If 0.0015*10^m/0.03*10^k=5*10^7, then m - k =

Question

If  \frac{0.0015*10^m}{0.03*10^k}=5*10^7 , then m - k =

(A) 9
(B) 8
(C) 7
(D) 6
(E) 5

Bunuel · Accepted Answer

\frac{0.0015*10^m}{0.03*10^k}=5*10^7 ;

\frac{15*10^{-4}*10^m}{3*10^{-2}*10^k}=5*10^7 ;

5*\frac{10^{m-4}}{10^{k-2}}=5*10^7 ;

10^{(m-4)-(k-2)}=10^7 ;

(m-4)-(k-2)=7 ;

m-k=9 .

Answer: A.

ScottTargetTestPrep · Answer

We start by simplifying the numerator and denominator of the given fraction. First, we simplify the numerator:

(0.0015)(10^m)

It will be helpful to convert 0.0015 to an integer. To do so we must move the decimal point in 0.0015 four places to the right. Since we are making 0.0015 larger by four decimal places we must make 10^m, smaller by four decimal places. Thus, 10^m now becomes 10^(m-4). Thus, the numerator becomes (15)(10^(m-4)).

Next we can simplify the denominator:

(0.03)(10^k)

It will be helpful to convert 0.03 to an integer. To do so we must move the decimal point in 0.03 two places to the right. Since we are making 0.03 larger by two decimal places we must make 10^k, smaller by two decimal places. Thus, 10^k now becomes 10^(k-2). The denominator can thus be re-expressed as (3)(10^(k-2)).

So now we are left with:

/  = 5(10^7)

Dividing 15 by 3 on the left hand side of the equation, we have 15/3 = 5. Recall that when we divide powers of like bases, we subtract the exponents, so 10^(m-4)/10^(k-2) = 
10^((m-4) – (k-2)) = 10^(m-k-2). Therefore, we have

5(10^(m-k-2)) = (5)(10^7)

5 will cancel out from both sides of the equation, leaving us with:

10^(m-k-2)=10^7

Because we are left with a base of 10 on both the right-hand side and the left-hand side of the equation, we can drop the base and set the exponents equal and hence determine the value of m – k:

m – k – 2 = 7

m – k = 9

The answer is A.

BrentGMATPrepNow · Answer

Another approach is to  assign k a "nice" value.

Let's see what happens when  k = 0

We get: ( 0.0015  x  10^m ) / ( 0.03  x  10^ 0  ) =  5  x  10^7

Simplify: ( 0.0015  x  10^m ) / ( 0.03  x  1 ) =  5  x  10^7

Simplify: ( 0.0015  x  10^m ) / (0.03) =  5  x  10^7

Multiply both sides by 0.03 to get:  0.0015  x  10^m  =  0.15  x  10^7

Eliminate blue decimals by multiplying both sides by 10,000 to get:  15  x  10^m  =  1500  x  10^7

Divide both sides by 15 to get:  1  x  10^m  =  100  x  10^7

Rewrite 100 as 10^2:  1  x  10^m  =  10^2  x  10^7

Simplify: 10^m = 10^9

So, m = 9

In other words, when k = 0, m = 9
So, m - k = 9 - 0 = 9

Answer: A

Cheers,
Brent

maibhihun · Answer

Rearrange the decimals : (15*10^-4*10^m)/(3*10^-2*10^k) = 5*10^7

After adjusting and cancelling 5 from each side : 10^(m-k-2) = 10^7
=> m-k-2=7 => m-k = 9

sambam · Answer

rewrite the expression like this:

15(10)^-4 (10)^m / 3(10)^-2 (10)^k = 5(10)^7

When you simplify the expression you are left with: (10)^m / (10)^k = 10^9

Therefore m-k = 9
Answer: A

josemnz83 · Answer

I solved this question by cross multiplying.

15*10^-4m=15*10^-14k
-4m-14k=0
-4(7)-14(-2)=0
-28+28=0
m-k
7-(-2)= 9

Im2bz2p345 · Answer

Hello josemnz84 - I was looking at the alternate ways to solve this problem and I don't quite understand what you did here. Can you please explain? Specifically how did you get 10^{-14k} . I feel this is incorrect, but please correct me if my calculations are off. If you want to cross multiply, the problem would be done this way: \frac{15*10^{-4 + m}}{3*10^{-2+k}} = 5*10^7 15*10^{-4 + m}=15*10^{5 + k} - when you multiple powers of 10, you multiply the whole numbers and add the powers of 10 (you seemed to have multiplied the exponents rather than adding to get 10^-14 in your answer). See this: https://www.dummies.com/how-to/content/m ... ation.html After cancelling out the "15 x 10^" (essentially 10^1) from both sides, you are left with: -4 + m = 5 + k m - k = 5 + 4 m - k = 9 Answer: A ~ Im2bz2p345

PareshGmat · Answer

Removing decimal point & solving ahead,

5 . 10^(m+2-4-k) = 5 . 10^7

Equating , m-k = 9 = Answer = A

ShekharRavi · Answer

Why So Complicated, Keep it simple bro.

\frac{0.0015*10^m}{0.03*10^k}=5*10^7 ;

\frac{15}{10000} * \frac{100}{3} * {10^{m-k} = 5*10^7 ;

\frac{5}{100} * {10^{m-k} = 5*10^7  ;

{10^{m-k-2} = 10^7  ;

m-k = 9  ;

dave13 · Answer

generis please tell me if my solution is correct, i got 9 but negative 9

where am i wrong \frac{0.0015*10^m}{0.03*10^k}=5*10^7 multiply 0.0015 by 10^m and 0.03 by 10^k \frac{0.015*10^m}{0.3*10^k}=5*10^7 now divide 0.015^m by 0.3^k 0.05^{m-k} = 5*10^7 now i need to convert this 5*10^7 into 0.05 to have the same base 0.05^{m-k} = 0.05^{-9} now equate bases m-k = -9

generis · Answer

dave13 Wow - SO close.

Your arithmetic is hard! Despite that degree of difficulty, you had everything correct until the highlight. Please show me your steps? How did you get from 5 * 10^7 to .05 * 10^{-9} ?

dave13 · Answer

generis great to hear from you

many thanks for your question

i am happy to respond 5 * 10^7 to .05 * 10^{-9} ? 5 * 10^7 means 5+7 zeros --> 50 000 000 so now i need to get 0.05 so i move 9 decimal points left

If i am doing something wrong here 0.05^{m-k} = 0.05^{-9} - than how to equate bases?

abhimahna · Answer

Hey  dave13  ,

As mentioned by  generis  , your calculation is incorrect here :  5 * 10^7  to  .05 * 10^{-9} .

When you are converting 5 to 0.05, it means you are multiplying and dividing 5 with 100. That means 5/100 would become 0.05 and two extra zeros at the top(when you multiplied by 100) will be added to 7, thereby giving you 9.

Hence, your conversion will be  5 * 10^7  to  .05 * 10^{9} .

Does that make sense?

generis · Answer

dave13 - Whoops! Opposite direction. Now I see what happened. You say that .05 * 10^{-9} = 50,000,000 But if I move 9 decimal digits to the left, with 50,000,000, I get .050000000. That is not correct. LHS now = .05 (that's all - zeros to right of 5 don't matter) In fact, .05 * 10^{-9}= .000000005 It's an easy and common mistake. We need to ask, how do I write .05 so that it equals 50,000,000? (.05 times WHAT = 50,000,000?) .05 * x = 50,000,000? .05: as abhimahna mentions, move the decimal 9 places to the right = 50,000,000, so RHS: .05 * 10^9 = 50,000,000 I'm going back to this formulation of yours for LHS: \frac{0.015*10^m}{0.3*10^k} Divide decimals first, then 10s. \frac{.015}{.3} =.05 \frac{10^m}{10^k} = 10^{m-k} LHS altogether is .05 * 10^{m-k} So now you have 0.05 * 10 ^{m-k} = 0 .05 * 10^9 m - k = 9 I have an idea that might work better. Try to change things into integer bases. JMO, but it's easier. So when you get to .05 * 10^{(m-k)} Use powers of 10 to change .05 to 5, so that LHS (5) = RHS(5) .05 = 5 * 10^{-2} Whole thing now is: 5 * 10^{-2} * 10^{(m-k)} = 5 * 10^7 Consolidate powers of 10 on LHS Thus: 5 * 10^{(m-k-2)} = 5 * 10^7 Now equate powers of 10 . . . see what you get.

Convert bases to integers, IMO. It's easier to keep track of direction. You did just fine.

dave13 · Answer

thank you

abhimahna generis you know there some points I dont understand you say "When you are converting 5 to 0.05, it means you are multiplying and dividing 5 with 100" why are you multiplying and dividing

we are talking about this expression as one whole number right 5 * 10^7 ? and even if I multiply 5 by 100 and result divide by 100 I get 500

so what ? here is a rule I learnt

but I still have question regarding the rule. it says if"When the number is 10 or greater" which number the power

To figure out the power of 10, think "how many places do I move the decimal point?" When the number is 10 or greater, the decimal point has to move to the left, and the power of 10 is positive. When the number is smaller than 1, the decimal point has to move to the right, so the power of 10 is negative. so i get back ti intial problem 5 * 10^7 = 50.000.000 from this i need to get 0.05 (now it says "When the number is 10 or greater, the decimal point has to move to the left" ) so since 50.000.000 is greater than 10 the decimal point has to move to the left

am i right ? so i count 9 decimal points to the left with positive power of 10

0.05^9 where are the fireworks

?

pushpitkc · Answer

Hey  dave13

Lets try and break-down this conversion:  5 * 10^7  to  .05 * 10^{9} .

5 * 10^7  can also be written as  5 * 1 * 10^7  (Here 1 =  \frac{100}{100} )

We replace 1 and the expression becomes  5 * \frac{100}{100} * 10^7 = \frac{5}{100} * 100 * 10^7

The final step to this conversion is  0.05 * 10^2 * 10^7 = 0.05 * 10^{2+7} = 0.05 * 10^9  because   x^m * x^n = x^{m+n}

Hope it's clearer now!

olga55555 · Answer

A simpler way to solve this without complicated algebraic manipulations. First notice that this ugly equation equals to 5 times some power of 10. Hence it gives us a clue that we must get 5 somehow, it is easy because we see that numerator has 0,0015 and denominator has 0.003. Now think what we must multiply 0.0015 to get 15, we must multiply it by 10,000, which is 10^4, then what we must multiply 0.03 by to get 3, we must multiply it by 100, which is 10^2. Ok so no we got 0,0015 * 10^4 in the numerator and 0.03 *10^2 in the denominator which will give us 5, but we must have 5 *10^7, so we must add 10^7 to the numerator, this way we will get (0.0015*10^4*10^7)/0.03*10^2 ==>(0.0015*10^11)/0.03*10^2 ==>11-2=9

mathislife · Answer

First simplify .0015/.03.

4 decimal places in numerator, 2 decimal places in denominator = needs two decimal places.
 .05

.05*10^x=5*10^7

The RHS tells us 7 decimal places after 5.0

The LHS tells us we need 2 decimal places to get to 5, and then an additional 7 to equal the RHS.

Therefore exponent needs to = 9, or m-k = 9.

kokus · Answer

0.0015 * 10^m / 0.03* 10^k = 5* 10^7 
15*10^m = 300 * 10^k = 5 * 10^7
10^m / 20^10k = 5 * 10^7
10^m / 10^2*10^k = 10^7 -> m-k must equal 7 plus 2.

If 0.001510^m/0.0310^k=5*10^7, then m - k =

Prep Toolkit

Top 5 GMAT Debrief Videos

615 to 715 in 15 Days (Ria's Strategies)

More than Quant/Verbal Abilities: Speed vs. Accuracy

745 with Only Self-Prep and GMAT Club

From 555 to 765: 210-point Improvement

Perfect 805 Score Debrief by Julia

My Rewards

GMAT Problem Solving (PS) Questions