Quantitative Practice Questions - 4 | Mock Test series for GMAT Classic Edition PDF Download

We need both statements to find out how much money Mary made.
Statement 1 gives the type and number of hours, and statement 2 gives the amount she made per hour. Both together are sufficient, but neither is sufficient alone.

−6x − 20 = −2x + 4(1 − 3x) 
−6x − 20 = −2x + 4 − 12x
−6x − 20 = −14x + 4
−6x + 14x = 4 + 20
8x = 24
x = 3

Solve using the quadratic formula:

For the first 5 games the bowler has averaged 215. The equation to calculate the answer is

where x is the score for the sixth game. Next, to solve for the score for the 6th game (x) multiply both sides by 6:
(215⋅5) + x = 1,320
which simplifies to:
1,075 + x = 1,320 
After subtracting 1,075 from each side we reach the answer:
x = 1,320 − 1,075 = 245

Let  M be the number of pounds of Mocha Madness coffee beans Mike uses. Then he will use  (40 − M)  pounds of Sumatra Sweetness coffee beans. 
The Mocha Madness coffee costs $12 a pound times  M  pounds, or  $12M  dollars; similarly, the Sumatra Sweetness coffee costs  $20(40 − M) . The total cost of the coffee will be  14⋅40 = $560 . Since the beans will sell for the same price as unmixed, we can add the prices to obtain and solve this equation:
12M + 20(40 − M) = 560 
12M + 800 − 20M = 560 
−8M + 800 = 560 
−8M + 800 − 800 = 560 − 800 
−8M = −240 
−8M ÷ (−8) = −240 ÷ (−8) 
M = 30 
30 pounds of the Mocha Madness will go into the mixture.

We derive the equation of base of a triangle from the area of a triangle formula:

V = πr²h = π(10²)30 = 3000π

For statement (1), she spent
1/3 x 3000 = $1000 
on rent, and
15 x (3000 − 1000) = $400  
on clothes.
So she spent  $1000 + $400 = $1400  in total.
Therefore, she has  $3000 − $1400 = $1600  left.
For statement (2), we can set Clara’s salary to be  x , then we have

Then simplify the equation:

Therefore,  x = $3000 .
Now we can just follow what we did for the first statement to calculate the money he had left.

Suppose we know   a,b ∈ {1,3,5,7,9} , but we do not assume the second statement.
If c = 2 c=2  and d = 4 d=4 , then A ∩ B = {a, b, 2, 4}, a four-element set. If If c = 2 c = 2  and d = 12 d = 12 ,
Therefore, we cannot make a conclusion about the size of A∩B . A similar argument can be used to show that assuming only the second statement also does not allow a conclusion.
If we know both statements, however, A∩B = {a, b, c, d }, and we can prove that A∩B  has four elements.
The answer is that both statements together are sufficient to answer this question, but neither statement alone is sufficient.

The total time is calculated by the equation  Time = distance / rate .
Statement 2 provides the rate, but we have no information regarding distance, therefore, the quesiton is impossible to solve without more information.

Statement 1 gives us the number of tickets sold but not the price. Insufficient.
Statement 2 gives us the price of the tickets but not the number sold. Insufficient.
Together, the two statements give us both the number of tickets sold AND the price of each ticket. From this we can calculate the total revenue.
Note: We are only trying to determine if we have enough information to answer the question. We don't have to actually do the computations!

Using statement 1, we can find that he has 10 green boxes and 10 brown. We solve mixture problems using 2 equations. If we let  A  be the number of green boxes, and  B  be the number of brown boxes. We can set the equations
1)  A + B = 20  and
2)  10A + 30B = 400 
Solving equation 1) for  A , we get  A = 20 − B .
Subsituting for  A  into equation 2) results in
10(20 − B) + 30B = 400 
Simplifying we get  200 + 20B = 400  or  B = 10 
Therefore, this answer would be 10 green boxes and 10 brown.
The second statement is irrelevant in finding our answer. 

Using statement 1, it is easy to see how much the total retail amount should have been. The total retail amount discounted by 25% is the amount that Susan spent. So, we name a variable. Let x be the total retail amount. Then x - .25x = $122. We can rewrite this as x(1-.25) or x(.75) = 150 thus, x = 200. So we subtract the known retail prices of the skirt and the purse to get the retail price of the jacket. So 200 - 100 - 40 = $60 is the retail price of the jacket.
Now, we should check statement 2. Using statement 2, we can calculate the savings we had on the jacket. We can first calculate how much we saved on the skirt. So 30% of the $40 retail price is $12. After we find this, we use the information from statement 2 to find the savings we had on the jacket. So $12 + $6 = $18 saved on the jacket. 
Now, we need to figure out how much we spent on the jacket. We do this by taking the total amount we spent and subtracting the discounted price of the purse and the discounted price of the skirt. So, a $100 purse, at 20% off, is $80. We calculated our savings on the skirt earlier, so we know we spent $28 on the skirt. So $150 - $80 - $28 = $42.
Now combining these two pieces of information, we see we spent $42 and saved $18 so 42+18 = $60 retail price for the jacket. 

We can see that either statement alone is sufficient to solve the problem.

We need both statements to get the answer.
Statement 1 alone is not enough. Having 24 total cookies does not tell us the proportions of each type of cookie. We could have 1 chocolate chip cookie and 23 oatmeal raisin, or vice versa. 
Statement 2 alone is not sufficient either. Knowing that there are 8 more chocolate chip cookies than oatmeal raisin without knowing the total amount in the bag does not help. It could be 1 oatmeal raisin cookie and 9 chocolate chip, or it could be 100 oatmeal raisin cookies and 108 chocolate chip cookies (although that is a pretty big bag!)
Only by having both statements together can we find the right proportion. If we let  A  be the number of chocolate chip cookies, and  B  be the number of oatmeal raisin, we can set two equations and find the right ratio. We know that
A + B = 24 
from statement 1, and
A = B + 8  
from statement 2.
Substituting  A  into the first equation we get
(B + 8) + B = 24 
2B = 16 
B = 8 
Therefore, we get 8 oatmeal raisin cookies, and 16 chocolate chip cookies. The answer is
16 / 24 = 2 / 3  
or 66.67% chance that the monster pulls out a chocolate chip cookie. 

From Statement 1 alone, you know that Patty spent  $3.09 × 3 = $9.27 . It is possible, however, that Mickey spent more (for three espressos, for example), just as much (on three Turkish coffees), or less (for three cappucinos, for example). Statement 2 alone gives you even more scant information, since it only tells you one beverage Mickey bought.
Suppose you know both statements. Then you know Patty spent  $9.27 . You do not know how much Mickey spent, but the most he could have spent was  $2.49 + 3.19 + 3.19 = $8.87  (for an Americano and two esperessos). Therefore, the two statements together are necessary and sufficient to answer the question.

(1) This statement gives us the length of one side of the triangle. This information is insufficient to solve for the perimeter.
(2) This statement gives us the length of one side of the triangle. This information is insufficient to solve for the perimeter.
Additionally, since we don't know which one of the sides ( AB  or  BC ) is one of the equal sides, it's impossible to determine the perimeter given the information provided.

The side opposite the angle of greatest measure is the longest of the three, so if we can determine which angle is the longest, we can answer this question.
It follows from Statement 1 by definition that  m∠A + m∠B = 90^∘ , so, since the measures of the three angles total  180∘ ,  m∠C = 90^∘ , making  ∠C  right and the other two acute. This proves  ∠C  has the greatest measure of the three.
It follows from Statement 2 that  ∠C  is either right or obtuse; therefore,  m∠C ≥ 90^∘ .
Subsequently, the other two angles are acute, so again,  ∠C  has the greatest measure of the three.
From either statement alone, we can therefore identify as the side of greatest measure.

A right triangle must have its two acute angles complementary; if Statement 1 is assumed, then this is false, the triangle is not a right triangle, and  ∠C  is not a right angle.
If Statement 2 is assumed, then we apply the converse of the Pythagorean Theorem to show that the triangle is not right. The sides of a triangle have the relationship 
c² = a² + b²  
only in a right triangle. If  c = 260 , then the statement to be tested would be 
260² = 70² + 240² 
67,600 = 4,900 + 57,600 
67,600 = 62,500 
This statement is false, so the triangle is not a right triangle, and  ∠C  is not a right angle. 

For a triangle to be isosceles, two of the sides must be equal.  To determine wheter this is true, we must have the three side lengths.  Statement 2 gives us those three side lengths.  However, Statement 1 also gives us all of the information we need by giving us the three vertices.  By using the distance formula, we can easily get the three triangle sides from the vertices.  Therefore both statements alone are sufficient.

Statement (1) informs us that the triangle is an equilateral triangle, with all sides equal to 6. Therefore, the height would divide the triangle into two 30-60-90 triangles, which have side lengths in a ratio of  1: √3 :2 . For this triangle the hypotenuse would be 6 and the base would be 3. The height would therefore be  3√3 . SUFFICIENT
Statement (2) also lets you deduce that the triangle is an equilateral triangle. Since the triangle is either isoscles or equilateral (at least 2 sides are equal), that means that two angles are equal. Therefore, if one angle is   60∘ , the other two also must be   60∘ . See above. SUFFICIENT

This can be illustrated by showing this triangle inscribed inside a rectangle whose vertices are  (0, 0), (5, 0), (0, 6), (5, 6) :

The area of the white triangle  AW  is the one whose area we calculate. To do this, we need the area of the square:

A = 5  x 6 = 30
The area of the red triangle:
A₁ = 1/2 x 5 x 2 = 5 
The area of the green triangle:
A₂ = 1/2 x 6 x 2 = 6 
And the area of the beige triangle:
A₃ = 1/2 x 4 x 3 = 6 
The area of the white triangle will be as follows:

Statement 1 alone gives a proportion between two pairs of corresponding sides of the triangles. This is not enough to prove the triangles similar without the third side proportionality (SSS Simiilarity statement) or the congruence of the included angles (SAS Similarity statement).

Statement 2 gives two congruencies between corresponding angles, which by the Angle-Angle statement is enough to prove the triangles similar.

Statement 1: An isosceles triangle has two equal angles. Since the interior angles of a triangle always sum to 180^∘, the only possible angles the other sides could have are 35^∘ .
Statement 2: This does not provide any information relevant to the question. 

It's actually easier to solve for the complement first.  Let's solve \left | 2x-5 \right |<3 |2x−5|<3 .  That gives -3 < 2x - 5 < 3.  Add 5 to get 2 < 2x < 8, and divide by 2 to get 1 < x < 4.  To find the real solution then, we take the opposites of the two inequality signs.  Then our answer becomes x ≤ 1 or x ≥ 4.

The equation that will be used is (rate * number of days = number of fields plowed). From the first part of the question, number of fields plowed \dpi{100} \small (x) is calculated as:
rate⋅5 = x 
To solve for rate both sides are divided by 5.
rate = x / 5 
This rate is used for the second part of the problem.  x / 5 x days = y . To solve for days, both sides are divided by  x / 5 , which is the same as multiplying by 5 / x , cancelling out the  x / 5  and giving the answer of  days = 5y / x .

Putting these together,

In this case, we need to solve for the volume of the water tank, so we set the full volume of the water tank as  x . According to the question,  4/7 -full  can be replaced as  4/7x .  13/14 -full  would be  13/14x . Therefore, we can write out the equation as: 

Then we can solve the equation and find the answer is 14 gallons.

7 < 2x − 3 < 15 
3 is added to each side to isolate the x term:
10 < 2x < 18 
Then each side is divided by 2 to find the range of x
5 < x < 9 
The only integers that are between 5 and 9 are 6, 7, and 8.
The answer is 3 integers.

5z + 9 = 10 
5z = 10 − 9
5z = 1
z = 1 / 5

To factor this, we need two numbers that multiply to 4 x  -15 = 60 and sum to 4. The numbers -6 and 10 work.

2y(2y − 3) + 5(2y − 3)
= (2y − 3)(2y + 5)

Think of this in terms of "greenhouses per minute", not "minutes per greenhouse". Converting hours and minutes to just minutes, Steve can paint 1 / 280  greenhouses per minute; Phil can paint 1 / 360  greenhouses per minute. 
If we let  t  be the time in minutes that it takes to paint the greenhouse, then Steve and Phil paint  1/ 280t and 1 / 360t greenhouses, respectively; since one greenhouse total is painted, then we can add the labor and set up this equation:

Simplify and solve:

or 2 hours, 38 minutes.

A  includes all multiples of 2; B includes all multiples of 3. A∪B comprises all multiples of either 2 or 3.
Knowing n is a perfect square is neither necessary nor helpful, as, for example, 9 ∈ B ⊆ A ∪ B, but 25 ∉ A ∪ B (as 25 is neither a multiple of 2 nor a multiple of 3).
If you know that n is a multiple of 99, then it must also be a multiple of any number that divides 99 evenly, one such number is 3. This means n ∈ B ⊆ A ∪ B

The rate formula  RT=D  - or, actually, the equivalent equation  T = D/R  - will help.
Let  x, y, z  be the speed limits along the interstate, the first half of the state route, and the second half of the state route, respectively.
Then Tom can drive I-10 in  55/x  hours; the first half of SR 34,  10/y ; the second half of SR 34,10/z .
The time  t  it takes, in hours, for Tom to make the entire trip is the sum of these fractions: 

To calculate  t , we need  x, y, z . Statement 1 only tells us x ; Statement 2 only tells us y and z . Therefore, both together, but neither alone, are enough to allow us to calculate  t .

Suppose we only know that 200 ml of the 40% solution is used. Then we can call  x  the amount of 25% solution used, and the total amount made is   200 + x . The solution equation becomes, 
200(0.40) + x(0.25) = (200 + x)(0.35) 
80 + 0.25x = 70 + 0.35x 
10 = 0.10x 
x = 10 ÷ 0.10 = 100 
So 100 ml of the 25% solution is used.
Similarly, if we only know that 100 ml of the 25% solution is used, then we can call  y  the amount of 40% solution used, and the total amount made is   100+y  . The solution equation becomes, 
y(0.40) + 100(0.25) = (100 + y)(0.35) 
0.40y + 25 = 35 + 0.35y 

0.05y = 10 
y = 10 ÷ 0.05 
y = 200 
So 200 ml of the 10% solution is used.
Either way, we get that
100 + 200 = 300 
and 300 ml of solution is created.
The answer is that either statement is sufficient to answer the question.

If there are  N  wumps in a zump, then there are  N²  square wump in a square zump, and   N³  cubic wump in a cubic zump. Therefore, given the first statement, you can take the square root of the conversion factor to get the number of wumps in a zump; given the second, you can take the cube root of that conversion factor to the same end. In both cases you get 19.

In order to calculate the percent discount, both the original price and the sale price must be known. From statement 1, we know the sale price and with the additional information from statement 2, we can calculate the original price and then overall percent discount.

Each statement alone gives information about only one order, so at the very least, both statements will be necessary.
From Statement 1, it is known that Ralph will pay  $3.19 × 2 = $6.38  for his drinks, since he will only pay for the two espressos. 
From Statement 2, it cannot be determined exactly what Janet will pay. However, she will pay for all three drinks, two of which are cappucinos; the least she will spend is the price of two cappucinos and an Americano, which is  $2.79 + 2.79 + 2.49 = $8.07 .
From both statements together, it is definite that Janet will pay more.

Knowing that the triangle has a right angle indicates the question can be solved using the Pythagorean Theorem, however it is unclear which side is the hypotenuse. For example, if 8 is not the hypotenuse, the length of the third side is 10. If 8 is the hypotenuse, the length of the third side is 5.3.
Additionally, if you only know that 8 is the longest side, the length of the third side could be anything greater than 2 and less than 8. Therefore, having both pieces of data will allow you to solve the problem.

If Statement 1 is assumed, then by the converse of the Pythagorean Theorem, the triangle is a right triangle with right angle  ∠B , which is explicitly stated in Statement 2. If  ∠B  is a right angle, then the other two angles are acute, since a triangle must have at least two acute angles. A right angle measures  90^∘  and an acute angle measures less, so from either statement, we can deduce that  ∠B  is the angle with greatest measure.