction of the US people with mumps is:

`\frac{2600}{309.3 \text{ million}}`

`(2600)/(309.3 \cdot 10^6)`

You can put this in the calculator and then multiply the result by 100 to convert it to a percent.

This is the way I would put it in a calculator:

(2600) ÷ (309.3 × 10 ^ 6)

After this answer is displayed on screen type ×100 to convert to a percent.

You should get:

0.00084%

Now they wrote this answer in scientific notation.

Count the number of spaces you would need to move over to get between 8 and 4.

0.00084

^ ^^ ^

The decimal has been moved 4 times to the right:

`8.4 * 10^(-4)\%`.

8.4 is close to 8

`8 * 10^(-4)\%` (roughly)

Now if you were suppose to do this without a calculator and figure out a pretty rough figure:

`(2600)/(309.3 \cdot 10^6)`

By law of exponents
`(1)/(10^6)=10^(-6)`:

`(2600)/(309.3) \cdot 10^(-6)`

I rounded up 309.3 to 310 so I could cancel out a common factor of 10 in the first fraction:

`(2600)/(310) \cdot 10^(-6)`

Cancel the common factor of 10 in the first fraction:

`(260)/(31) \cdot 10^(-6)`

I round 31 down to 30 to find another factor of 10 to cancel in the first fraction:

`(260)/(30) \cdot 10^(-6)`

Canceled a common factor of 10:

`(26)/(3) \cdot 10^(-6)`

I know that 3 divides 24 and 26 isn't too far from 24:

`(24)/(3) \cdot 10^(-6)`

Reduced 24/3 to 8:

`8 \cot 10^(-6)`

Looking at above I know these aren't equivalent but I knew that my choices are too far apart for exactness to not matter too much.

If you did actually put the following in your calculator they shouldn't deviate too much relatively:

`(2600)/(309.3)`

`(2600)/(310)`

`(260)/(31)`

`(260)/(30)`

`(26)/(3)`

`(24)/(3)`

For fun let's put these in our calculator. You get the following in order of the fraction mentioned: 8.4 , ≈8.38 , ≈8.38 , ≈8.67 , ≈8.67 , 8

So all of the first fractions stayed 8 something.

To convert as a percent multiply by 100 or
`10^(2)`.

`8 \cdot 10^(-6) \cdot 10^(2)`

`8 \cdot 10^(-6+2)`

`8 \cdot 10^(-4)`