Fractions

[chughbrajesh]

(Q. from Thursdays with Ron video)

If the positive integer b is 5% greater than the positive integer a, which of the following expressions must be equal to an integer?

I. b/3
II. b/5
III. b/7

A. None
B. II only
C. I and II only
D. I and III only
E. II and III only

I broke this question down like this:
b=.05 +a or b-0.5=a

But I read a solution on GMAT club to this problem

"I would pick D

b=a(1+(5/100))
b=a+(a/20)

looks like b will be comprised of a positive multiple of 20 + the multiple

b = a + (20a)
b=21a

I believe this makes b positive multiple of 21, so it will have a (7)(3) in its factors."

How are they getting 1 in this equation? b=a(1+(5/100))

Thank you.

[georgepa]

I am also getting D - but I tried to solve it by picking numbers.

When I looked at this problem, I decided to try picking numbers instead of algebra - since the question had the word "must".

Try cases - where 5% of a is 1,2,3,4,5,...
(1) a = 20, then b = 21
(2) a = 40, then b = 42
(3) a = 60, then b = 63
(4) a = 80, then b = 84
(5) a = 100, then b = 105
(6) a = 120, then b = 126

You can see b is always divisible by 3 and 7 so answer is

D. I and III only


Note: smallest +ve (a,b) is (20,21) - which is why I started with that pair

My algebra method is as follows

b = 1.05a
100 b = 105a
20b = 21a

Since a and b have to be integers, a has to be a multiple of 20 and b has to be a multiple of 21. Therefore 7 and 3 divide b

[RonPurewal]

I broke this question down like this:
b=.05 +a or b-0.5=a

Both of these are incorrect because they fail to take into account the fact that "percent greater than" is a relative relationship.

I.e., "5 percent" is not a pure number, just 0.05. Rather, "5 percent" is 5 percent OF something. It's 0.05 times whatever you're actually taking 5 percent of.

"5 percent of x" is 0.05x.

"5 percent MORE than x" is x PLUS 5 percent of x.
That's x + 0.05x, or 1.05x.
Once you do this once, you can just remember, forever, that "5 percent greater than..." can be achived in one step, by multiplying by 1.05. (Similarly, "44 percent greater than..." is multiplication by 1.44, and so on.)

The second version is also incorrect because it says 0.5 instead of 0.05, although I assume that's just a typographical error.

[RonPurewal]

b=a(1+(5/100))
b=a+(a/20)

This is the same as what's above, except in that it uses a fraction form (5/100) rather than a decimal form (0.05).

[arunspanda]

If the positive integer b is 5% greater than the positive integer a, which of the following expressions must be equal to an integer?

I. b/3
II. b/5
III. b/7

A. None
B. II only
C. I and II only
D. I and III only
E. II and III only

Given that b = a*(105/100)
Or, b = a*(3*7)/(4*5)
Or, a = b*(4 * 5)/(3 * 7)

Since a is an integer, 3 & 7 must be prime factors of b.
Answer: (D)

[RonPurewal]

Yep.