Which of the following fractions has the greatest value

[ogbeni]

I'm not following the official explanation to this question and I think there is a typo in the official explanation. And yes I bombed the question and would like for one of you smart folks to show me how to solve this problem!!! Thanks!!

Which of the following fractions has the greatest value?
a) 3/8!

b) 6(4!)/5(9!)

c) 4(3!)/9!

d) 3/5(7!)

e) 14/5(8!)

If you'd like for me to post the official explanation, let me know and I will.

[chandanthakur]

In this case,we can multiply all the option by common denominator. Looks like 5.9! here. And then check

[sunny.jain]

To compare different fractions, they need to have the same denominator.

Let's compare, for example, the denominator in (A) (8!) with the denominator in (C) (9!).
8! = 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
9! = 9 * 8 * 7 * 6 * 5 * 4 * 3 * 2 * 1
You notice that the common multiple between 8! and 9! is 9!.

If you try to find the common multiple of all the denominators, you'll find that it's 5(9!). Now we can change all the fractions to have 5(9!) as the denominator.

(A) 3/8! = 3*5*9/(5(9!)) = 135 / 5(9!) [Multiply top and bottom by 5*9]
(B) 6(4!)/5(9!) = 144 / 5(9!)
(C) 4(3!)/9! = 5*4*3! / 5(9!) = 120 / 5(9!) [Multiply top and bottom by 5]
(D) 3/5(7!) = 3*8*8 / 5(9!) = 216 / 5(9!) [Multiply top and bottom by 9*8]
(E) 14/5(8!) = 14*9 / 5(9!) = 126 / 5(9!) [Multiply top and bottom by 9]

Here, (D) is the largest.

[Ben Ku]

One neat approach to factorial problems is to recognize something like this:

9 * 8! = 9!

So let's suppose you have something in 7! and you're trying to get 9!. You would multiply the first by 9*8.

9 * 8 * 7! = 9!

This will help us find a common denominator for the answer choices. From among the answer choices, you will find that 5(9!) is the greater common denominator. To change the answer choices to this denominator, you must:

(a) multiply numerator and denominator by 5*9, so the denominator is 5(9!)
(b) the denominator is already 5(9!)
(c) multiply numerator and denominator by 5, so the denominator is 5(9!)
(d) multiply numerator and denominator by 9*8, so the denominator is 5(9!)
(e) multiple numerator and denominator by 9, so the denominator is 5(9!)

If you perform the above, the answer choices become:

(a) 3/8! = 135 / 5(9)!
(b) 6(4!)/5(9!) = 144 / 5(9!)
(c) 4(3!)/9! = 120 / 5(9!)
(d) 3/5(7!) = 216 / 5(9!)
(e) 14/5(8!) = 126 / 5(9!)

Here, (D) is the largest fraction.