Test your GMAT skills with this GMAT problem solving practice question!

The general equation for a line is: ax + by + c = 0 for all values (x,y) that are on that line. In this form, the slope is –a/b and the y-intercept is –c/b.

The most common equation for a line is called slope-intercept form: y = mx + b, for all values (x, y) on the line. Here m is the slope and b is the y-intercept.

A modified version of slope-intercept form is called point-slope form: y – y₁ = m (x – x₁) + b. This equation is helpful if you are given two points on the line, (x, y) and (x₁, y₁).

For most GMAT questions involving lines, you can use any of these three equations to represent the line. You may find that one is easier depending on the information you are given and what the question is asking, so it’s helpful to practice manipulating all three. Let’s look at an example GMAT problem from Grockit:

Does the line y = ax + b pass through the point (2,5)?

(1) When it is reflected around the x-axis, the line passes through the point (1,-6).

(2) When it is reflected around the y-axis, the line passes through the point (-3,4).

Notice how y = ax + b looks almost exactly like the slope-intercept form, except here a = slope. To know whether (2,5) is a possible (x,y) for this equation, we would need to know the slope AND the y-intercept.

To find the slope of a line two known points are required. Slope = (y₂ – y) / (x₂ – x). The y-intercept is the point at which the line crosses the y-axis. As a pair, the y-intercept can always be expressed as (0, b) since the x-coordinate when a line crosses the y-axis is always zero. If we know the slope AND one pair of coordinates, it is possible to solve for the y-intercept. To answer this yes/no Data Sufficiency question, we need to know two points.

(1) If (1,-6) is on the reflection around the x-axis, then the point (1,6) is on the original line. However, we would not be able to find the slope or the y-intercept with only one coordinate pair.

(2) If (-3,4) is on the reflection around the y-axis, then the point (3,4) is on the original line. This is not sufficient for the same reason as the first statement.

Combining the statements, we see that we have two points on our line. This will be sufficient. Two distinct points are always enough to find the equation of the line. Keep in mind that for DS, we don’t have to solve, but just to make sure:

y = (6-4)/(1-3)x + b
y =( -2/2)x + b
y = -x + b

We can then plug in either of the points to find the value of b.
4 = -(3) + b
4 = -3 + b
7 = b

The equation of the line is y = -x + 7

(5) = -(2) +7
5 = 5

The answer is that the line does pass through the point (2,5).

You can check out more articles on Coordinate Geometry on the Grockit blog or in the Beat the GMAT Library!

Check out this GMAT graph question for more GMAT practice!

1. Analyze the data first. Don’t jump to the question and then go back to the data to look for an answer. Read all of the labels on the presented graph or table. What is in each column? What is in each row? What is the range of values? Does the data have a direct or indirect relationship? Do the lines have positive or negative slopes? Where was there the most change or growth? Where was there the least?  With this kind of discipline, you will already understand what is being presented. The question becomes almost an afterthought, and you’ll know exactly where to look for the correct information.

2. Ask yourself: what data do I need to solve? Graphs and tables often give us extraneous information. You will not need everything to solve. Chances are the correct answer hinges on just 1 or  2 numbers from the data. Most people get Data Interpretation questions wrong because they do not know what data to use to solve, not because the Math involved is difficult. You will likely only be performing calculations as simple and easy as addition, subtraction, multiplication and division.

3. Approximate when possible. Let the answer choices be your guide. If they are very close together, then you will need to be more accurate in your approximation. However, if the answer choices are markedly far apart, then by all means round to the nearest integer.

Let’s apply these test day tips to a Grockit question:

Analyze: When we analyze this chart, we can see that it represents figures for 5 books. In general, all of the books sold more in paperback then they did in hardcover.

In the table to the left, the amount of hardcover and paperback copies sold in a given year for novels L, M, N, P, and Q is provided. For which novel is the ratio of hardcover copies sold to the paperback copies sold the greatest for that year?

What data do I need?: We’ll need to divide the number in the 1^st columns by the number in the 2^nd columns to express the “ratio.”

Approximate: Round each number to make the comparison easier. L is 800,000/1,300,000, or 8/13. Following the same logic of approximating and estimation, M is 3/6. N is 8/10. P is 3/7. Q is 10/25.

The question asks for the greatest ratio. Notice how 3/6, 3/7 and 10/25 are all around 50% or less. That leaves only 8/13 and 8/10. When the numerator stays the same and the denominator shrinks, we know we are getting a larger fraction of the total. Therefore, 8/10 is greater, so the answer is Title N.

For more practice following these 3 steps for Data Interpretation, create a custom game in the Grockit lobby using only the skill tag “Interpretation of Graphs and Tables.”

1. A firm increases its revenues by 10% between 2008 and 2009. The firm’s costs increase by 8% during this same time. What is the firm’s percent increase in profits over this period, if profits are defined as revenues minus costs?(1) The firm’s initial profit is $200,000.(2) The firm’s initial revenues are 1.5 times its initial costs.

In this question from Grockit, we can start with our most basic Profit formula:

Profit = Revenue – Cost

Try this GMAT problem solving question for more GMAT practice!

Using Statement (1), we can say that 200,000 = R – C.

(1.1)r – (1.08)c = 200,000(1 + x), where x equals the amount of the increase. We still do not know R and C so we can’t find x. Insufficient.

Using Statement (2), 1.5c – c = p and (1.1)(1.5)c – (1.08)c = (1 + x)P. Here we can simplify.

.5c = p

.57c = (1 + x)p

Without continuing to solve, we can see that we can solve for x using substitution. .57c = (1 + x)(.5c), and dividing both sides by c will cancel out that variable and allow us to isolate x. Statement 2 is sufficient. Now to a more challenging question!

1. A store purchased 20 coats that each cost an equal amount and then sold each of the 20 coats at an equal price. What was the stores gross profit on the 20 coats?

Statement 1. If the selling price per coat had been twice as much, the store’s gross profit on the 20 coats would have been $2400.
Statement 2. If the selling price per coat had been $2 more, the store’s gross profit on the 20 coats would have been $440.

This GMAT Prep question asks about gross profit.

Gross Profit = Selling price – Cost

For the value Data Sufficiency question, we need to know the price of each coat and the selling price of each coat. From the given information, we can use our known formula to set us the equation: P = 20 (s – c). So either we’ll need a value for s and a value for c, or we’ll need the value of (s – c).

Statement (1) tells us that $2400 = (20(2s – c)) or 2400 = 40s – 20c. We can divide both sides by 20 and simplify it to: 120 = 2s – c.  We still don’t know s and c. Insufficient.

Statement (2) tells us that 440 = 20(s + 2 – c). Let’s simplify: 440 = 20s + 40 – 20c.  400 = 20s – 20c.  400 = 20 (s – c). 20 = s – c. Sufficient. Even though we didn’t solve for s and c separately, we were able to find the value of (s – c). Sometimes DS will surprise you!

1. If the cost price of 20 articles is equal to the selling price of 25 articles, what is the % profit or loss made by the merchant?A. 25% loss
   B. 25% profit
   C. 20% loss
   D. 20% profit
   E. 5% profit

Profit/Loss % = (Sales Price – Cost Price) / Cost Price x 100

The question asks about % profit or loss. It tells us that 20c = 25s, or 4c = 5s. So the ratio of the sales price to the cost price is 4/5.

Let’s simplify our Profit/Loss % formula by dividing each term by the cost price:

Profit/Loss % = (S/C – C/C) x 100

P/L% = (S/C – 1) x 100

We know that S/C = 4/5 for this problem. So we can plug in and solve:

P/L% = (4/5 – 1) x 100

P/L% = (-1/5) x 100

P/L% = -20%.   The answer is a 20% loss.

Find out how you can collaborate with your peers to reinforce your knowledge and theirs on Grockit today.

PEMDAS is an acronym for the order of operations, which stands for Parentheses, Exponents, Multiplication, Division, Addition, and Subtraction. Always start with what is inside the parentheses then address any exponents to simplify an expression. Next, move left to right, doing all division and multiplication. Finally, again moving from left to right, do any addition or subtraction.  PEMDAS is important to memorize as it will tell you how to simplify expressions correctly.

Try this GMAT problem solving question for more GMAT practice today!

For example, let’s look at this problem: 4 + 6 / 2 = ?

A GMAT student who didn’t know about PEMDAS, might try to solve from left to right, first adding, then dividing. That would give us a solution of 5. However, we know that division must come before addition! The correct answer is 7.

What about 7 + (2 × 4² + 1)? Using PEMDAS, first we will focus on the exponent.

7 + (2 x 16 + 1)

Multiplication comes before addition:

7 + (32 + 1)

And what is inside parentheses comes before what is outside:

7 + (33)

Then we add the final integers to solve:

40 is the answer!

Let’s look at a more challenging expression: –(x – (1 – (4 – 3x)) + 6x). With so many parentheses, where do we start? Begin with the inner-most, and work your way out!

–(x – (1 – (4 – 3x)) + 6x)

-(x –(1 – 4  + 3x) + 6x)

- (x – (1 + 3x) + 6x)

- (x – 1 – 3x + 6x)

- (4x – 1)

- 4x + 1 is the correct simplification. Let’s look at an example GMAT function.

If f(x) = 3x + 4, what does f(6) = ?

Here, we plug what is inside the parenthesis for x.

f(x) = 3x + 4
f(x) = 3(6) + 4
f(x) = 18 + 4
f(x) = 22

Notice how we multiplied the 3 and the 6 BEFORE we added the 4. Another way the GMAT could present this problem would be to ask:

If f(x) = 3x + 4, what is x when f(x) = 16?

Here, we would plug in 16 for f(x).

f(x) = 3x + 4
16 = 3x + 4
12 = 3x
4 = x

The GMAT can make Functions questions harder by having more than one equation and requiring multiple steps to solve. Let’s see an example.

g(x) = x + 4

w(x) = 5x – 8

What is w(g(2)) = ?

Just like any algebra problem, we start with the innermost parentheses. We will first solve for g(2) by plugging the 2 into the “g” function.

g(x) = x + 4
g(x) = 2 + 4
g(x) = 6

Now we have, w(6) = ? So we will plug the 6 into the “w” function.

w(x) = 5x – 8
w(x) = 5(6) – 8
w(x) = 30 – 8
w(x) = 22

The answer is 22.

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Check out your data sufficiency skills on this question! Good luck!

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Distance = rate x time

Rate = distance / time

Time = distance / rate

Let’s check out an example to start learning the strategies for distance problems.

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1. How much longer, in seconds, does it take to drive 1 mile at 50 miles per hour than at 70 miles per hour?

First, let’s figure out how long it takes to drive 1 mile at 50 mph:

T= 1/50   hour →  1/50  hour  x 60 minutes  = 1.2 minutes

Now, do the same thing for 1 mile at 70 mph:

T= 1/70  hour  → 1/70  hour x 60  minutes  = (approximately) .857 minutes

So the difference in minutes is 1.2 – .857 = .343 minutes  →

.343 minutes x 60 seconds = 20.58 seconds

Note: Many distance problems require you to manipulate units of measurement as in the above problem. This step is often forgotten by students.

2. John drove home to Jake’s house at 60 miles per hour. Returning over the same route, the traffic caused him to slow down to 40 miles per hour. If the return trip took 1 hour longer, how many miles did he drive each way?

This kind of problem requires some heavy algebra. The best thing to do in this situation is to organize all your information and see where you can form equations. To begin, let x = the number of hours John took going to Jake’s house. Let’s organize our information in a table.

The important thing to realize here is that John drove the same distance to and from Jake’s house. That will give us the equation:

60x = 40(x+1) → 60x=40x+40 → 20x = 40 → x= 2

Once you’ve solved for a variable, ask yourself what that variable represents. Remember, x represents the number of hours John took going to Jake’s. The question, though, is asking how many miles did he drive. Just plug in 2 for x in one of the distances (say, 60x), and you have 60 * 2 = 120 miles.

Finally, remember that distance problems are not just “distance” problems—they are “rate” problems. The ‘distance’ could stand for any type of work that is performed at a given rate. The above problem, for example, could have been about John’s painting output, where John painted 120 square feet at a rate of 60 square feet an hour. As long as a rate and an output is involved, considered it a ‘distance problem.’

Try this GMAT problem solving question and test your GMAT skills today!

Try this GMAT geometry practice question and test your skills today!

Parallel lines can be tested in either Problem Solving or Data Sufficiency and though the concept itself is relatively simple, the test questions presented can appear somewhat complex. Don’t make assumptions about lines that look parallel but may in fact not be. Let’s look at an example of how the “eyes can deceive” from GMATPrep:

This question tells us that x < 90 and y < 90 and that PS and QR are parallel. We need to be able to answer whether or not PQ < SR. We cannot assume that PQ and SR are parallel simply because they look parallel, or that x = y. Statement (1) tell us that x > y. Since x is greater than y, then PQ will be slightly shorter (more vertical) than SR. Thus, (1) is sufficient. Statement (2) tells us that the sum of the angles is greater than 90, but does not tell us the value of each angle. It is insufficient.  Let’s try a question from Grockit:

If the two horizontal lines in the figure provided are parallel, what is (a + f + d) – (e + h)?

1)  a + f = 180

2)  a – f = 60

This question solely relies on our knowledge of angle measures and parallel lines. Angles a and b are complementary (add up to 180, a straight line), and angles b and f are congruent, so we already know that a + f = 180 without Statement (1) telling us. (a + f + d) – (e + h) = a + f + d – e – h = 180 – h. Since we do not know the exact value of h (the difference between the obtuse and acute angles), Statement (1) is not sufficient. Statement (2) gives us the difference between the obtuse and acute angle, and so it is sufficient.

Follow me on Beat the GMAT to read more of my posts, or message me via Grockit for more help with Parallel Lines questions!

Let z € y be defined by the equation z € y = z2 – 1/y. What is the value of 3 € 1?

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Here we plug in 3 for z and 1 for y to solve.

3² – 1/1

9 – 1

8

2. Replace abstracts with numbers. For questions with functions in the answer choices, use the given definitions to substitute in for x.

For which of the following functions f is f(x) = f(1 – x) for all x?

A) f(x) = 1 – x
B) f(x) = 1 – x2
C) f(x) = x2 – (1 – x)²
D) f(x) = x2(1 – x)²
E) f(x) = x / 1-x

For instance, with here f(x) = f(1-x), one can easily and quickly notice that f(4)=f(-3).

If you notice that, then it very easy to find the solution, replace each function with 4 and -3 instead of x, and see if f(4)=f(-3).

A. F(-3) = 1 – (-3) = 4

F(4) = 1  – (4) = -3

They are NOT equal. Eliminate. Continue this method with the other choices, until you find the function for which the value you get when you plug in 4 is the same as when you plug in –3. That choice is D:

D. F(-3)  = (-3)2 (1 – (-3)² = (9)(16)
F(4) = (4)2 (1 – (4))² = (16)(9)
F(-3) = F(4). This is the correct answer.

3. Use the answer choices.  Let the function $x be defined as $x = (x + 3)(x – 3).

If $a = a + 3, what is one possible value for a?

A  9
B  6
C  4
D  3
E  0

Plug in the answer choices into the function to see which one will yield a + 3. Since the answer choices are listed numerically, let’s start with answer choice C. If our answer is too large, we will be able to eliminate A and B as well. If it’s too small, we’ll eliminate D and E.

C  $4 = (4 + 3)(4 – 3)
$4 = (7)(1) = 7

Need more GMAT practice? Try this GMAT quantitative practice problem. 

Based on the figure, is this is the correct graph for the equation:  y = ^{(x3 + x)}/_{x ?}

For a question like this, don’t get nervous if the equation given does not match the standard equation for a parabola. This equation simplifies to y = x² + 1. We can see from the graph that the y-intercept is indeed 1. When we plug in the x-coordinates of the other two given points, we know this must be the correct equation. When in doubt, plug in!

The x-intercepts are also called the “roots” or “solutions” of a parabola. On the GMAT, parabolas will often be referred to as “functions” interchangeably. The x-intercepts can be found using the quadratic formula:

To find the number of x-intercepts a given parabola has, calculate what is called the discriminant:  or the information underneath the radical in the quadratic formula.

If the discriminant is positive, the parabola has two intercepts with x-axis; if it is negative there are no intercepts with the x-axis, and if the discriminant is equal to zero there is one intercept with the x-axis.

The vertex represents the maximum (or minimum) value of the function. Think of it as the starting point of the function.

The vertex of the parabola is located at point  for the standard equation. If you are given the standard equation, you can find the vertex and the x-intercepts.

The standard equation of a circle is (x – h)² + (y – k)² = r² where (h, k) is the center point of the circle and r is the radius. For example, on test day you might see a circle plotted on a graph like so:

The question will ask you to find the equation of the circle. All we have to find is the center point (0, 4) and the radius (4), and then plug it into our equation.

(x – h)² + (y – k)² = r²

(x – 0)² + (y – 4)² = 4²

(x)² + (y – 4)² = 16

Need more Coordinate Geometry help? Schedule a 1-hour lesson with one of Grockit’s GMAT tutors to focus only on coordinate geometry questions from the Grockit GMAT question bank, or check out the Coordinate Geometry questions in Grockit’s Academy, SAT, ACT, or GRE question banks for even more practice!

QUESTION #3: Tracey ran to the top of a steep hill at an average pace of 6 miles per hour. She took the exact same trail back down. To her relief, the descent was much faster; her average speed rose to 14 miles per hour. If the entire run took Tracey exactly one hour to complete and she did not make any stops, how many miles is the trail one way?

ANSWER: For the way up the hill, we know that D = 6mph x T.

For the way down the hill, we know that D = 14mph x T.  Since we went know that the distance up the hill was the same as the distance down the hill, we can pick a number for D. Let’s choose “84″ since it is a multiple of both 6 and 14.  If 84 = 6mph x T, then we know that T = 14 hours. If 84 = 14mph x T, then we know that T  = 6 hours.

Now we can use another formula, the Average Rate formula, to find the average speed for the WHOLE trip. Average Rate = Total Distance / Total Time

Using our Picked Number of 84, we know that the Total Distance traveled would be 168 miles. The Total Time is 14 hours + 6 hours = 20 hours.  So the Average Rate = 168 miles / 20 hours = 8.4 mph.

It doesn’t matter that Tracey didn’t “really” go 168 miles, or that we know she didn’t “really” go 20 hours. We Picked a Number just so that we could find the ratio of the Total Distance to the Total Time in order to calculate the Average Rate of the ENTIRE journey.

Now that we have found the Average Rate for the whole trip, we can plug it in to the “DIRT” formula to find the ACTUAL distance for the entire journey.

D = R x T

D = 8.4mph x 1 hour

We know that T = 1 hour because the problem told us so. Therefore, the actual distance for the entire trip was 8.4 miles. The problem asks how many miles the trail was one way. 8.4 / 2 = 4.2. The answer to the question is 4.2 miles.

You could also solve this problem in other ways, including using a system of equations and substitution, but it’s nice to know that you can pick a number for the Distance traveled and use it to find the Average Rate for the whole journey! Be on the lookout for those trips where the distance there and back is the same!

We’ve joked about it, but let’s actually try one of those “train”  questions:

QUESTION #4: A train runs over a straight route from town A to town D. It is scheduled to depart town A at 7am and arrive at town D at 2pm, with 10 minute stops in towns B and C. The train’s top speed is 60mph. The entire length of the route is 320 miles. Will the train arrive on time?

(1) The train experiences a 30 minute delay in town B in addition to its scheduled stop.

(2) The train travels at its top speed for exactly 75% of the trip.

ANSWER: This is a yes/no data sufficiency. In order to answer yes or no, we need to know the time it takes the train to make its journey. From the question, we can see that the train is supposed to take 7 hours to go from Town A to Town D with two 10-minute stops. Thus, the total travel-time of the train is 7 hours – 20 minutes = 6 hours, 40 minutes, or 6.67 hours. A train that takes the full 6.67 hours to travel 320 miles would need to travel at a speed of approx 48mph or faster to make its timetable. Remember that this is what the train is supposed to do. Let’s see how each statement affects the time-table.

With a 30 minute delay, the train’s travel-time is now 6 hours, 10 minutes, or 6.167 hours. With that delay, the train needs to travel at approx 51mph to make its timetable. However, what is missing from this statement is proof that the train actually did increase its speed to make its timetable. Just because it was possible the train arrived on time, doesn’t mean it did. Statement (1) is insufficient.

Let’s look at Statement (2). 75% of the trip is 240 miles out of the total 320 miles. This means the train has to travel the remaining 80 miles in a little less than 3 hours, an average speed of approximately 25 miles an hour in order to make the timetable. This is clearly within the train’s limits, but we do not know anything about the train’s speed for these remaining 80 miles. Statement (2) is insufficient.

Combining the two statements, we know that the train only has 2 hours and 10 minutes to travel the remaining 80 miles. This requires traveling almost 40 mph. This is possible, but we have no certainty that the train accomplished this. The answer, therefore, is (E).

For more practice with Distance and Rate questions, you can schedule a group study game in the GMAT Lobby on Grockit. Use Grockit’s message system to invite other students with high scores and Grockit GMAT tutors to RSVP. Studying together is more fun than studying alone.