Last Updated on November 27, 2023
Whether you are taking the current version of the GMAT or the new GMAT Focus, GMAT Problem Solving questions are the majority of the GMAT quant questions you will see. Thus, to get a great score on the GMAT, you must be able to crush this question type. In this blog, we will discuss the essence of GMAT Problem Solving questions and look at some GMAT Problem Solving sample questions and their solutions. (If you need data sufficiency help, we’ve covered that in a separate article.) If you need more practice after completing what we offer in this article, please check out the Target Test Prep online GMAT course.
Here are the topics we’ll cover:
- What is a GMAT Problem Solving Question?
- The GMAT Quant Topics
- GMAT Problem Solving Topic 1: Number Properties – Factorial Divisibility
- GMAT Problem Solving Topic 2: Inequalities – Combining Equations and Inequalities
- GMAT Problem Solving Topic 3: Rates – Converging Rate Questions
- GMAT Problem Solving Topic 4: Statistics – Finding the Median of a Large Set of Data
- GMAT Problem Solving Topic 5: Ratios – Three-Part Ratios
- In Summary
- What’s Next?
Let’s begin with a discussion of what a GMAT Problem Solving question is.
What is a GMAT Problem Solving Question?
The good news is that GMAT Problem Solving questions are identical to the multiple-choice questions you’ve seen since your days of doing basic math questions. As such, a Problem Solving (PS) question presents the answer choices A, B, C, D, E and has just one correct answer. On the current version of the exam, PS questions make up about two-thirds of the questions in the GMAT quantitative section. So, you’ll see about 21 PS questions.
On the GMAT Focus, PS questions are actually their own section, which consists of 21 GMAT math questions. So, whether you are taking the traditional GMAT or GMAT Focus, you need to know GMAT Problem Solving questions!
KEY FACT:
There are 21 Problem Solving questions on GMAT Focus and around 21 Problem Solving questions on the standard GMAT.
Now, let’s discuss the quant topics you may see covered in GMAT Problem Solving questions.
The GMAT Quant Topics
If you are somewhat new to the exam, you may wonder, what the heck is tested in the GMAT quant section? Most of what is tested on the GMAT is math that you likely saw at one time in your life. So, rather than learning things from scratch, you can build back up the quant muscles you previously had. Sure, those concepts are tested slightly differently on the GMAT, but in general, there should not be many math topics that are completely new to you.
TTP PRO TIP:
There is a high likelihood that you are familiar with most of the math topics tested on the GMAT.
Let’s list the topics tested:
- GMAT Arithmetic Questions
- Fractions and Decimals
- Number Properties
- GMAT Algebra Problems
- Quadratic Equations
- GMAT Number Properties
- Exponents and Roots
- Inequalities
- Absolute Values
- Word Problems
- Rate Problems
- Work Problems
- Unit Conversions
- Ratios and Proportions
- Percents
- Statistics
- Overlapping Sets
- Permutations and Combinations
- Probability
- GMAT Geometry Questions
- Coordinate Geometry
- Sequences
- Functions
Note that Geometry is not included in the GMAT Focus.
Now that we are familiar with the basics of a GMAT Problem Solving question and the topics those questions may involve, let’s get into our GMAT problem-solving practice.
In the sections that follow, we will first present a topic, and then show how it can be presented in a GMAT PS question.
GMAT Problem Solving Topic 1: Number Properties – Factorial Divisibility
The first example is based on the topic of factorial divisibility, which is one of many integer properties. The nice thing about factorial divisibility is that, although it appears to be a difficult topic, it’s actually quite simple once we learn to use a very cool strategy for this type of question.
For example, let’s say you need to determine the maximum value of n for the expression (14!) / (2^n) such that the result is an integer. To determine the max value of n, we do the following:
First, divide 14 by 2, and note the quotient while ignoring any remainder:
14/2 has a quotient of 7.
Next, divide 14 by 2^2 = 4, and note the quotient while ignoring any remainder:
14/4 has a quotient of 3.
Next, divide 14 by 2^3 = 8, and note the quotient while ignoring any remainder:
14/8 has a quotient of 1.
Next, divide 14 by 2^4 = 16, and note the quotient while ignoring any remainder:
14/16 has a quotient of 0.
Since we have found a quotient of zero, we can stop. The final step is to add up all the quotients; that sum is the maximum value of n. So, the maximum value of n is 7 + 3 + 1 = 11.
TTP PRO TIP:
Use the strategy provided above to solve factorial divisibility problems.
Let’s practice with an example.
GMAT Problem Solving Example 1:
What is the greatest integer j, such that 240! / 4^j is an integer?
- 15
- 31
- 60
- 63
- 127
Solution:
First, we divide 240 by 4^1, noting the quotient and ignoring the remainder:
240 / 4 = 60
Now we divide the quotient 60 by 4^2, noting the quotient and ignoring the remainder:
60 / 4^2 = 60 / 16 = 3
Now we divide the quotient 3 by 4^3, noting the quotient and ignoring the remainder:
3 / 4^3 = 3 / 64 = 0
Because the quotient is 0, we stop.
The value of j is the sum of all the quotients, so we have:
60 + 3 + 0 = 63
This tells us that there are 63 fours in 240!
Thus, we know that the largest value of j that allows 240! / 4^j to be an integer is j = 63.
Answer: D
Next, let’s discuss a topic from inequalities.
GMAT Problem Solving Topic 2: Inequalities – Combining Equations and Inequalities
One of the first things you will learn on the GMAT is solving for the value of the two variables contained in two equations, you often use the substitution method, which functions just as it sounds like it would. We also use this process when we have one equation and one inequality containing two variables.
For example, let’s say we have the following:
Equation: y = 2x – 1
Inequality: 3x + 4y > 25
If we want to know what is true about x, we do the following:
Since y = 2x – 1, we can substitute 2x – 1 for y in the inequality 2x + 4y > 25. Doing so gives us:
2x + 4(2x – 1) > 25
2x + 8x – 4 > 25
10x > 29
x > 29/10
x > 2.9
Thus, we know that x is greater than 2.9.
TTP PRO TIP:
When working with inequalities and equations, we can substitute the equation into the inequality.
Let’s practice with one more example.
GMAT Problem Solving Example 2:
If 2x – 4y = -10 and 5x – 3y < 3, then which of the following must be true?
- y < 28 / 13
- y < 53 / 17
- y < 4
- y > -4
- y > -28 / 13
Solution:
The answer choices indicate that we need to get an answer for y, so we will first solve the equality to get x in terms of y.
2x – 4y = -10
2x = 4y – 10
x = 2y – 5
We now substitute 2y – 5 for x into the inequality and solve for y:
5(2y – 5) – 3y < 3
10y – 25 – 3y < 3
7y < 28
y < 4
Answer: C
GMAT Problem Solving Topic 3: Rates – Converging Rate Questions
As you study rates on the GMAT, you will discover that there are many ways in which rate questions may be presented. Thus, you’ll want to become familiar with each type and know the associated formula for each one. If you can apply the appropriate GMAT math strategies to rate questions, you’ll be in a great place come test day.
We do not have the time to cover each type of rate question in this article, but we will focus on a common type, the converging rate question.
A converging rate is when two people or things head toward each other on a parallel path. An important characteristic of converging rates is that when two objects converge (or meet), the total distance that originated between them is equal to the sum of the individual distances of the two objects. Thus, we use the following formula:
Distance of Object 1 + Distance of Object 2 = Total Distance Traveled
TTP PRO TIP:
When two objects meet, the sum of their individual distances is equal to the total distance they traveled from their respective starting points.
Let’s practice how we would use this formula with an example.
GMAT Problem Solving Example 3:
The distance between Philadelphia and Boston by train is 311 miles. Train A departs Philadelphia at 12:00 PM, traveling to Boston at a constant speed of 50 miles per hour. Train B departs Boston at 12:30 PM and heads toward Philadelphia on a parallel track at a constant speed of 60 miles per hour. How far has Train A traveled at the moment the trains meet?
- 140
- 150
- 155
- 156
- 171
Solution:
We see that this is a converging rate question, as two trains are traveling toward each other (on parallel tracks!).
Let’s let r1 = Train A’s rate, t1 = Train A’s time, and d1 = Train A’s distance traveled.
Similarly, we will let r2 = Train B’s rate, t2 = Train B’s time, and d2 = Train B’s distance traveled.
The individual distance traveled by Train A will be:
r1 x t1 = d1 (Equation 1)
The individual distance traveled by Train B will be:
r2 x t2 = d2 (Equation 2)
Because the distance between the two cities is 311 miles, we can say that: the sum of the individual distances is equal to the total distance:
d1 + d2 = 311
We can substitute Equation 1 and Equation 2 into Equation 3, as follows:
r1 x t1 + r2 x t2 = 311
Substituting the known information for the rates of the two trains, we have:
50 x t1 + 60 x t2 = 311 (Equation 4)
We have two variables and only one equation, so we need additional information about the relationship between the two times. Because Train B left half an hour after Train A, its travel time is half an hour less than Train A’s. Thus, we see that t2 = t1 – 0.5, and we substitute this into Equation 4 and solve:
50 x t1 + 60 x (t1 – 0.5) = 311
50 x t1 + 60 x t1 – 30 = 311
110 x t1 = 341
t1 = 3.1
Since Train A traveled for 3.1 hours, we substitute this value into Equation 1:
50 x 3.1 = d1
155 = d1
Train A traveled 155 miles.
Answer: C
Next, let’s discuss how to find the median of a large set of data.
GMAT Problem Solving Topic 4: Statistics – Finding the Median of a Large Set of Data
If you have ever studied how to determine the median of a set of data, you may recall that it’s a pretty simple process when you have a small set of data, as you can manually calculate it pretty easily. However, what do you do when you have a large set of data? Don’t worry; there’s an excellent way to determine the median, even in a large set!
To determine the place where the median falls in a set of data in ascending or descending order, we use the following formula, where n represents the total number of values in the set:
position of the median = (n + 1) / 2
Keep in mind that this formula works when the number of data points is odd and when it’s even, but in slightly different ways. Let’s do two quick examples that illustrate the difference.
Median Example With an Odd Number of Numbers
What is the median of -4, -3, 0, 1, 4, 6, 10, 11, 15?
Since there are nine numbers in the ordered set, we can determine the position of the median as follows:
position of the median = (n + 1) / 2
position of the median = (9 + 1) / 2 = 5
So, the median is the 5th number in the set when counting from lowest to highest. Thus, the median is 4. Now let’s look at a set with an even number of numbers.
Median Example With an Even Number of Numbers
What is the median of -4, -3, 0, 1, 2, 6, 10, 11, 15, 19?
Since there are ten numbers in the set, we can determine the position of the median as follows:
position of the median = (n + 1) / 2
position of the median = (10 + 1) / 2 = 5.5
Since the median cannot be in the “5.5 position” of the set, we calculate the average of the number in the fifth and sixth positions. The number in the fifth position is 2, and the number in the sixth position is 6. The average of those two numbers is 8/2 = 4. So, the median is 4.
TTP PRO TIP:
The position of the median of an ordered data set is found by using the formula: position of median = (n + 1) / 2, where n is the number of values in the set.
Let’s practice with one more example.
GMAT Problem Solving Example 4:
At a candy shop, there are sixteen candies costing $1 each, twenty candies costing $2 each, and forty candies costing $3 each. What is the median cost of the candies?
- 2
- 2.5
- 2.75
- 3
- 3.50
Solution:
Let’s calculate the position of the median for the 76 candies:
Position of median = (n + 1) / 2 = (76 + 1) / 2 = 77 / 2 = 38.5
We know that the median is the average of the 38th and 39th data values.
We don’t have the 77 data values listed individually, but we know that we are looking for the 38th and 39th data values. We see that the first 16 values are all $1, and the next 20 values (the 17th through 36th values) are all $2. The next 20 values (the 37th through the 66th values) are all $3.
Thus, we see that both the 38th and the 39th data values are each $3. Thus, the median is $3.
Answer: D
Next, let’s discuss one final GMAT Problem Solving topic: three-part ratios.
GMAT Problem Solving Topic 5: Ratios – Three-Part Ratios
Three-part ratios are one of the more challenging topics tested in ratios. Three-part ratio problems generally present two two-part ratios with a shared term represented by different numbers in each ratio.
For example, we may be given the following two ratios concerning the number of cartons of white milk, chocolate milk, and strawberry milk in a New York deli.
White to Chocolate = 3 to 2
White to Strawberry = 5 to 4
In the first ratio, the number of cartons of white milk is represented by 3. In the second ratio, the number of cartons of white milk is represented by 5. However, we need the number of cartons of white milk to be represented by the same number in both ratios before we can combine the two ratios into a single three-part ratio.
The LCM of 3 and 5 is 15. Thus, we’ll be able to combine the ratios if we create two equivalent ratios such that both have the number of cartons of white milk represented by 15:
White to Chocolate = 3 to 2 = 3 × 5 to 2 × 5 = 15 to 10
White to Strawberry = 5 to 4 = 5 × 3 to 4 × 3 = 15 to 12
Now that both equivalent ratios have the same number, 15, representing the number of cartons of white milk, we can create the following three-part ratio:
White : Chocolate : Strawberry = 15 : 10 : 12
TTP PRO TIP:
To convert two two-part ratios to one three-part ratio, use the LCM of the common item shared by both ratios.
Let’s now try a Problem Solving example dealing with this concept.
GMAT Problem Solving Example 5
At a farm, the ratio of horses to ponies is 10 : 7, and the ratio of goats to ponies is 3 : 2. If there are 60 horses at the farm, how many goats are there?
- 20
- 28
- 35
- 42
- 63
Solution:
The common connection in the two ratios is ponies, so we need to find the LCM of the two “ponies” numbers, which are 7 and 2. Thus, the LCM is 7 x 2 = 14.
We can now convert the first ratio of horses to ponies, 10 : 7, to one in which the number of ponies in the ratio is 14, by multiplying by 2:
Horses to ponies = 10 to 7 = (10 x 2) to (7 x 2) = 20 to 14
Similarly, to convert the goats : ponies ratio, currently 3 : 2, such that the number of ponies in the ratio is 14, we multiply the ratio by 7.
Goats to ponies = 3 : 2 = (3 x 7) to (2 x 7) = 21 to 14
The three-part ratio can now be stated as horses : ponies : goats = 20 : 14 : 21.
There are 60 horses at the farm. If we multiply the three-part ratio by 3, we obtain the equivalent ratio of horses : ponies : goats as 60 : 42 : 63. Thus, there are 63 goats at the farm.
Answer: E
In Summary
Whether you are registered for the GMAT or the GMAT Focus Edition, Problem Solving questions will constitute a large part of the quantitative portion of your exam. The more exposure you have to the various topics tested by PS questions, the better prepared you’ll be on test day.
In this article, we have focused on 5 examples of PS questions you might encounter on the GMAT or the GMAT Focus. They have come from the major topics of Number Properties, Inequalities, Rates, Statistics, and Ratios. They represent only a small proportion of the topics and subtopics that you need to master in order to get a great score on the GMAT.
What’s Next?
The PS questions we covered in this article represent only a small proportion of the topics and subtopics that you need to master in order to get a great score on the GMAT. Check out our article that introduces additional GMAT PS math questions for more practice and expert tips!
