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Decimals and Exponents

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by GMATGuruNY » Sat Oct 10, 2015 1:19 pm
if (0.0015 x 10^m) / (0.03 x 10^k) = (5 x 10^7), then m - k = ?

a. 9
b. 8
c. 7
d. 6
e. 5
Let k=0.
(0.0015 * 10^m) / (0.03* 10^0) = (5 * 10^7)

Convert the decimals to integers:
(15 * 10^(-4) * 10^m / (3 * 10^(-2) * 1) = (5 * 10^7)

Divide the integers and subtract the exponents:
5 * 10^(-2) * 10^m = 5 * 10^7

Cancel the 5's and divide each side by 10^(-2):
10^m = 10^9
m = 9.

Thus:
m-k = 9-0 = 9.

The correct answer is A.
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by Brent@GMATPrepNow » Sat Oct 10, 2015 2:37 pm
if (0.0015 x 10^m) / (0.03 x 10^k) = 5 x 10^7, then m - k = ?

a. 9
b. 8
c. 7
d. 6
e. 5
Mitch's approach is great. Here's another (longer) approach:

First rewrite the equation as:
(0.0015/0.03) x (10^m)/(10^k) = 5 x 10^7
Simplify to get: (0.0015/0.03) x 10^(m-k) = 5 x 10^7

Aside: A quick way to simplify 0.0015/0.03 is to multiply top and bottom by 10,000 to get 15/300.
Simplify to get 1/20

So, we now have (1/20) x 10^(m-k) = 5 x 10^7
Isolate 10^(m-k) by multiplying both sides by 20 to get:
10^(m-k) = (20)(5) (10^7)
Simplify: 10^(m-k) = (100)(10^7)
Simplify: 10^(m-k) = (10^2)(10^7)
Simplify: 10^(m-k) = 10^9

So, m-k must equal 9

Answer: A

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by ceilidh.erickson » Tue Oct 13, 2015 3:26 pm
Here's another way to visualize it:


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by Matt@VeritasPrep » Wed Oct 14, 2015 10:55 pm
One last approach:

.0015/.0300 * 10��� = 5 * 10�

15/300 * 10��� = 5 * 10�

15 * 10��� = 5 * 300 * 10�

10��� = 5 * 20 * 10�

10��� = 100 * 10�

so (m - k) = 9
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by oquiella » Fri Oct 16, 2015 7:31 am
Matt@VeritasPrep wrote:One last approach:

.0015/.0300 * 10��� = 5 * 10�

15/300 * 10��� = 5 * 10�

15 * 10��� = 5 * 300 * 10�

10��� = 5 * 20 * 10�

10��� = 100 * 10�

so (m - k) = 9

HI THERE CAN YOU EXPLAIN THE LAST FEW STEPS BELOW? i THINK I MAYBE MISSING A FUNDAMENTAL HERE. PLEASE EXPLAIN AND GIVE ANOTHER EXAMPLE DEMONSTRATING THE SAME STEP YOU USED. THANK YOU i REALLY APPRECIATE IT.

10��� = 100 * 10�

so (m - k) = 9[/quote]
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by [email protected] » Fri Oct 16, 2015 9:36 am
Hi oquiella,

It's common in 'exponent' questions for you to have to 'rewrite' a given number to get to the correct answer.

For example:

(9^2) can be rewritten as (3^2)^2, which can then be rewritten as 3^4.

In this prompt, when we get to...

(5)(20)(10^7)

We can multiply the 5 and the 20 to get...

100, which can be rewritten as 10^2

When we multiply (10^2) and (10^7), we end up with 10^9

So 10��� = 10^9

(m-k) = 9

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