Ratios are one of those concepts that can pop up in just about any GRE problem. You could have the ratio between two angles that form from intersecting lines. You could have the ratio of rates a person drove at. With such a broad application, ratios are one of the concepts it is imperative you learn now.

To see how ratios work – and to see the type of problems the GRE will devise to try to trick you – let’s try a few simple problems first and then some challenging ones.

## GRE Ratio Practice Problem

1. Tom is selling apples and oranges. The ratio of apples to oranges in his cart is 3:2. If he has 12 oranges, how many apples does he have?

(A) 2 (B) 3 (C) 8 (D) 18 (E) 30

**Solution:**

A ratio is basically a fraction that has been reduced as much as possible. In this problem the ratio 3:2, can be represented as 3/2. One way to solve this problem is to set up a simple equation:

Notice I placed the 12, the number of oranges, in the denominator. We have to make sure that the number 12 corresponds to 2, the oranges in the ratio. Solving for x, we get 18 (D).

An even quicker way is to notice that we have (x6) the oranges (from 2 we go to 12) so we just have to (x6) the apples in the ratio: 3 x 6 =18.

Now’s let’s try the same question but with a spin:

2. Tom is selling apples and oranges. The ratio of apples to oranges in his cart is 3:2. If he has a total of 30 fruits, how many apples does he have?

(A) 2 (B) 3 (C) 12 (D) 18 (E) 30

**Solution:**

This question, while essentially the same, is the one that gives students a lot more trouble. The problem is combining two concepts: ratio and total. To do so simply add the ratios. We have 3:2 so the total is 5.

One way to solve the problem is to set up the table. Tables are great both from a teacher’s and beginner’s standpoint. In this case, I get to show you a nice, tidy way of solving the problem and you have an easy way both to conceptualize and solve the problem.

However, once you become used to tables, in the interest of time, learn to solve a ratio without one (I’ll show you how to do so in a second!).

Apples | Oranges | Total | |
---|---|---|---|

Ratio | 3 | 2 | 5 |

M(x) | |||

Actual | ? | ? | 30 |

What do we multiply the total ratio by to get the actual total? (x6).

So in the middle row in the total column we can place a 6.

Apples | Oranges | Total | |
---|---|---|---|

Ratio | 3 | 2 | 5 |

M(x) | 6 | 6 | 6 |

Actual | 18 | 12 | 30 |

Notice the (M)x, which stands for multiply (you can dispense with the M, I just didn’t want anyone thinking there is this random variable x floating around).

Now we multiply the apples and oranges by 6 to get 18 and 12, respectively.

Remember the faster way I mentioned?

- Add the ratio
- Figure out the x6
- Multiply 3 x 6

18 (D). Also remember not to mix up apples and oranges. A classic trick on the GRE is they reverse the order.

## Practice Questions

Okay, you’ve got the hang of ratios. Now you want something a little more challenging. Voila – it’s the ratio challenge workout.

The problems below appear superficially similar. Yet, the math behind each is different. Nonetheless, each of them deals with ratios, and will put your ability to the test. The problems tend to get progressively harder. So if you struggled at the end, don’t despair. As long as you can nail the first three, you’re doing pretty well on ratios.

Good luck!

1. A jewel necklace contains only emeralds, rubies, and diamonds. If the ratio of emeralds to diamonds is 2:7 and the ratio of diamonds to rubies is 3:2, then which of the following could not be the number of jewels on the necklace?

(A) 41

(B) 81

(C) 82

(D) 123

(E) 205

2. A tiara is studded with a mixture of gems. The ratio of sapphires to emeralds is 3:1. If 6 emeralds are added, the tiara will contain an equal number of sapphires and emeralds. How many emeralds must be added to the original tiara so that the ratio between emeralds and sapphires is 3:1?

(A) 9

(B) 12

(C) 18

(D) 24

(E) 27

3. An imperial scepter is mounted with diamonds, rubies, emeralds, and sapphires in the ratio of w : x : y : z. If w, x, y, and z are distinct single digit primes, then how many gems could be on the imperial scepter if it only contains the gems listed above?

(A) 19

(B) 58

(C) 85

(D) 97

(E) Cannot be determined from the information provided above.

4. (Okay – enough with the jewels!) Marty has a coin collection, which consists of only New World and Old World coins, in a ratio of 3:1. Marty’s friend Kyle swaps 28 of his Old World Coins for 28 of Martin’s New World coins. If Marty now has as many Old World coins as he does New World coins, how many coins did Marty originally have in his collection?

(A) 28

(B) 84

(C) 112

(D) 130

(E) 390

5. (Okay, back to jewel necklaces). Marty has a necklace that contains a total of 36 rhinestones, zirconium, and obsidian “gems.” If the ratio of obsidian to zirconium is 2:5, then which of the following could not be the number of rhinestones in Marty’s necklace?

(A) 8

(B) 12

(C) 15

(D) 22

(E) 29

## Answers:

1. B

2. D

3. C

4. C

5. B

Got all the problems right! Took some time though:-(

could you help me out with question no 3 im stuck

Hi there 🙂

In this problem, we’re told that w, x, y, and z are distinct single digit primes. With that information, we can list the values for these variables:

2, 3, 5, 7.

It doesn’t matter which variable goes with which prime number. The important aspect of having four unique variables is that each one has a different value. So, once we have the four values, we know that the ratio of the gems is

2:3:5:7

If there were only 2, 3, 5, and 7 of the different gems, then the total would be the sum of this ratio:

2+3+5+7 = 17

However, 17 is not an answer choice. This indicates that the ratio 2:3:5:7 is a simplified ratio and that there is a common factor, n, that we must multiply the ratio by to get the actual number of gems.

2n:3n:5n:7n

If we were to find the sum of the number of gems using the ratio above, then, we would have

2n + 3n + 5n + 7n = 17n,

which shows us that the actual number of gems must be a multiple of 17. Out of the possible answer choices, only C (17*5 = 85) is a multiple of 17. So the correct answer is C 🙂

I hope this helps!

Hi Chris,

In problem 5 I’ve been trying to figure out how to place the number 22 (answer D) in the ratios as to discard it, but I just can’t. How is it 22 is not an answer like 12?

You can have 4 obsidian, 10 zirconium, and 22 rhinestones. Thus, a ratio of 2:5:11.

OMG I just got it!! They switched Emeralds and sapphires! ugh

Great! Glad to hear that :). Sometimes, it’s just seeing that one little thing that opens up the entire problem.

I think I’m getting confused on question # 2 because of the way I’m reading it. I think misinterpretation on the math will hurt me more than math itself. Can you help? The passage says:

“A tiara is studded with a mixture of gems. The ratio of sapphires to emeralds is 3:1. If 6 emeralds are added, the tiara will contain an equal number of sapphires and emeralds. How many emeralds must be added to the original tiara so that the ratio between emeralds and sapphires is 3:1?”

I’m reading it as you have a tiara where the ratio of sapphires to emeralds is 3:1. Good so far. But then 6 Emeralds are added to this tiara throwing the ratio off making it 1:1. We should have more sapphires than Emeralds to stick with the original ratio of 3:1. Now here is where I get stuck. If you added 6 emeralds and it was already to much. Making the ration 3:3 or 1:1 why would you add more? I understand the math works and it comes out to 24. But when I read the question it makes no since to me. If adding 6 emeralds makes the the ratio go up why would we add 24? That’s where I’m confused. Not so much the math. But I can’t tie it back to the question to make real world sense. There must be something I’m not reading right.

Chris thank you for all your explanation but I am stuck in number 1# do you have anything else to support it that will help me understand better. If I have to elements to compare I am fine and I do well but having a third element become difficult to me.

Thank you

Yes, the three ratios are difficult because there is a “mismatch”. You have to think of the three ratios as a fraction in which you have to find a common denominator. So with question 1):

Emeralds : Diamonds is 2 : 7

Diamonds : Rubies 3 : 2.

Next, look for what the two pairs have in common: Diamonds. Notice that in the first pair the number 7 corresponds to diamonds, and in the second set 3 corresponds to diamonds. 3 and 7 are totally different numbers. How do we make them similar? By finding their lowest common multiple = 21. Therefore, multiply the top pair (emeralds and diamonds) by 3, and the bottom pair by 7″

Emeralds : Diamonds = 6 : 21

Diamonds : Rubies = 21 : 14

Now we can put all three stones in one row (because diamonds in both cases equals 21).

Emeralds : Diamonds : Rubies = 6 : 21 : 14.

Hope that helps set up the question!

Chris Lele you are the best!!! it really help me out to understand better I hope I can get the others exercise. I practiced last night to 3 am and I could not get through it much. Now with your help I will face the others exercises I hope i can make it this time.

Thank you

Thanks Yamil for the kudos! I am happy the explanation helped — even if it was in the middle of the night :).

hey Chris. These are awesome posts. I just have a couple of questions on number 1, especially on what you explained to the other chris on september 16th, 2013. I understand that you have to make a similar denominator, which is 21, but why do you have to multiply one row by 3 and the other by 7? I get that 3 and 7 are the multiples of 21, and those are the diamonds and that that is the only thing we have in common in this problem, but why multiply these factors to these rows? for what?

Also, before you stated that if we were capable of doing the first 3 problems than we have an “ok” understanding of ratios. So if I was’t even capable of doing the first one, doesn’t that say something? :/ where am i slacking? or where can i find simpler problems so that i can proceed to the first one with more ease…

Thanks again Chris, you guys are great!

Thank you Tom!

The table method works great in every ratio problem. you helped me a lot!

For the question 5, can we solve it this way:

R:0:Z= unknown:2:5 = 36.

We subtract each of the option from 36 and see whether the resultant is perfectly divisible by 7. if it is not, then that is the answer.

In this case , when we subtract option b from 36, we get the output of 24 which is not perfectly divisible by 7.

Therefore, answer is B

Hi Ve,

Yes, that is a good way of doing it :).

Hi Chris,

I haven’t understood question no 5. Can you please explain it once again? The entire sum..?

Hi Anki,

When we have a ratio, we want to make sure to differentiate between the ratio and the sum. For instance, if the ratio of gems is 2:2:5, the total is still going to be 36. But that doesn’t mean the number of gems is 2, 2, and 5. You will have to multiply each by 4, to get 8, 8, and 20 stones, which equals 36.

Of course with this problem there are various combinations that you have to figure out, besides the 2:2:5. For instance, the ratio can also be 2:5:5, 2:5:11, and 2:5:29. The final ratio actually equals 36, which means there are 2, 5, and 29 gems.

Hope that helps!

Thank you

Yes, I believe the last vid is a repeat of #4, and I am stuck on #5. Thanks.

I’ve added in the 5th video, I hope it helps!

Hi,

Thank you for these great explanations! However, I notice that video #4 and #5 are the same exact videos.

It’s been fixed, you can watch the explanation for the 5th video now. Thanks for letting me know!