Given a positive number N, when N is rounded by a certain method (for

Given a positive number N, when N is rounded by a certain method (for convenience, call it Method Y), the result is \(10^n\) if and only if n is an integer and \(5*10^{n − 1} ≤ N < 5*10^n\). In a certain gas sample, there are, when rounded by Method Y, \(10^{21}\) molecules of \(H_2\) and also \(10^{21}\) molecules of \(O_2\). When rounded by Method Y, what is the combined number of \(H_2\) and \(O_2\) molecules in the gas sample?

This means RANGE of both \(H_2\) and \(O_2\) will be \(5*10{21-1}<10^{21}<5*10^{21}\)

(1) The number of \(H_2\) molecules and the number of \(O_2\) molecules are each less than \(3*10^{21}\).
We know the range of both as \(5*10{21-1}<10^{21}<5*10^{21}\).
If both are less than \(2.5*10^{21}\), answer will be \(10^{21}\) as the TOTAL will always be less than \(5*{21}\).
But if both are, say, \(2.6*10^{21}\), then total is \(5.2^{21}\), which will get rounded off to \(10^{22}\).
Insufficient

(2) The number of \(H_2\) molecules is more than twice the number of \(O_2\) molecules.
Again various possibilities..
If \(H_2\) molecules = \(2.6*10^{21}\) and \(O_2\) molecules=\(1.3^{21}\)....total will become \(3.9^{21}\), rounded off to \(10^{21}\).
But if \(H_2\) molecules = \(4.8*10^{21}\) and \(O_2\) molecules=\(2.4^{21}\)....total will become \(7.2^{21}\), rounded off to \(10^{22}\).
Insuff

Combined..
Let us take the MAX value of \(H_2\) ~ or just less than \(3*10^{21}\), so max value of \(O_2\) will be just less than \(1.5*10^{21}\).
Total = \(4.5*10^{21}\), rounded off to 10^21..
Thus answer will always be \(10^{21}\)

C
_________________

\(5*10^{20} ≤ N_h < 5*10^{21}\)
\(5*10^{20} ≤ N_o < 5*10^{21}\)

Statement 1-
\(5*10^{20} ≤ N_h < 3*10^{21}\)......(1)

\(5*10^{20} ≤ N_o < 3*10^{21}\)......(2)

Combining (1) and (2), wen can get the range of \(N_h\)+\(N_o\).

\(10^{21} ≤ N_h+N_o < 6*10^{21}\)

Case 1- If \(10^{21} ≤ N_h+N_o < 5*10^{21}\)
\(N_h\)+\(N_o\)is rounded off to \(10^{21}\)

Case 2- \(5*10^{21} ≤ N_h+N_o < 6*10^{21}\)
\(N_h\)+\(N_o\) is rounded off to \(10^{22}\)

Insufficient

Statement 2-
\(10^{21} ≤ N_h < 5*10^{21}\)......(1)

\(0.5*10^{21} ≤ N_o < 2.5*10^{21}\)......(2)

Hence,

\(1.5*10^{21} ≤ N_h+N_o < 7.5*10^{21}\)

Case 1- If \(1.5*10^{21} ≤ N_h+N_o < 5*10^{21}\)
\(N_h\)+\(N_o\) is rounded off to \(10^{21}\)

Case 2- \(5*10^{21} ≤ N_h+N_o < 7.5*10^{21}\)
\(N_h\)+\(N_o\) is rounded off to \(10^{22}\)

Insufficient

Combining both statements

\(10^{21} ≤ N_h < 3*10^{21}\)......(1)

\(0.5*10^{21} ≤ N_o < 1.5*10^{21}\)......(2)


\(1.5*10^{21} ≤ N_h+N_o < 4.5*10^{21}\)

Hence, \(N_h\)+\(N_o\) rounded off to \(10^{21}\)

Sufficient

Video solution from Quant Reasoning starts at 00:24
Subscribe for more: https://www.youtube.com/QuantReasoning? ... irmation=1

_________________

Hello Experts,

Would it be good strategy to not waste time over this question in exam scenario where it would be difficult to think through the question.
_________________

I second this. It's taking too much time to even understand the stem. Also, is the question posted incomplete by any chance? The sentence structure seems to be all over the place.

I did understand until «Combining (1) and (2), wen can get the range of \(N_h\)+\(N_o\). \(10^{21} ≤ N_h+N_o < 6*10^{21}\)», but I don’t understand after that.

How did you arrive to make a conclusion that “If \(10^{21} ≤ N_h+N_o < 5*10^{21}\), \(N_h\)+\(N_o\)is rounded off to \(10^{21}\), and If \(5*10^{21} ≤ N_h+N_o < 6*10^{21}\), \(N_h\)+\(N_o\) is rounded off to \(10^{22}\).”

And at the statement 2, \(10^{21} ≤ N_h < 5*10^{21}\), where did 5 go??? Wasn’t it \(5\)\(*10^{20} ≤ N_h < 3*10^{21}\)??? How did you arrive to \(10^{21} ≤ N_h < 5*10^{21}\), and \(0.5*10^{21} ≤ N_o < 2.5*10^{21}\)[/b]. Hence, \(1.5*10^{21} ≤ N_h+N_o < 7.5*10^{21}\) from \(N_h>2*N_o\)???

Seems like everyone understood what your saying but me....

Please help me 🙏🏻

Thank you in advance!

Posted from my mobile device

Hi
I know i'm very late , you probably don't need it anymore but this might help someone. (PS- i tried my best explaining the answer as it took me also a very long time to understatnd it)
This question has been made this way so that most of the test takers skip it
If you dive deep into its logic, it's pretty simple
I'll try to explain

Normal approximation:

Lets take a number , for example- 1.7
In normal approximation , you would round it off to 2 because it lies in the range 1.5 < NUMBER GIVEN < 2.5 and the numbers from 1.5 to 2 would be rounded up to 2 and numbers greater than 2 to numbers < 2.5 would be rounded down to 2.
The numbers in the given range after approximation will become equal to the mid value of the range i.e. 2

BACK TO THE ORIGINAL QUESTION

the result is 10^n if and only if n is an integer and 5∗10^n−1≤N<5∗10^n
This is the same kind of approxmation I have explained above
The numbers in the given range after approximation will become equal to the mid value of the range i.e. 10^n
The range is a complicated because it involves exponents and variables but look through it, it's just a number with many many zeros in it and range for our approximation is large

the method Y is nothing but our usual approximation but with a very large range

Now , THE EXACT NUMBER OF oxygen and hydrogen molecules lies in this given range ( 5∗10^n−1≤N<5∗10^n )

THE QUESTION WANTS US TO ( STEP 1) FIND OUT THE EXACT NUMBER OF O2 AND H2 molecules----->2 ADD THEM UP ---->3 Find combined number of H2 and O2 molecules in the gas sample AFTER APPROXIMATION/ROUNDING UP

since we want a unique answer and the answer is a rounded up value, we don't really need an exact value of number of molecules, WE WANT VALUE THAT GIVES A UNIQUE ANSWER


STATEMENT 1: The number of H2 molecules and the number of O2 molecules are each less than 3∗10^21
CASE 1
Step 1: if No. EXACT NUMBER of O2 and H2 molecules is 2.5*10^21 each
Step 2: ADDING THEM UP : TOTAL= 5*10^21
Step 3: Round them--> since 5*10^21 lies outside the range given in the question and it actually lies in the next possible range i.e 5∗10^21≤N<5∗10^22 whose mid-value is 10^22 hence after rounding up the value of the number of molecules willl be 10^22
CASE 2
Step 1: if No. OF EXACT NUMBER of O2 and H2 molecules is 2*10^21 each
Step 2: ADDING THEM UP : TOTAL= 4*10^21
Step 3: Round them--> since 4*10^21 lies inside the range given in the question , approxmated value is 10^21
2 POSSIBLE ANSWERS , HENCE NOT SUFFICIENT

STATEMENT 2 The number of H2 molecules is more than twice the number of O2 molecules.
CASE 1
Step 1: if No. OF EXACT NUMBER of O2 molecules is 1*10^21 and H2 molecules is 2.1*10^21 ( 2.1 is more than 2*1=2)
( Also notice, the no of molecules of O2 and H2 satisfies the range given for each of them)
Step 2: ADDING THEM UP : TOTAL= 3.1*10^21
Step 3: Round them--> since 3.1*10^21 lies intside the range given in the question, approximated value will be 10^21
CASE 2
Step 1: if No. EXACT NUMBER of O2 molecules is 2.3*10^21 and H2 molecules is 4.7*10^21 ( 4.7 is more than 2*2.3=4.6)
Step 2: ADDING THEM UP : TOTAL= 7*10^21
Step 3: Round them--> since 7*10^21 lies outside the range given in the question and it actually lies in the next possible range i.e 5∗10^21≤N<5∗10^22 whose mid-value is 10^22 hence after rounding up the value of the total number of molecules willl be 10^22
2 POSSIBLE ANSWERS , HENCE NOT SUFFICIENT

COMBINING 1 & 2
Step 1: if EXACT NUMBER of O2 molecules is 1.4*10^21 and H2 molecules is 2.9*10^21 ( 2.9 is more than 2*1.4=2.8)
( Also notice, we are taking the maximum permissible values)
Step 2: ADDING THEM UP : TOTAL= 4.3*10^21
Step 3: Round them--> since 4.3*10^21 lies inside the range given in the question, approximated value will be 10^21
we can try the same with other values as well but we won't get another answer as we have already tried out the maximum possible values . Taking minumum values will complicate the calculations so i just took the maximum possible values
WE ARE GETTING A UNIQUE ANSWER HENCE ---> answer : C

Hope it helps !!!

First off, this is like too tough. Honestly, it took me 5 minutes to understand the equation itself.
But, I reattempted this today and I used MGMAT Guessing Strategy on this one. Luckily, I must say, I marked C only.
So here's how I came to this.

(A) Gives out some information about H2 and O2 molecules both being less than 3*10^21. So, those who are familiar with MGMAT Strategies would understand that if a statement gives a range it is mostly insufficient *DO NOT GENERALISE THIS PLS* - A and D out

(B) It just tells the relationship between 2 variables. I gulped here, because GMAT also said that both these are integers and often an equation is suitable for 2 variables in only case when they are integers. (Again Don't Generalise)
But read finely, there are same when rounded off, there possibly can't be just one solution and the gap is too large (5*10^20 < N < 5*10^21) - B is out too

If you can solve till here with the reasoning I did, just hold on to nerves, it's gonna be fine
C and E are left.

Now if you are guessing, it can be 50-50, but here's why I chose C over E

There is a large gap from which there are two integers to be rounded off (which are same when rounded up) B lowers it significantly. If I combine, there's a chance I can find the sum and round it up.

I know this is really shaky and might just leave some mathematicians baffled, but it worked for me.
Choose (C) - Just because it can give you an answer and choosing E will create bigger problems because the answer actually tries to make sure there is an answer. Trust the Exam Maker, for he won't bring you so far to give a no answer (at least here)

Specific value question, we need to know whether we can get a unique value or not.
(Note: We don’t need to actually find what this unique value is).

We first need to understand the method of rounding.
Using n = 2 as an example, the first sentence says that if a number lies between 5 X 10^1 (i.e. 50) and 4.99 X 10^2 (i.e. 499), then it is rounded to 100.

We’ll continue using n = 2 (instead of n = 21) to understand the second sentence:
We are told that the rounded number of molecules of H2 and O2 is 10^2 (i.e., 100) each.
That means the actual number of each molecule lies in a range between 5 X 10^1 (i.e. 50) and 5 X 10^2 (i.e. 499).
So the actual range will always lie between 5(followed by n-1 zeros) and 4.99(followed by n zeros).
Therefore:
Maximum possible actual sum = 500 + 499 = 999; Rounded total sum = 1,000 = 10^3
Minimum possible actual sum = 50 + 50 = 100; Rounded total sum = 100 = 10^2
(Obviously, the question stem alone is not sufficient to answer the question).

The question is asking us to find the sum of the actual numbers and then round this off using the rounding method.
We can find a unique value of the rounded sum only if we know for sure that the actual sum lies within the upper and lower end of the range.

Statement 1 says the actual number of each molecule is less than 3 X 10^2 (i.e. 300).
So the maximum actual total sum is capped at 6 X 10^2 (i.e. 600)…
and the minimum actual sum remains at 10^2 (as seen above).
Therefore the rounded sum could be either 10^3 or 10^2
Insufficient.

Statement 2 says the actual number of H2 molecules is more than double that of O2 molecules.
Let’s pick values for each of these within the permissible range:
Scenario 1: Minimum O2 = 50; therefore minimum H2 = 101; therefore total = 151. Rounded sum = 10^2.
Scenario 2: Maximum H2 = 499; therefore maximum O2 = 249; therefore total = 748. Rounded sum = 10^3.
Insufficient.

Combining both statements:
Scenario 1: Minimum O2 = 50, therefore minimum H2 = 101. Total = 151. Rounded sum = 10^2.
Scenario 2: Maximum H2 = 300 therefore maximum O2 = 149. Total = 449. Rounded sum = 10^2.
Sufficient.

Answer choice C.

Moral of the story: Use smaller dummy numbers to understand all possible scenarios!

This question will be much easier to solve if it is imagined that n = 2 and 10^21 = 10^2.