Decimals and Exponents

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oquiella: Master | Next Rank: 500 Posts; Posts: 164; Joined: Tue May 12, 2015 9:06 am; Thanked: 3 times

by oquiella » Sat Oct 10, 2015 1:05 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

Please explain in detail

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Source: Beat The GMAT — Problem Solving | Sat Oct 10, 2015 1:05 pm

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

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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Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

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

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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ceilidh.erickson: GMAT Instructor; Posts: 2095; Joined: Tue Dec 04, 2012 3:22 pmFollowed by: 247 members; Thanked: 1443 times

by ceilidh.erickson » Tue Oct 13, 2015 3:26 pm

Here's another way to visualize it:

Ceilidh Erickson
EdM in Mind, Brain, and Education
Harvard Graduate School of Education

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Matt@VeritasPrep: GMAT Instructor; Posts: 2630; Joined: Wed Sep 12, 2012 3:32 pmFollowed by: 119 members; Location: East Bay all the way; Thanked: 625 times; GMAT Score:780

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

One last approach:

.0015/.0300 * 10áµ�â�»áµ� = 5 * 10â�·

15/300 * 10áµ�â�»áµ� = 5 * 10â�·

15 * 10áµ�â�»áµ� = 5 * 300 * 10â�·

10áµ�â�»áµ� = 5 * 20 * 10â�·

10áµ�â�»áµ� = 100 * 10â�·

so (m - k) = 9

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oquiella: Master | Next Rank: 500 Posts; Posts: 164; Joined: Tue May 12, 2015 9:06 am; Thanked: 3 times

by oquiella » Fri Oct 16, 2015 7:31 am

Matt@VeritasPrep wrote:One last approach:

.0015/.0300 * 10áµ�â�»áµ� = 5 * 10â�·

15/300 * 10áµ�â�»áµ� = 5 * 10â�·

15 * 10áµ�â�»áµ� = 5 * 300 * 10â�·

10áµ�â�»áµ� = 5 * 20 * 10â�·

10áµ�â�»áµ� = 100 * 10â�·

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.

10áµ�â�»áµ� = 100 * 10â�·

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 10áµ�â�»áµ� = 10^9

(m-k) = 9

GMAT assassins aren't born, they're made,
Rich

Contact Rich at [email protected]

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