The useful life of a certain piece of equipment is determine

Question

The useful life of a certain piece of equipment is determined by the following formula: u =(8d)/h^2, where u is the useful life of the equipment, in years, d is the density of the underlying material, in g/cm3, and h is the number of hours of daily usage of the equipment. If the density of the underlying material is doubled and the daily usage of the equipment is halved, what will be the percentage increase in the useful life of the equipment?

A. 300%
B. 400%
C. 600%
D. 700%
E. 800%

A. 300%
B. 400%
C. 600%
D. 700%
E. 800%

I plugged the following numbers:
d=4
h=4
Formula:
u=\frac{(8d)}{h^2}

First part
u=\frac{(8*4)}{4^2}
u=\frac{32}{16}
u=\frac{2}{1}
u=2

Second part

Density doubled
d=4*2
Usage halved
h=4/2

d=8
h=2

u=\frac{(8*8)}{2^2}
u=\frac{64}{4}
u=\frac{16}{1}
u= 16

Third Part
So my initial value is 2 and my value after the change is 16.
16/2=8, so there is an 8 fold increase, which is 800%

This is wrong. In the spoiler is the explanation, which uses the exact same method but with other numbers and its result is different. What am I missing here?

One of the most effective ways to solve problems involving formulas is to pick numbers. Note that since we are not given actual values but are asked to compute only the relative change in the useful life, we can select easy numbers and plug them into the formula to compute the percentage increase. Let’s pick d = 3 and h = 2 to simplify our computations:

Before the change: d = 3, h = 2; u = (8)(3)/22 = 24/4 = 6 
After the change: d = (2)(3)= 6, h =2/2 =1; u = (8)(6)/12 = 48

Finally, percent increase is found by  first calculating the change in value  divided by the original value and then multiplying by 100:
(48 – 6)/6 = (42/6) = 7
(7)(100) = 700%

The correct answer is D.

Bunuel · Accepted Answer

Algebraic approach:

u_1=\frac{8d}{h^2} 
Density of the underlying material is  doubled  and the daily usage of the equipment is  halved  -->  u_2=\frac{8(2d)}{(\frac{h}{2})^2}=8*\frac{8d}{h^2}  -->  u_2=8u_1  --> 8 times more = increase by 700%.

Answer: D.

shrouded1 · Answer

You are on the right track, its just the last bit of the calculation.

The useful life goes from x to 8x (8 times). Now the increase is therefore 7x. And percent increase is 7x/x * 100 or 700%

Remember : %-age increase =  \frac{New-Old}{Old} * 100

Bunuel · Answer

Hi, and welcome do Gmat Club.

8 times more means increase by 700% the same way as twice as much means increase by 100%.

x(1+p/100)=x(1+700/100)=x(1+7)=8x

Hope it's clear.

u0422811 · Answer

As its asking for an increase you have to remember to subtract 100% from the original answer. For example, lets say you had $2 and you invest it and get back $2. Your return isn't 100% as would be found if we had taken $2/$2 = 100%, you need to subtract the original value or this can be accomplished by just subtracting 100%. So the equation would read ($2/$2)-1 = 0%. Taken one step further if we invested $2 and got back $4 our return would be ($4/$2)-1 = 100%.

Hope that helps.

Hope that helps.

RudeyboyZ · Answer

Why is the result different if I plug in 1 for both d and H? the result is 300% in which case, even by following the rule of (Increase-original/100)

Bunuel · Answer

You have to show your work, the result should and IS the same.

RudeyboyZ · Answer

Dear  Bunuel ,

You are right there isn't a way to get different answers, I forgot to square the h in the second part of the work. You're the best!

Regards,

Rudraksh.

Anonamy · Answer

I'm confused because I'm getting a different answer. Where am I going wrong?

Original: u=8d/h(sqrd) 
With changes: u=8(2d)/

I tried the plugging in approach. d=4 and h=4:

Original: 32/16=2
With changes: 64/8=8

Therefore the answer would be 300%, which is not correct, so I then tried d=6 and h=4:

Original: 48/16=4
With changes: 96/8=12

Therefore the answer would be 200%... What is going on here? Thanks

davedekoos · Answer

Hi Anonamy,

The original formula is  u=\frac{8d}{h^2} , so when the daily usage (h) is halved, you must replace  h  with  h/2 . All of  h/2  is then squared.

So the new formula looks like this:
 u=\frac{2*8d}{(\frac{h}{2})^2}

What you had was

u=\frac{2*8d}{\frac{h^2}{2}}

Do you see the difference?

Cheers

Anonamy · Answer

Hi davedekoos,

Ah! Yes I see my error now. Thanks for your help!

singhabhijeet · Answer

I used the plug in approach for this question.

u= \frac{8d}{h^2}  --- eq 1

Case 1  :

u = ?
d = 10
h = 2

Substitute above values in eq 1

u =  \frac{8 * 10}{2 * 2} 
u = 2*10 = 20

Case 2  : If the density of the underlying material is doubled and the daily usage of the equipment is halved.

u = ?
d= 2 * 10 = 20
h = 2/2 = 1

Substitute above values in eq 1

u =  \frac{8 * 20}{1 * 1} = 160

% Increase = \frac{Change}{original Value}  * 100

% Increase =  \frac{160 - 20}{20 * 100}  = \frac{140}{20} * 100 = 700

Ans:

AkshdeepS · Answer

My Solution :

sashiim20 · Answer

u=\frac{(8d)}{h^2}

Let initial values of  d = 1  and  h = 1

Then;  u = \frac{8 * 1}{1^2} => 8

Density is doubled;  d = 2

Daily usage of the equipment is halved;  h = \frac{1}{2}

New value of  u = \frac{8*2}{(1/2)^2} = \frac{16}{(1/4)} =16*4 = 64

Percentage Increase  = \frac{64-8}{8}*100 = \frac{56}{8}*100 = 700 %

Answer D

sakshamchhabra · Answer

posting 2 approaches :

]original :  \frac{8d}{h^2}

Now, d---> 2d and h----> \frac{h}{2}

substituting:  \frac{8*2d}{h/2^2}  ----->  \frac{16d}{h^2/4}

on simplifying:  \frac{64d}{h^2}

To calculate % increase:  \frac{New-old}{old}  X 100

new - old -->  \frac{64d}{h^2}-\frac{8d}{h^2}  =  \frac{56d}{h^2}

now change / old X100 =  \frac{56d}{h^2}  X  \frac{h^2}{8d}  X100

which gives ---> 700%

by plugging values :
 
original :  U=\frac{8d}{h^2}

let, d=1 and h=2

U= \frac{8}{4}=2

After change: d=2 and h=1

U=\frac{8*2}{1}  = 16

To calculate % increase:  \frac{New-old}{old}  X 100

change ---> 16-2 =14

% change--->  \frac{14}{2} X100 = 700%

CAMANISHPARMAR · Answer

If the density of the underlying material is doubled and the daily usage of the equipment is halved, then the new useful life will be 8 times the existing useful life or there will be 700% percentage increase in the useful life of the equipment. Correct answer is D

ManishaPrepMinds · Answer

Original formula: U = 8d/h2 
Changes:
d → 2d (doubled)
h → h/2 (halved)
 New formula: 
U New = 8(2d)/(h/2)2
= 8(2d)/(h2/4)
= 8(2d)(4/h2)
= 64d/h2
= 8(8d/h2)
= 8U
 Increase: 
8u - u = 7u
7u/u × 100 = 700%

Answer: D 
The useful life increased by 700%

bumpbot · Answer

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The useful life of a certain piece of equipment is determine

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