Problem Solving

Data for a certain experiment is given below:

Time --- Amount
1pm --- 10.0 grams
4pm --- x grams
7pm --- 14.4 grams

If the amount of bacteria increased by the same fraction during each of the two 3-hour periods shown, how many grams of bacteria were present at 4pm?

(A) 12.0 (B) 12.1 (C) 12.2 (D) 12.3 (E) 12.4

I am very interested getting views and feedback on the following:
- primary and secondary approaches to solving this. Please illustrate your logic.
- guessing strategies you would use on this type of question, if you were running out of time, or if you simply had to guess on this question
- the traps and tricks which are built into this question (if you can see any)
- any key take-aways, strategies to keep in mind for identiying future similar problems etc.

Thanks.

my logic for this question:
since the bacteria increased by the same fraction during each interval,
10:x = x:14.4

so, 10/x = x/14.4
or x^2 = 144
x = 12 or x = -12

since the number of bacteria increased and is a positive value, the answer is 12. (A) Is that correct?

I accept with the above answer 12 ! but wat is wrong in the below approach ?

in 6 hrs the bac increased to 4.4 grms
in 1 hr the bac increased to (4.4/6) gms
in 3 hrs the bac increaseed to 2.2 gms

Therefore the total bac at 4 pm is 12.2 gms

what is that I am missing ?


Thanks
Senthil

Since the increase is linear; I would simply take the average (10 + 14.4)/2 = 12.2

Hi,

I think the easiest is to consider this is as a problem similar to compound intrest wherein :
A=P(1+(r/100))^n

In this prob . P=10;A=14.4; n=2(sice we have 2 3 hr gaps cannot take it as 6)

Solving for r :

14.4 = 10(1+r/100)^2
1.44= (1+r/100)^2
1.2 =(1+r/100)
i.e r = 20%

Substituting this back

we have to get A after the 1st period :

i.e A= 10(1+(20/100))
A = 12.

Hope this makes sense.

I liked dipinj's approach, simple and quick.

I've used something similar. By looking at values, I could figure that 14.4 is somewhere near 20% increase in value of X, reason being 10+20% = 12 and 12+20% equals 14.4.

So, it took me 20 seconds to solve this problem without using paper. I ofcourse plugged value back to verify if it was correct.

The fraction of increase should be same for every 3 hours. So the best approach is to PLUG IN the values.

Ex: check if 14.4/12.2 = 12.2/10 similarly for all values. (u need not calculate once u understand these values)

Answer is A

How can the answer be A?

You have 6hrs with a total growth of 4.4. I like the simple way dipinj used. Started with 10 ended with 14.4, and because the question ask about what the growth is exactly half way...we would take the average which is 12.2. If it is 12, then you are saying the growth rate is 1 per hr. that would mean that the final rate @ 4pm would be 16. not correct.

sorry for the typo. the final rate if the answer were 12 would be 16 @7pm

If the amount of bacteria increased by the same fraction during each of the two 3-hour periods shown

I think the question is asking for a fractional increase every 3 hrs and not a liner increase every hour. If it is for every hour then 12.2 is the right answer.


Only in case of A;
14.4/12.0 = 12.0/10.0 (fractional increase for evry 3 hrs)
In rest all options, the fractional increase is not same for evry 3 hrs

Hi II
Whats the OA.

envyramana,

I get your reasoning. Makes perfect sense. @ 4 we have a 20% increase from the original 10, and we have another 20% increase from 12 that gives us 14.4. I guess I fell into the GMAT trap. This is why I scored a very miserable 480 last Sat. FOCUS, FOCUS, FOCUS...after all, this is only high school level.

thanks

Guys ... the official Answer is (A) !

I guess most of you are falling into the trap answer and selecting 12.2. Read the question carefully, especially the following part:

"Amount of bacteria increased by the same fraction during each of the two 3-hour periods shown".

The key to answering this question is to find the rate of increase.

There are several approaches which you can follow.

(1) Use Compount Interest Formula to find the growth rate:
This problem can also be viewed in the same way as a compound interest problem, where the interest is compounded.

The key piece of information to find here is the rate of growth. The question mentions that the amount of bacteria increases by the same fraction during each period. Once we know the rate of growth, we can then calculate the amount of bacteria present at 4pm.
Use the compound interest formula:
o A = P (1 + r)^t
Using the information from the question, we can find the value of r (rate of growth):
- A = 14.4 (final amount)
- P = 10.0 (initial amount)
- r = the rate of growth is unknown
- t = 2 (since there were 2 growth periods)
- 14.4 = 10 (1 + r)^2
- 1.44 = (1 + r)^2
- 1.2 = 1 + r
- 0.2 = r --> 20% growth rate --> 1/5 fraction

Now that we have r, we can find x (the amount of bacteria after the first growth period)
- Increase 10 by 20% = 10 * 1.2 = 12
- Or use the formula to find A after 1 growth period: A = 10 (1 + 0.2)^1 --> 10 * 1.2 --> 12
Answer is A

(2) Assigning variable, setting up equations:
What is the unknown here? rate of growth and the amount of bacteria present at 4pm. Lets assign variables to these:
o r = rate of growth
o x = amount of bacteria at 4pm
Write equations:
o So how can we get x, using the information available? 10 is the initial amount. If this is multiplied by the rate of growth, then we can find the amount of bacteria after one period. x multiplied by the rate of growth, will give us the amount of bacteria after 2 growth periods. This gives us the 2 distinct equations we need to solve for the 2 variables (unknowns)
- (1) 10r = x
- (2) xr = 14.4
So we can take (1) which tells us that x = 10r, and substitute this into (2)
- 10r(r) = 14.4
- 10r^2 = 14.4
- r^2 = 1.44
- r = 1.2
Now that we have r, we can substitute this into (1) to find x.
- 10(1.2) = 12.0
Answer is A

(3) Use the answer choices ... plug these into the table, and see if they derive the same fracion (Note: r is the unknown rate):
So if we start with (A).
- 10r = 12 --> r = 1.2
- 12 * 1.2 = 1.44
So this is correct

Lets take a look at (B) just for confirmation:
- 10r = 12.1 --> r = 1.21
- Is 12.1 * 1.21 = 1.44 ? No it isnt, so we can cross out this answer choice. We can continue down this route for the rest of the answer choices to confirm that A is the correct answer.