Lesson: Operations with Fractions

Comment on Operations with Fractions

To solve this problem:

Let p = the product of all the odd integers between 500 and 598, and let q = the product of all the odd integers between 500 and 602. In terms of q, what is the value of 1/p + 1/q?

Are we to memorize the prime numbers between 500 and 1000? If yes then what are the set of numbers for which we should memorize the prime numbers for?
gmat-admin's picture

I think you might be mixing up ODD numbers with PRIME numbers.

Here's my full solution to your question: https://gmatclub.com/forum/let-p-the-product-of-all-the-odd-integers-bet...

ASIDE: As far as memorizing prime numbers goes, I think it's fine to remember the primes from 2 to 97

Sorry to mention this, but there is a typo in your answer to this question:
in this statement:
So, we know that x must be greater than 1/3 AND less than 1/3
I am certain you meant it to be:
So, we know that x must be greater than 1/3 AND less than 1/2
gmat-admin's picture

Good catch!! Thanks Aladdin!
I have edited my response.


Can you please explain the concept of the smallest number that denominator divides into? How to find that?
gmat-admin's picture

The technique for finding the Lowest Common Denominator is the same as finding the Least Common Multiple (LCM).

So, for example, when adding 1/12 and 7/15, we must first find the LCM of 12 and 15.

Here's a video on finding the LCM: https://www.gmatprepnow.com/module/gmat-integer-properties/video/835


nice tricks!

Hi Brent,
Need help with the following question:
When a certain tree was first planted, it was 4 feet tall, and the height of the tree increased by a constant amount each year for the next 6 years. At the end of sixth year, the tree was 1/5 taller than it was at the end of 4th year. By how many feet the height of the tree increase each year?
A 3/10
B 2/5
C 1/2
D 2/3
E 6/5
gmat-admin's picture

I have seen examples where the numerators and denominators both have 2 fractions (total of 4 fractions). For example: 1/2 divided by 10/33 is the numerator, and the denominator is 1/3 divided by 25/11.

I assume the reciprocal approach applies here as well, and do it 3 times, or is there a better technique? Thanks!
gmat-admin's picture

I would evaluate the numerator and the denominator separately, and then complete the last division. Three divisions in total.

Yulia's picture

Hi Brent,
To be honest I didn’t understand the solution for this question https://gmatclub.com/forum/let-p-the-product-of-all-the-odd-integers-between-500-and-154251.html

Let p = the product of all the odd integers between 500 and 598, and let q = the product of all the odd integers between 500 and 602. In terms of q, what is the value of 1/p + 1/q?

A. 1/600q
B. 1/359,999q
C. 1,200/q
D. 360,000/q
E. 359,999q

However, I answered correctly and approach it by using a number sense if I can say like this. I could understand that the product of odd integers would be a large number therefore 1 in numerator can’t be thus A and B choices is out. E is out too because sum operation could not result in 359,999q (at least I could not see any ways to get such a reply). I discarded C because the numerator is too small since we have integers between 500 and 598 and 500 and 602, so for sure LCM will be large resulting also the numerators to increase.

I looked through the solutions but honestly could not understand them. Any advice on it?
gmat-admin's picture

Question link: https://gmatclub.com/forum/let-p-the-product-of-all-the-odd-integers-bet...

That's good that you got the correct answer, but I'm not sure I follow your rationale for eliminating A and B.
Can you elaborate?

Aside: The correct answer (360000/q) simplifies to be approximately 1/100000000000......, where the denominator has over 100 zeros,

Yulia's picture

Hi Brent,
Thank you for your reply.

What does "In terms of q, what is the value of 1/p+ 1/q" mean? Dont really understand expression "in terms of q"

In your solution I understand this:
p = (501)(503)(505)...(597)
q = (501)(503)(505)...(597)(599)(601)

After that I am lost. Why q = (p)(599)(601)?
gmat-admin's picture

"in terms of q" tells us to rewrite the mathematical relationship using only the variable q (and other numbers).

For example, let's say that we're told: x + y = 7
If we want to express x in terms of y, we can write: x = y - 7

We already know that: p = (501)(503)(505)...(597).
This means we can replace (501)(503)(505)...(597) with p, since the two values are equivalent.
So, we can take: q = (501)(503)(505)...(597)(599)(601)
and replace (501)(503)(505)...(597) with p to get: q = (p)(599)(601)

Does that help?

Yulia's picture

Thank you Brent.

Now it is very clear.


For this question, if my brain is just fool enough to get how I can do this, what should I do?
gmat-admin's picture

On test day, if you're stuck on a question, my best advice is to make your best guess and move on. The GMAT isn't about getting every question correct; it's about maximizing your score.

That said, for this particular question, you can start with my solution here: https://gmatclub.com/forum/let-p-the-product-of-all-the-odd-integers-bet...

Let me know if you have any questions after reading the solution.

Hi Brent,

Do you recommend write eliminated choices into the note sheet just so I have a clear mind what to choose? but sometimes I feel like it is waste of valuable times since for some answer choices I can eliminated very fast.

Especially for the verbal section, I think I don't have to on math one.
gmat-admin's picture

The answer to your question will vary from student to student. Some students mentally keep track of which answer choice they've eliminated, while other students prefer to use their notepad to cross off each answer choice they've eliminated.

Hi Brent, silly question on this one probably.


How did you get (0.1667)≈1/6 ?

When I try to convert 0.1667 to fraction as approximate 1600/10000 but not getting 1/6? Could you kindly clarify? Thanks
gmat-admin's picture

Question link: https://gmatclub.com/forum/of-the-following-which-best-approximates-2813...

If you calculate 1/6 by using long division to divide 1 by 6, you'll get 0.166666....
If we round 0.166666.... to the to the nearest ten thousandth, we get 0.1667

There's a fraction/decimal conversion chart to be memorized at 6:12 on this video lesson: https://www.gmatprepnow.com/module/gmat-arithmetic/video/1066

Be sure to have this memorized for test day.

Thanks Brent. Presumed this is due to the repeating number therefore when converting back to fraction for 0.166666, I will not get 1/6 from 1666/1000 unless I know that's 1/6 beforehand. On test day, is there a way around if I don't remember and not sure if 0.166666... = 1/6? Thanks
gmat-admin's picture

The key word here is APPROXIMATES, as in "Of the following, which best approximates..."
Since 0.1667 is very 0.1666....it's perfectly fine to use that approximation.

Thanks Brent for the clarification. Since is approximates therefore 0.16 = 16/100 ~ 1/6. Is my understanding correct? Thanks Brent.
gmat-admin's picture

In most cases, 1/6 will be a safe fractional approximation of 0.16 and/or 0.17

Great thanks Brent and noted. Really appreciate for all your clarificatin and great helps always.


Hi @BrentGMATPrepNow, wondering why approximation won't work here when I can get result that close as 3.428 or 3.43? Have I missed something here? Thanks

0.2 * 0.8 * 0.3/ 0.2 * 0.7 * 0.1 = 24/7 = 3.42
gmat-admin's picture

Question link: https://gmatclub.com/forum/of-the-following-which-best-approximates-2813...

In order to be able to approximate, the answer choices must be sufficiently spread apart (for more on, this read the following article: https://www.gmatprepnow.com/articles/simplify-your-calculations)

Notice that answer choices B, C, and D are very close together.
(B) 2.40
(C) 2.43
(D) 2.50
In fact 2.5 is only 4% greater than 2.4
In your approximation, you approximate 0.1667 to be 0.2
When you do this, you are increasing that value by almost 20%
So that one approximation (out of several approximations) already increases the value of the given fraction by 20%.
Since B, C and D are all within 4% of each other, that one approximation (from 0.1667 to 0.2) will make your final answer far too unpredictable.

Ah that make sense. Thanks Brent for the clarification.

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