Last year, a certain company began manufacturing product X and sold every unit of product X that it produced.

Last year, a certain company began manufacturing product X and sold every unit of product X that it produced. Last year the company's total expenses for manufacturing product X were equal to $100,000 plus 5 percent of the company's total revenue from all units of product X sold. If the company made a profit on product X last year, did the company sell more than 21,000 units of product X last year?

(1) The company's total revenue from the sale of product X last year was greater than $110,000.

(2) For each unit of product X sold last year, the company's revenue was $5.

This is a very tough and time consuming question from the GMAT Official Guide (video solution below). It has a few things that make it extra challenging:

  1. It's a long word problem so there's a lot to chew through.
  2. It has multiple variables.
  3. It's an inequality/threshold question

That said, conceptually it's not hard at all. The basic idea: find the break even point or threshold and see if the statements help nail down whether you are above or below.

As always start by reading carefully. Do not rush to the statements. Data Sufficiency success is all about setup!

Let's go ahead and define the question: Did the company sell more than 21,000 units of product X last year?

You might be tempted to stop there. But: keep going! How will we determine whether the company sold more than 21,000 units? What does that 21,000 depend on? What's that you say???? The price? YES! This all hinges on the price per unit. So let's get the inequality defined so we can figure out what price per unit of product X gets us above 21,000 units.

Here's this part translated into an inequality: Last year the company's total expenses for manufacturing product X were equal to $100,000 plus 5 percent of the company's total revenue from all units of product X sold.

#*$ - (100,000 + 5/100*#*$) > 0

Next add the number of units for which you want the dollar amount. We want to know what $ comes out to with # = 21,000.

21,000*$ - 100,000 - 5/100*21,000*$ > 0

21,000*$ - 100,000 -1050*$  > 0

19950*$ -100,000 > 0

$ > 100,000/19950

That comes out to a little more than 5. So at 21,000 units that price is a bit more than $5. So if we're 5 or below we've sold more than 21,000 units.

Let's take a look at statement (1) The company's total revenue from the sale of product X last year was greater than $110,000.

Always write out the algebra: #*$ > 110,000,

We need to know if # > 21,000. With statement (1) we can't separate # from $. So:

Yes: 110,000 units for $1 each.

No: 1 unit for $110,000

Insufficient.

Statement (2) For each unit of product X sold last year, the company's revenue was $5.

This helps. We defined the break even for 21,000 units at 100,000/19950 or a shade above 5. So if we know that revenue per unit X is $5 then the company has sold more than 21,000 units (as price goes down units sold goes up). Sufficient.

B.

Last year, a certain company began manufacturing product X and sold every unit of product X that it produced. Last year the company's total expenses for manufacturing GMAT Explanation Diagram

Last year, a certain company began manufacturing product X Video Solution

Additional GMAT Data Sufficiency Thresholds, Yes/No, Inequality Practice Questions

These two questions are a bit different in content and not as wordy as the above but they share the bigger concept of defining the threshold/breakeven/inequality.

GMAT Data Sufficiency Thresholds/Inequalities Question of the Day 1

GMAT Question of the day Thresholds/Inequalities/Statistics

 

GMAT Question of the Day - Data Sufficiency - Probability

Roy has a bowl with 12 red and j blue marbles. If Roy picks two random marbles out of the bowl it is more likely that he will pick one of each color than only two blue marbles. Are there an equal number of red and blue marbles in the bowl?

(1) j > 11

(2) The chance of picking two blue marbles is greater than 5/23

AnswerShow

GMAT Question of the Day Solution

On GMAT Data Sufficiency questions take your time to understand the given information. Avoid rushing into the statements. In this case you need to unravel this statement: If Roy picks two random marbles out of the bowl it is more likely that he will pick one of each color than only two blue marbles. This is critical. Now you might be thinking that this question is very open ended because you don't know how many blue marbles you have but the given information severely limits the options. The number of blue marbles must be less than or equal to 25. Here's the math to calculate that.

This is based on this statement: If Roy picks two random marbles out of the bowl it is more likely that he will pick one of each color than only two blue marbles.

j/(12+j) * (j-1)/(12+j-1) *(2!/2!)  < j/(12+j)*12/(12+j-1)*(2!/(1!*1!))

j<25

Statement (1) Could be a Yes, 12, or a no with anything greater than 12. Statement (2) give you the same information but in a slightly different way. We need to know whether we have 12 blue marbles so just try plugging in 11 marbles or 12 marbles see which one (or maybe both) agrees with the information in the statement. We can see that when picking 11 you get exactly 5/23. So there must be more than 11 blue marbles. That is the same info as statement 1. You could have the same number of marbles 12 and 12 or you could have more blue marbles, up to 24.

 

GMAT Question of the Day - Data Sufficiency - Algebra/Threshold

In a certain company 2/5 of the employees are either engineers or scientists. What is the ratio of engineers to scientists?

(1) There are 75 employees in the company

(2) There are more than 3 times as many engineers as there are scientists

AnswerShow

GMAT Question of the Day Solution

It's good policy on GMAT Data Sufficiency to spend as much time as necessary to understand the question before moving on to the statements. The questions (and the information given in the question) are your friends. In this question of the day there isn't a ton of information in the question but still it might help to write out that 2/5's of the company is made up of Engineers and Scientists and that we are looking for E/S.

Statement 1 tells us that that E + S = 30. But we don't know what the mix is. Insufficient.

Statement 2 tells us that the ratio is greater than 3 to 1. So the ratio could be 3:1 or 4:1 or 7:2. Many possibilities. Insufficient.

Putting the statements together provides some limitations but not enough of them to narrow the ratio down. We could have 29 engineers and 1 scientist or 28 engineers and 2 scientists. Both ratios are greater than 3 to 1. Multiple possibilities. Insufficient.

 

GMAT Question of the Day - DS - Ratio

GMAT Question of the Day - Data Sufficiency - Ratio/Weighted Average

The ratio of the number of students in the math department, history department, and science department is 3 to 7 to 10 respectively. Is the average height of students from all of the departments less than 60 inches?

(1) The sum of the heights of all the students is less than 100 feet.

(2) This math department has 7 fewer students than the science department.

AnswerShow

GMAT Question of the Day Solution

This GMAT question of the day presents another threshold question. In this case we want to know whether we are above or below 60 inches. From the given information we know that the minimum number of students is 20 (3 + 7 + 10).

Statement (1) If we divide max height (100 feet or 1,200 inches) by min students (20) we get 60. That's right at the threshold. Because we have an inequality we must be below this threshold. Sufficient.

Statement (2) From this we can infer the number of students but have no information on their heights. Insufficient.

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