Lesson: Introduction to Ratios

Comment on Introduction to Ratios

Awesome thanks for the lesson. I hear ratios is a big part of the gmat
gmat-admin's picture

Ratios are important, but I wouldn't say they're tested as often as the super popular concepts like percents, geometry, statistics, integer properties and overlapping sets.

I always mix up the ratios and my solution is always exactly the other way round, e.g. my solution is the ratio x:y but actually the right answer would be y:x. Do you have any idea what I am doing wrong?
gmat-admin's picture

This is a common error.

To help avoid making this mistake, it's useful to jot down the order in which you are presenting your info. Here's what I mean.

Let's say we're told that the ratio of girls to boys is 2 to 3.

Some students will take this information and write 2 : 3 (or 2/3) on their notepad. The problem with this is that it isn't clear what the two numbers represent.

Instead, I suggest that you first write: girls/boys to help you keep track of what each number represents. Then, when you write 2/3, you will be less likely to mix up the values.

You'll see that I do this throughout the videos.

In a certain district, the ratio of the number of registered Republicans to the number of registered Democrats was 3/5. After 600 additional Republicans and 500 additional Democrats registered, the ratio was 4/5. After these registrations, there were how many more voters in the district registered as Democrats than as Republicans?

(A) 100
(B) 300
(C) 400
(D) 1,000
(E) 2,500
. Please help can't figure out .
gmat-admin's picture

There are some nice approaches to that question here: http://www.beatthegmat.com/ratio-shorter-method-t114168.html and here https://gmatclub.com/forum/in-a-certain-district-the-ratio-of-the-number...

Let me know if you'd like me to solve as well.

I solved this problem and I figured out 400. What I'm doing wrong??
Let X = Registered Republicans
Let Y = Registered Democrats

X=(3/5)Y equation (1)

After Additional:
(X+600)/(Y+500)=4/5 Using (a/b)=(c/d) -> ad=bc
5X+3,000=4Y+2,000 equation (2)

Replacing equation (1) in equation (2)
Solving: Y=1,000
Putting Y in the equation (1): X = (3/5)*1,000


Help me to find what I'm doing wrong.


gmat-admin's picture

Everything is perfect . . . . . except we're not done yet.

You determined that X = 1000 and Y = 600.

However, X = Registered Republicans ORIGINALLY, and Y = Registered Democrats ORIGINALLY.

The question asks, "AFTER these registrations, there were how many more voters in the district registered as Democrats than as Republicans?"

So, AFTER the registrations...
X + 500 = number of Democrats, and...
Y + 600 = number of Republicans

So, AFTER the registrations...
1000 + 500 = 1500 = number of Democrats, and...
600 + 600 = 1200 = number of Republicans

So, there are 300 more Democrats than Republicans

Wow, I forgot to sum after the registrations!

Thank you.

Hey, thanks for all the awesome videos. I was wondering something though, at 01:20 it's said that ratio can be written in 3 ways. The two first ways of writing it makes sense, but the last one confuses me a little. When reading 2 divided on 5 (2 over 5) somehow that seems wrong to me. If I read it that way then I read it as 2 out of 5 being boys, and the remaining (3 out of 5) being girls. Instead of reading it as there being 2 boys for every 5 girls. Any simple explanation to keep me from getting it mixed up or for it to make sense to me?
gmat-admin's picture

Here's another way to look at the a/b notation.

Rather than say the ratio a/b represents "a out of b", we can say "For every b things there are a things"

For for example, if 2/3 of the students are girls, we can say "For every 3 students, 2 are girls"

In this case, the ratio of girls to total students is 2:3 (or 2/3)

Does that help?


For the reptiles and birds question, after you get 7/2 = R/28, you cannot cross simply 7 and 28 to get 1/2 = R/4 correct? I tried doing that and got 2 just to test this. I knew this was incorrect, but that makes me curious about cross multiplying and when it was okay to use that technique.
gmat-admin's picture

Good question.

Cross simplifying only works when we are multiplying fractions.

However, cross MULTIPLYING is something completely different.

If 7/2 = R/28, then we can solve for R by first cross multiplying to get: (2)(R) = (7)(28), and then solving this equation for R


Also for the cookies and nuts questions, I looked over the work you did to see if I could come up with a quicker way. I noticed, for example, in the cookie question, adding 2:1 to get 3, you can then divide 3 into 15 (the total amount of cookies) to get 5, which are the amount of cookies in each bag, then just multiply the corresponding number in the ratio to the person receiving the cookie by that 5, in this case, 10 for Kendra (2 times 5) and 5 for Patty (1 times 5). I tried this approach for the nuts questions and got the same result and found it quicker than drawing bags out.
gmat-admin's picture

Your approach is perfectly valid.

I start by drawing bags/buckets of values to show why the technique works. I imagine many students, once they're familiar with the related concepts, gradually transition away from drawing bags.

Hi Brent, can you please provide a simple solution to this problem:
Yesterday, the ratio of males to females working at Gigacorp was 2:3. Today, Gigacorp hired an additional 81 employees, and the ratio of males to females is now 6:5 (no employees left or were fired). What is the least number of males that could have been working at Gigacorp yesterday?

A) 16
B) 24
C) 39
D) 40
E) 42
gmat-admin's picture

Hi Jalal,

Here's my step-by-step solution: https://gmatclub.com/forum/yesterday-the-ratio-of-males-to-females-worki...


Hi Brent, I have a question regarding this:

In a certain physics class, the ratio of the number of physics
majors to non-physics majors is 3 to 5. If two of the physics
majors were to change their major to biology, the ratio would
be 1 to 2. How many physics majors are in the class?
(A) 16
(B) 18
(C) 24
(D) 30
(E) 32

If I use the same approach of the others questions the answer I get that the answer is 16. A). Since:
(3x-2)/(5x+2) =1/2
3*(6)-2= 16. They are asking how many physics majors ARE in the cl...OH.. I just realize the question is asking about actual physics and not physics in a theorical scenario.
Thank you anyway.
Cheers, Alex
gmat-admin's picture

Nice catch! You're certainly not the first person to confuse the hypothetical information with the given information :-)

Hi Brent,

Regarding the portioning of ratios part, what should we do when we are unable to divide the quantity in equal parts?

For instance, in the nuts mix question, what if we had to find the proportion of walnuts in 47 pounds of nut mix?
gmat-admin's picture

Great question!

In general we can say:
If the terms of the ratio add to x, then the number of "things" to be distributed in that ratio must be divisible by x (otherwise the distribution is impossible).

Consider this example:
Someone wants to distribute cookies to Joe and Sue in a 1 : 2 ratio, in which Joe receives 1 cookie for every 2 cookies that Sue receives.
1 + 2 = 3
So, for every 3 cookies to be distributed, Joe receives 1 cookie and Sue receives 2 cookies.

For example, if there are 15 cookies in total, then Joe receives 5 cookies and Sue receives 10 cookies. Notice that our distribution works perfectly, since 15 is divisible by 3.

Conversely, if there are 4 cookies in total, it's impossible to distribute those 4 cookies in the 1 : 2 ratio. Notice that our distribution doesn't work in this case because 4 is NOT divisible by 3.

Does that help?

Cheers, Brent

yes it does. Thank you.

Hi Brent,

For the following question - https://gmatclub.com/forum/oil-vinegar-and-water-are-mixed-in-a-3-to-2-to-1-ratio-to-make-salad-273726.html

We have been asked the maximum number of cups of salad dressing he can make. If 6x makes ONE cup of salad dressing and x is 8/3 --> shouldn't we be able to make (1*8/3) cups of salad dressing?

I may be misinterpreting what the question is asking. Please help.
gmat-admin's picture

Question link: https://gmatclub.com/forum/oil-vinegar-and-water-are-mixed-in-a-3-to-2-t...

The question: Oil, vinegar, and water are mixed in a 3 to 2 to 1 ratio to make salad dressing. If Larry has 8 cups of oil, 7 cups of vinegar, and access to any amount of water, what is the maximum number of cups of salad dressing he can make with the ingredients he has available, if fractional cup measurements are possible?

Given the notation in your question, I assume you're referring to Scott's solution (at https://gmatclub.com/forum/oil-vinegar-and-water-are-mixed-in-a-3-to-2-t...)

In his solution, Scott let's x = cups of water, 2x = cups of vinegar and 3x = cups of oil

So, x + 2x + 3x = 6x
So, 6x = the total CUPS of dressing.

So, if x = 8/3, then we're combining:
- 8/3 cups of water
- 16/3 cups of vinegar
- 24/3 cups of oil
For a total of 16 cups of dressing.

Does that help?

I think I messed up in understanding 6x as TOTAL cups of dressing. Thanks.

Hi Brent, thanks for previous clarification I will post my questions on the proper place and with all options for the answer.

The ratio of cups of flour to cups of milk in a cake is 3:2. 6 cups of milk are added to the cake, and the new ratio is 3:4, how many cups of flour are there in the cake?
A) 6
B) 8
C) 9
D) 10
E) 12
gmat-admin's picture

Let F = the number of cups of flour in the mixture
Let M = the number of cups of milk in the mixture
So we can write: F/M = 3/2
Cross multiply to get: 2F = 3M

After we add 6 cups of milk...
F = the number of cups of flour in the mixture
M + 6 = the number of cups of milk in the mixture
So we can write: F/(M + 6) = 3/4
Cross multiply to get: 4F = 3(M + 6)
Expand: 4F = 3M + 18

We now have the following system of equations:
2F = 3M
4F = 3M + 18

Solve to get: F = 9

Answer: C


I think there are some ambiguity here because someone could understand it as "the original values add up to 117?
gmat-admin's picture

Given: IF k is added to each number the new ratio will be 4 to 5, and the sum of the numbers will be 117

The key words/phrases here are IF and WILL BE.
These words set up a hypothetical situation.
So, the sum of 117 refers to a hypothetical situation if it were true that k was added to each number.

While it's possible someone may misread the question, the words themselves are not ambiguous.

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