Seed mixture X is 40 percent ryegrass and 60 percent bluegrass by weight; seed mixture Y is 25 percent ryegrass and 75 % fescue. If a mixture of X and Y contains 30% ryegrass, what percent of the weight of the mixture is X? GMAT Explanation, Video Solution, and More Practice!

Seed mixture X is 40 percent ryegrass and 60 percent bluegrass by weight; seed mixture Y is 25 percent ryegrass and 75% fescue. If a mixture of X and Y contains 30% ryegrass, what percent of the weight of the mixture is X?

A. 10%
B. 33 1/3%
C. 40%
D. 50%
E. 66 2/3%

Correct Answer: B

This is a tough weighted average/mixture question from the GMAT Official Guide that comes up a bunch in GMAT tutoring session.

Define the Question

what percent of the weight of the mixture is X?

We know the proportion of rye grass in each mixture, X and Y, and we know the proportion of rye grass adding x and y together. What we want to infer is the actual weight of seed mixture x.

Setup 1

There are two ways I'd approach this. The first way is faster and my preference.

25 (Y)----------30-----------------------40 (X)

So, just looking at the above, any idea which mixture X or Y is in greater abundance? The weighted average, 30, is closer to 25 (Y) so there must be more of Y. That means the mixture is less than 50% X so you can eliminate D and E. Based on the layout you could also eliminate A in that 10% is too extreme.

Helpful but that's not the actual method.

25 (Y)----------30-----------------------40 (X)

To solve with certainty take the difference of each part with the total. So 30 - 25 and 40 - 30. 5 and 10. Add them. 15. The 15 is the denominator. Now take the differences 5 and 10 and pop them in as numerators. 5/15 and 10/15. So x = 1/3 and y = 2/3. That's it. B.

Setup 2

The alt. way is using work/rate T's. Weighted average/mixture questions are the same as average rate questions so you can solve them in the same way.

Video Solution: Seed mixture X is 40 percent ryegrass and 60 percent bluegrass by weight; seed mixture Y is 25 percent ryegrass and 75 % fescue. If a mixture of X and Y contains 30% ryegrass, what percent of the weight of the mixture is X?

Additional GMAT Weighted Average/Mixture Practice Questions

Here's another tough weighted average/mixture question from the GMAT Official Guide: Jackie has two solutions that are 2 percent sulfuric acid and 12 percent sulfuric acid by volume, respectively. If these solutions are mixed in appropriate quantities to produce 60 liters of a solution that is 5 percent sulfuric acid, approximately how many liters of the 2 percent solution will be required?

And one from the GMAT Prep Tests: Three grades of milk are 1 percent, 2 percent and 3 percent fat by volume. If x gallons of the 1 percent grade, y gallons of the 2 percent grade, and z gallons of the 3 percent grade are mixed to give x+y+z gallons of a 1.5 percent grade, what is x in terms of y and z?

During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine's average speed for the entire trip? GMAT Explanation, Video Solution, and More Practice!

During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine's average speed for the entire trip?

A. (180−x)/2

B. (x+60)/4

C. (300−x)/5

D. 600/(115−x)

E. 12,000/(x+200)

Average speed, distance, miles per hour... Yes, it's a GMAT work and rate question. In this case we have an average rate with variables in the question and in the answer choices. That's a signal for considering picking numbers. Why pick numbers? To make life easier. How do I judge if picking numbers will make life easier? If the question would be a simple 1-2-3 if you had the actual numbers that's a pretty good indicator. Here, if you just had to calculate the average speed, assuming you know how to do a weighted average, things would be smooth. Here's another GMAT word problem from the GMAT Official Guide on which you can pick numbers to make things much easier.

Let's get back to Francine and her trip! Let's define the question: what was Francine's average speed for the entire trip?

Now let's take inventory. What numbers are we missing? Distance! Yes, I'd go ahead and pick a distance for the trip. Pick something that works well with the speeds, 40 and 60. Why not go for 240 total distance and divide that into 120 and 120 for each leg of the trip? That way you've got easy division.

120 miles/40mph = 3 hours

120 miles/60mph = 2 hours

Total time = 5 hours

Total distance = 240 miles

240/5 = 48mph

Just as a sanity check for the 48. We spent more time (3 hours) traveling at the slower speed (40mph) so it makes sense that the average speed would be closer to 40mph than 60mph.

OK, so, with total distance at 240 the average speed is 48. Of course, that's not an answer. We have to plug in "x", the total distance at an average speed of 40 miles per hour, in order to yield 48. We can use our divisibility shortcut here. Since we know the correct answer is 48 we know that the numerator has to be a multiple of 3 (because 48 is a multiple of 3). So just focus on the numerator and eliminate answer choices not divisible by 3.

A. (180−x)/2.  180-50 = 130

B. (x+60)/4.  50 + 60 = 110

C. (300−x)/5.  300-50 = 250

D. 600/(115−x) Div by 3

E. 12,000/(x+200) Div by 3

You end up with D and E as possibilities. At this point you can just work out the entire expression for both choices. You could also:

  1. Continue using divisibility. 48 has four 2's in it. 600 only has 3. So that's out.
  2. Use Magnitude. It should be pretty clear that D. is way too small.

So, in terms of x, what was Francine's average speed for the entire trip? E. 12,000/(x+200)

I think picking numbers is a great way to go on this one. You can also do the algebra which could be a little faster depending on how speedy you are at setting it up but, at least in my mind, is less straight forward and I've seen many GMAT tutoring students screw it up. I'll do the algebra in the diagram and in the video for your reference.

During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour GMAT Explanation Diagram

During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour GMAT Explanation Shortcut Diagram

During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour GMAT Explanation Alt Diagram

 

Video Solution: During a trip, Francine traveled x percent of the total distance at an average speed of 40 miles per hour and the rest of the distance at an average speed of 60 miles per hour. In terms of x, what was Francine's average speed for the entire trip?

Additional GMAT Work and Rate/Weighted Average Practice

Here's a work and rate question from GMAT Question of the day which has the same basic structure and explains the rate t's a bit more.

And another average rate question from the GMAT Question of the day

Here's a very challenging average rate question from the GMAT Prep Tests. Same basic premise but on this one the follow through is tougher because it involves a quadratic.

GMAT Work and Rate question from the Official Guide to practice picking numbers. It's a cooperative rate so the follow through is a bit different than the above "Francine" average rate but you can still use rate T's and again pick numbers: Working simultaneously at their respective constant rates, Machines A and B produce 800 nails in x hours

Last month 15 homes were sold in Town X. The average (arithmetic mean) sale price of the homes was $150,000 and the median sale price was $130,000. Which of the following statements must be true?

I. At least one of the homes was sold for more than $165,000.
II. At least one of the homes was sold for more than $130,0000 and less than $150,000
III. At least one of the homes was sold for less than $130,000.

(A) I only
(B) II only
(C) III only
(D) I and II
(E) I and III

This question is from the GMAT prep tests 1 and 2 so if you haven't done those yet skip this one for now. You're given average and median and with those two pieces of information need to make some inferences. Know your basic GMAT content. There shouldn't be any confusion about fundamental statistics terms, average, median, mean, and standard deviation. For standard deviation you don't need to know the formula or how to calculate it but should have a strong understanding of what it measures.

OK, back to the question. What does it mean that the median sale prices was $130,000? Well, median is a spatial measurement. It's a location: the middle of the set. For an odd set it's just the middle number (in this case the 8th). For an even set it is the average of the two middle numbers. OK, so if you have a median of $130,000 in a 15 number set then it must be that 7 numbers are equal to or below $130,000 and 7 numbers are equal to or above above $130,000. With that information alone none of the statements MUST be true. They all COULD be true.

I. At least one of the homes was sold for more than $165,000. Sure. All of the houses above the median could have been sold for $1.2 million. Or not. They could all have been sold for $140,000. There's nothing limiting this.
II. At least one of the homes was sold for more than $130,0000 and less than $150,000 Same idea as statement one. Easy to have home in that range or not.
III. At least one of the homes was sold for less than $130,000. All of the homes could have been sold at the median or you could have 7 houses below and 7 above. There a variety of ways to make this true or not true.

The average sale price being $150,000 also doesn't allow us to commit to any of the statements. They still all COULD be true.

I. At least one of the homes was sold for more than $165,000. With the mean at $150,000 we know that the balance of things ends up at $150,000 but how we get there is completely unknown. You could have 14 houses sell for $1 and the 15 house sell for way above $150,000 so the average balances at $150,000. You could have all 15 houses sell for $150,000. Again, there many ways to do this.

II. At least one of the homes was sold for more than $130,0000 and less than $150,000 Same idea as statement one. Easy to have home in that range or not.
III. At least one of the homes was sold for less than $130,000. All of the homes could have been sold at the average ($150,000) or you could have a home sold for less than $130,000 with at least one of the other houses above the average to balance out to the average.

 

Putting the mean and median together it becomes clear that at least some of houses need to be above $130,000 in order to create a mean of $150,000. That puts a shadow over statement III. It doesn't seem that we any limitations on the downside. Our main concern is having enough $'s above the median to balance things out to $150,000. Statement II has the same issue. There's nothing to limit the sales prices that are above the mean to the range of $130,000 and $150,000. With that we can pick A.

But how do we prove that at least one of the homes was sold for more than $165,000? Let's say we maximize the value of the houses to the left of the median. To do that we'd make them all equal the median at $130,000. If you have 7 houses at $130,000 what do the 7 houses above the median have to be to balance to a mean of $150,000? There's an equal number of houses (7 and 7) so just think of it as one house at $130,000. What does the other house have to be: $170,000. Add those and divide by two and you get $150,000. But, you might say, what about the median? We didn't count that. No, we didn't. But will the median at $130,000 make it so the $170,000 is higher or lower? We're adding more weight to the $130,000 side so the $170,000 would need to be increased to compensate. At $170,000 we're already clearing the $165,000 line from statement I so adding the median just pushes us further in that direction. Again, A MUST be true.

Additional GMAT Statistics Example Questions!

Here's a GMAT statistics question (median and range) which will teach you how to organize a whole bunch of unknowns without using variable. Also a good one to work on maximizing a value.

A set of 15 different integers has median of 25 and a range of 25. What is greatest possible integer that could be in this set?

Here's a mini-statistics puzzle from the GMAT Official Guide. It has a somewhat simple solution. Still, it's surprising how often it comes up in GMAT tutoring sessions.

If Q is an odd number and the median of Q consecutive integers is 120, what is the largest of these integers?

Here's a challenging statistics, max/min question which we review in just about every GMAT preparation. It's a great one to review not only to understand how to organize a whole bunch of potential variables but to understand how to put context on a max/min scenario.

Seven pieces of rope have an average (arithmetic mean) length of 68 centimeters and a median length of 84 centimeters. If the length of the longest piece of rope is 14 centimeters more than 4 times the length of the shortest piece of rope, what is the maximum possible length, in centimeters, of the longest piece of rope?

Meat and potatoes GMAT statistics question. Not the toughest but it is still very important. Be ready to interpret a chart/graph. What isn't challenging in practice can quickly become a mess on test day.

The table above shows the distribution of test scores for a group of management trainees. Which score interval contains the median of the 73 scores?

Middle difficulty DS statistics range question from GMAT question of the day

And here's a very challenging GMAT statistics range question from GMAT question of the day that also incorporates a bunch of algebra and number properties. Stay organized!

Good luck and happy studies!

GMAT Question of the Day - Problem Solving - Average/Word Problem

For the past 300 days machine X has produced an average of 78 units of part K. Today machine X produced Z units of part K bringing machine X's total production of part K to 23,468 units. What is the value of Z?

A. 34

B. 36

C. 68

D. 72

E. 74

AnswerShow

GMAT Question of the Day Solution

GMAT weighted average questions are very similar to GMAT Rate/Work questions. For both question types you can use a T to organize the information. The T will help you make inferences. It can also help you to define the question. In a classic GMAT weighted average question you will have two scenarios and a total. You can organize those scenarios into three T's. Once you've done that this question of the day breaks down to simple arithmetic. Here's an Official GMAT question on which we also use the T method to organize the information: A boat traveled upstream a distance of 90 miles at an average speed of (v-3) miles per hour and then traveled the same distance downstream at an average speed of (v+3) miles per hour

GMAT Question of the Day Problem Solving Average Word Problem Solution

More GMAT Average Rate Practice Questions:

Distance, Work and Rate - Average Rate 1

Distance, Work and Rate - Average Rate 2

For more GMAT distance, work, and rate practice visit GMAT Question of the Day.

GMAT Question of the Day - PS - Word Problem

GMAT Question of the Day - Problem Solving - Word Problem/Ratio/Algebra

Alice and Carl each completed a portion of a certain job. Alice completed 20% of the job while Carl completed the rest. If they were each paid the same amount, what proportion of her payment must Alice give to Carl so that they each are paid the amount of the total that is proportionate with the amount of work that they each performed?

A. 1/2

B. 2/3

C. 3/5

D. 3/4

E. 5/8

AnswerShow

GMAT Question of the Day Solution

The GMAT Quant section has a whole bunch of word problems so it's important to feel comfortable with them. These questions tend to have a tough image but are often simple in concept (this question of the day has this feel). One thing that seems to help GMAT tutoring students: Taking time to carefully read the question. You do not need to solve the question as you read. You don't even have to take notes. The important thing is to process the information. To understand what the question is asking you to do. This is the time to make a broad plan for solving the question.

In this case Alice will need to give Carl part of her payment so that the pay per work is equal. Sounds like we can set up an equation with Carl on one side and Alice on the other. Pick a variable for the original payment (let's call it x) and pick a variable for the amount that Alice needs to lose from her side and that Carl will gain (let's call this y). Solve for y in terms of x and that will tell you the proportion of her payment that Alice needed to give to Carl.

You might be thinking: how do I deal with the different proportions of the job done by Alice and Carl? Well - look at it this way. We're dealing with 20% and 80% (.2 and .8). So the ratio is 1:4. Carl did 4 units of work to Alice's 1 unit of work. So Carl's money divided by his effort of 4 units must be equal to Alice's money divided by her effort of one unit.

GMAT Question of the Day Word Problem Algebra Solution Diagram

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