Problem Solving Questions

Question

BATCH #1 (10 QUESTIONS)

1. Absolute values:

1.1 For how many prime numbers x is |x - 1| + |4 - x| < |x + 11|?

(A) 6
(B) 7
(C) 15
(D) 17
(E) Infinite

There are three transition points at -11, 1 and 4 (transition points are points for which expression in modulus is equal to 0). So, four ranges to check:

1. x < -11
2. -11 ≤ x ≤ 1
3. 1 < x ≤ 4
4. x > 4

However, notice that we are only concerned with prime numbers x. Since prime numbers are positive integers and the least prime number is 2, we can ignore the first two ranges (since no prime numbers lie in those ranges) and only check the last two.

3. 1 < x ≤ 4

For this range:
x - 1 is positive, so |x - 1| = x - 1.
4 - x is positive (or 0), so |4 - x| = 4 - x.
x + 11 is positive, so |x + 11| = x + 11.

Thus, |x - 1| + |4 - x| < |x + 11| becomes x - 1 + 4 - x < x + 11, which in turn is x > -8. Since, we consider 1 < x ≤ 4 range, then finally for it we get 1 < x ≤ 4.

The prime numbers in this range are 2 and 3.

4. x > 4

For this range:
x - 1 is positive, so |x - 1| = x - 1.
4 - x is negative, so |4 - x| = -(4 - x).
x + 11 is positive, so |x + 11| = x + 11.

Thus, |x - 1| + |4 - x| < |x + 11| becomes x - 1 -(4 - x) < x + 11, which in turn is x < 16. Since, we consider x > 4 range, then finally for it we get 4 < x < 16.

The prime numbers in this range are 5, 7, 11 and 13.

Thus, the prime numbers that satisfy the inequality are 2, 3, 5, 7, 11 and 13, which is a total of 6 prime numbers.

Answer: A.

Takeaway:
When solving absolute value questions, remember the key property of absolute value:

|x| = x when x ≥ 0, and |x| = -x when x < 0.

This is what allows us to split the number line into ranges and rewrite the inequality correctly in each range.

What this question tests:
This question tests basic absolute value properties and their application in inequalities. In addition, it adds a number properties layer, since after solving the inequality, you must use the restriction that x is prime to identify the valid values.

Bunuel · Answer

1. Absolute values:

1. 2. What is the value of \(|\sqrt{2} - |\sqrt{3}-\sqrt{8}|| - ||\sqrt{2} - \sqrt{3}|-\sqrt{8}|\) ?

A. 2√3 − 2√8 − 2√2
B. 2√3 − 2√8
C. 2√3 − 2√8 + 2√2
D. 2√8 − 2√3
E. 2√8 + 2√3

To answer this question we should recall the property of the absolute value:

\(|x| = x\), when \(x \geq 0\);
\(|x| = -x\), when \(x < 0\).

So, we should evaluate the expressions in the modulus to see whether they are positive or negative.

STEP 1:

Since \(\sqrt{3}-\sqrt{8}<0\), then \(|\sqrt{3}-\sqrt{8}|=-(\sqrt{3}-\sqrt{8})=\sqrt{8}-\sqrt{3}\);
Since \(\sqrt{2} - \sqrt{3}<0\), then \(|\sqrt{2} - \sqrt{3}|=-(\sqrt{2} - \sqrt{3})= \sqrt{3} -\sqrt{2} \).

Thus, \(|\sqrt{2} - |\sqrt{3}-\sqrt{8}|| - ||\sqrt{2} - \sqrt{3}|-\sqrt{8}|\) will become:

\(|\sqrt{2} -(\sqrt{8}-\sqrt{3})| - |(\sqrt{3} -\sqrt{2})-\sqrt{8}|=|\sqrt{2} +\sqrt{3}-\sqrt{8}| - |\sqrt{3} -\sqrt{2}-\sqrt{8}|\).

STEP 2:

\(\sqrt{2} +\sqrt{3}-\sqrt{8}\) must be positive because \(\sqrt{2} +\sqrt{3} > \sqrt{8}\). Indeed, if we square both sides, we get \((\sqrt{2} +\sqrt{3})^2 = 2 + 2\sqrt{6} + 3 = 5 + 2\sqrt{6} > 5 + 4 = 9 > 8\). So, \(|\sqrt{2} +\sqrt{3}-\sqrt{8}| =\sqrt{2} +\sqrt{3}-\sqrt{8}\);

\(\sqrt{3} -\sqrt{2}-\sqrt{8}\) is obviously negative. So, \(|\sqrt{3} -\sqrt{2}-\sqrt{8}|=-(\sqrt{3} -\sqrt{2}-\sqrt{8})=\sqrt{8}+\sqrt{2}-\sqrt{3}\)

Thus, \(|\sqrt{2} +\sqrt{3}-\sqrt{8}| - |\sqrt{3} -\sqrt{2}-\sqrt{8}|\) will become:

\((\sqrt{2} +\sqrt{3}-\sqrt{8})-(\sqrt{8}+\sqrt{2}-\sqrt{3})=2\sqrt{3}-2\sqrt{8}\).

Answer: B.

Takeaway:
When solving expressions with several absolute values, simplify from the innermost modulus outward. At each step, first determine whether the expression inside the modulus is positive or negative, and then remove the absolute value. It is also helpful to know the approximate values of the square roots of positive integers up to 10, since this can be useful in many GMAT Quantitative and DI questions.

What this question tests:
This question tests basic absolute value properties and their application to nested absolute value expressions. In addition, it tests number approximations and number sense through several steps of simplification.

Bunuel · Answer

2. Algebra

2.1. If x ≠ 0 and  f(x) = \frac{1 - x^6}{x^3} , what is  f(-\frac{1}{x})  in terms of  f(x) ?

A.  f(x) 
B.  -f(x) 
C.  \frac{1}{f(x)} 
D.  -\frac{1}{f(x)} 
E.  (f(x))^2

We are given the function  f(x)=\frac{1-x^6}{x^3}  and asked to express  f(-\frac{1}{x})  in terms of  f(x) . To do that, we simply substitute  -\frac{1}{x}  for  x  in the original definition of the function and then simplify the resulting expression.

f(-1/x)=

=\frac{1 - (-\frac{1}{x})^6}{(-\frac{1}{x})^3} =

=\frac{1 - \frac{1}{x^6}}{-\frac{1}{x^3}} =

=(1 - \frac{1}{x^6})(-x^3) =

=-x^3 + \frac{1}{x^3} =

=\frac{1 - x^6}{x^3} =

=f(x)

Answer: A.

Takeaway: 
When a function question replaces x with a new expression, the main step is to plug that new expression into the original formula exactly where x appears and then simplify carefully. Many questions of this type become easy once the substitution is done correctly.

What this question tests: 
This question tests understanding of functional relationships and basic algebraic manipulation. In particular, it tests whether you can correctly substitute a transformed input into a function and simplify the result to match the given function form.

Bunuel · Answer

2. Algebra:  
  
 2.2. If m and n are distinct integer solutions of the equation  (2x + 1)(x - 5) + \sqrt{t^2} = (x + \sqrt{5})(x - \sqrt{5}) , where x is a variable and t is a constant, how many integer values can t take?

A. 4
B. 5
C. 8
D. 9
E. 10

First, let's simplify the equation:

(2x + 1)(x - 5) + \sqrt{t^2} = (x + \sqrt{5})(x - \sqrt{5}) 
 2x^2 - 9x - 5 + |t| = x^2 - 5 
 x^2 - 9x + |t| = 0

Next, to solve this question, apply Vieta’s theorem: for a quadratic equation of the form  ax^2 + bx + c = 0 , if the roots are m and n, then:

m + n = -\frac{b}{a} 
and
 m * n = \frac{c}{a}

So here, since the equation is  x^2 - 9x + |t| = 0 , we get:

m + n = -\frac{-9}{1} = 9 
and
 m * n = \frac{|t|}{1} = |t|

Since mn = |t|, and the absolute value is always positive or 0, then mn ≥ 0. Together with m + n = 9 = positive, this implies that both m and n must be non-negative numbers.

Now list all distinct non-negative integer pairs whose sum is 9:
0 and 9
1 and 8
2 and 7
3 and 6
4 and 5

These give the following values of  |t| :
0 * 9 = 0
1 * 8 = 8
2 * 7 = 14
3 * 6 = 18
4 * 5 = 20

So, the possible values of  |t|  are:
0, 8, 14, 18, 20

Now convert these into values of t.
If  |t| = 0 , then t = 0.
If  |t| = 8 , then t = 8 or t = -8.
If  |t| = 14 , then t = 14 or t = -14.
If  |t| = 18 , then t = 18 or t = -18.
If  |t| = 20 , then t = 20 or t = -20.

So the integer values of t are:
0, 8, -8, 14, -14, 18, -18, 20, -20

That is a total of 9 integer values.

Answer: D.

Takeaway: 
1. Vieta’s theorem is often the fastest way to relate the roots to the coefficients, and more generally to work efficiently with quadratic equations.
2. Manipulating the equation first is often the best way to simplify the problem and identify the next steps more clearly.

What this question tests: 
This question tests basic algebraic manipulation and understanding of quadratic equations. It also shows the importance of case testing in identifying the valid combinations that satisfy the given conditions.

Bunuel · Answer

3. Arithmetic:

3.1. A review of two digital archives led to major updates in their file counts. The estimated number of documents in Archive Nova was raised from 252 * 10^6 to 933 * 10^6. The estimated number of documents in Archive Atlas was also revised upward, and its new estimate is thrice its old estimate. Even after these updates, Archive Nova still has only 32% as many documents as Archive Atlas. Before the revisions, approximately how many times as many documents did Archive Atlas have as Archive Nova?

A. 2
B. 4
C. 6
D. 8
E. 10

This question involves ugly numbers, which are not easy to manipulate directly. In the Problem Solving section, we do not have access to a calculator, so in many questions approximation and estimation is the only practical way to solve the question in an acceptable amount of time.

First, let’s drop 10^6 from both quantities to simplify the calculations.
 
Next, consider using 900 instead of 933, and 30% instead of 32%. This gives us:

900 = 30% * (Archive Atlas’s new estimate)

Notice that the first approximation decreases the new estimate for Archive Atlas, while the second approximation increases it. Therefore, these two approximations partly balance each other out.

Thus, from 900 = 30% * (Archive Atlas’s new estimate) we determine that Archive Atlas’s new estimate is about 3,000.

Given that the new estimate for Archive Atlas is three times its old estimate, its old number of documents was about:

3,000/3 = 1,000.

Consequently, before the revisions, Archive Atlas had approximately 1000/252  ≈  4 times as many documents as Archive Nova.

Answer: B

Takeaway: 
When a question contains ugly numbers and the answer choices are well spread out, estimation is often the fastest and most practical approach. Also, when making more than one approximation, it is useful to check whether the approximations push the result in opposite directions, since they may partly balance each other out.

What this question tests: 
This question tests estimation and approximate calculations as a tool for faster solving. It also tests whether a student can recognize and drop unnecessary units or common factors in order to simplify the work.

Bunuel · Answer

3. Arithmetic

3.2. If  x = (-0.00083)(279^2) + (0.00083)^2(279)  , then  x  satisfies which of the following inequalities?

A.  x < -1,000 
B.  -1,000 < x < -100 
C.  -100 < x < -10 
D.  -10 < x < -1 
E.  -1 < x < 0

This question involves ugly numbers, plus the options are well spread apart, which is the direct sign to approximate:

(-0.00083)(279^2) + (0.00083)^2(279) =  
  \approx (-0.0008)(300^2) + (0.001)^2(300) =  
  =( -\frac{8}{10,000} )(90,000) + (\frac{1}{1,000,000})(300) =  
  =( -72) + (\frac{3}{10,000}) =

(\frac{3}{10,000})  is negligibly small compared to -72, so we can ignore it, and deduce that the whole expression is approximately -72 (just FYI, the actual value is about -65).

Thus,  x  is between  -100  and  -10 .

Answer: C

Takeaway: 
When a question contains ugly decimals and large numbers, and the options are well spread apart, approximation is often the fastest approach. Rewrite the decimal in a simpler form and then round the remaining values to numbers that are easy to work with mentally. Ignore terms whose magnitude is negligible compared with the main term to speed up.

What this question tests: 
This question tests estimation and approximate calculations as a tool for faster solving. It also tests whether you can identify the correct interval without doing unnecessary exact computation, by estimating the scale of the numbers.

Bunuel · Answer

4. Combinations

4.1. At an annual conference, each of five companies has sent two representatives. A four-member committee is to be formed from these ten representatives. What is the number of different committees that can be formed if two people from the same company cannot both serve on the committee?

A. 16
B. 40
C. 80
D. 120
E. 160

Approach 1:

Since the committee must not include both representatives from the same company, only one representative from each chosen company can be selected.

The number of ways to select four companies that will send one representative each to the committee is given by  C^4_5 .

For each of these four companies, there are two choices for the representative. This gives  2 * 2 * 2 * 2 = 2^4  possibilities.

Therefore, the total number of ways to form the committee is  C^4_5 * 2^4 = 5 * 16 = 80 .

Approach 2:

• The first person on the committee can be any one of the 10.
• The second person can then be selected from only 8 people, since the first person and the other representative from that same company can no longer both be included together.
• The third person can then be selected from only 6 people, since the first two selected people and the other representatives from their companies are excluded.
• The fourth person can then be selected from only 4 people, since the first three selected people and the other representatives from their companies are excluded.

So, this gives 10 * 8 * 6 * 4 ordered groups.

However, this number includes ordered arrangements, meaning it includes all possible orderings of the same committee, such as (ABCD), (ABDC), (ADBC), ... which all represent the same four-member committee. Since 4 distinct people can be arranged in 4! ways, we should divide 10 * 8 * 6 * 4 by 4!.

Therefore, the number of combinations is:

\frac{10 * 8 * 6 * 4}{24} = \frac{1920}{24} = 80

To explain more clearly why we divide by 4!, consider a smaller example. Suppose there are two companies, each sending two representatives:  A_1 ,  A_2  and  B_1 ,  B_2 . We want to choose a two-person committee such that both people are not from the same company. The possible committees are:

A_1, B_1 
 A_1, B_2 
 A_2, B_1 
 A_2, B_2

So only 4 such committees are possible.

But if we count in order, the first person can be chosen in 4 ways, and the second person can then be chosen in 2 ways. This gives 4 * 2 = 8. That count is too large because it includes duplicates. For example, choosing  A_1  first and then  B_1  gives the same committee as choosing  B_1  first and then  A_1 . So each committee is counted  2!  times, and we divide by  2! :

\frac{8}{2!} = 4

The same idea is why, in the original question, we divide by  4! .

Answer: C

Takeaway: 
Most GMAT combinations questions are fairly straightforward and can often be solved in more than one way. It is useful to be comfortable with different approaches, because a second method can help you double-check your work if you are not fully confident in the first one. Also, in combinations questions, always pay attention to whether your method is counting ordered groups or unordered groups, and make sure that matches what the question is actually asking for.

What this question tests: 
This question tests basic combinatorics under a restriction. It also tests whether you can recognize two valid counting methods for the same problem: counting in stages, or counting permutations first and then converting to combinations.

Bunuel · Answer

4. Combinations

4.2. A security system creates 5-digit access codes from the digits 0, 1, 2, 4, 5, 7, and 9, without repeating any digit. If a code cannot begin with 0 and must be divisible by 9, how many such codes are possible?

A. 120
B. 181
C. 192
D. 216
E. 240

We start by finding which 5 digits out of the given 7 can form a 5-digit number divisible by 9.

We have seven digits: 0, 1, 2, 4, 5, 7, 9, and their sum is 28.

For a number to be divisible by 9, the sum of its digits must be divisible by 9. So the sum of the 5 chosen digits must be a multiple of 9.

Each code uses 5 of the 7 digits, so 2 digits are left out. Therefore, if the 5 chosen digits must have a sum divisible by 9, then the 2 omitted digits must have a sum of 1, so that the sum of the remaining digits is 27, or a sum of 10, so that the sum of the remaining digits is 18.

Now check which pairs of digits from the set have these sums:

Sum 1:
0 + 1

Sum 10:
1 + 9

So the only 5-digit combinations that yield a number divisible by 9 are:

{2, 4, 5, 7, 9}
and
{0, 2, 4, 5, 7}

No other pair of omitted digits gives a remaining sum divisible by 9.

Next, we have two sets of digits:

{2, 4, 5, 7, 9} and {0, 2, 4, 5, 7}

Let us find how many 5-digit numbers can be formed from each set.

{2, 4, 5, 7, 9}

This set has no 0, so every permutation gives a valid 5-digit number. umber of such codes = 5! = 120

{0, 2, 4, 5, 7}

Here, per the restriction, 0 cannot be the first digit, so the desired number is the total number of permutations, 5!, minus the number of arrangements with 0 fixed in the first position. If 0 is fixed first, the remaining 4 digits can be arranged in 4! ways. Therefore, the number of valid codes is 5! - 4! = 120 - 24 = 96.

In total, the number of valid 5-digit access codes is:

120 + 96 = 216

Answer: D.

Takeaway 
In counting problems like this, breaking the problem into clearly defined cases simplifies the work and helps avoid missed scenarios. First identify which selections are even valid based on the condition, and then count arrangements within each case. Also, when restrictions are present, it is often faster and cleaner to count the total number of arrangements and subtract the invalid ones, rather than trying to build only valid cases directly.

What This Question Tests 
This question tests structured counting, specifically selecting valid combinations of digits and then counting their permutations under a restriction. The divisibility condition adds an extra layer, which makes the problem more nuanced than a standard permutation question.

Bunuel · Answer

5. Distance and Speed Problems

5.1. Lena is walking at a constant speed of v kilometers per hour along a straight trail when Marco passes her on an electric scooter, traveling along the same trail at a constant speed 3 times Lena’s speed. After riding for m minutes, Marco stops at a café on the trail and waits there for Lena. If Marco waits 50 minutes before Lena reaches the café, what is the value of m?

A. 10
B. 15
C. 20
D. 25
E. 30

Solution 1:

Lena moves at v kilometers per hour, while Marco moves at 3v kilometers per hour.

So Marco pulls away from Lena at a rate of

3v - v = 2v kilometers per hour.

In m minutes, or m/60 hours, the gap between them becomes

2v * m/60 = mv/30 kilometers.

After Marco stops, Lena closes that gap at her own speed of v kilometers per hour. So the time she needs is

(mv/30)/v = m/30 hours.

We are told that this catch-up time is 50 minutes, which is 5/6 hour. Therefore,

m/30 = 5/6

m = 25

Answer: D

Solution 2:

In the 50 minutes that Marco is waiting, Lena travels at v kilometers per hour for 5/6 hour, so she covers:

v * 5/6 = 5v/6 kilometers.

That distance must equal the lead Marco had built before stopping. Now, Marco was increasing his lead at a relative speed of 3v - v = 2v kilometers per hour. So in m minutes, or m/60 hours, the lead he built is:

2v * m/60 = mv/30 kilometers.

Set that equal to the distance Lena covers during the 50-minute wait:

mv/30 = 5v/6

m = 25

Answer: D

Takeaway 
In speed and motion problems like this one, the key idea is relative speed:

• When two objects move in opposite directions, the relative speed is the sum of their speeds. 
• When they move in the same direction, the relative speed is the difference of their speeds.

The underlying principle is the same in both cases: relative speed measures how quickly the distance between the two objects changes.

What This Question Tests 
This question tests speed and motion reasoning, especially the use of relative speed to track how the distance between two moving objects changes over time. It also tests whether you can connect two stages of motion, first the lead being created, then that same lead being closed, while handling a time conversion correctly.

Bunuel · Answer

5. Distance and Speed Problems

5.2. Natalie is standing at a point on a straight road between her apartment and the airport. To get to the airport, she has two options: she can either walk directly to the airport, or she can first walk back to her apartment and then take a taxi from her apartment to the airport. The time required to walk directly to the airport is twice the time required for the second option. If the taxi travels 10 times as fast as Natalie walks, what is the ratio of Natalie’s distance from her apartment to her distance from the airport?

A. 3:10
B. 4:11
C. 5:11
D. 9:11
E. 11:4

Let Natalie’s distance from her apartment be m, and her distance from the airport be n. We want to find the ratio m/n.

Assume Natalie’s walking speed is v.

If she walks directly to the airport, the time required is

n/v

If instead she walks back to her apartment and then takes a taxi to the airport, the total time is

m/v + (m + n)/(10v)

We are told that walking directly to the airport takes twice as long as the second option, so

n/v = 2

Multiply both sides by v:

n = 2

n = 2m + (m + n)/5

Multiply by 5:

5n = 10m + m + n

5n = 11m + n

4n = 11m

So

m/n = 4/11

Therefore, the required ratio is 4:11.

Answer: B

Takeaway 
In distance and rate problems like this, breaking the situation into separate parts is often the cleanest way to build the equation correctly. Here, each route must be expressed as a time expression first, and then those time expressions must be related according to the condition given in the question. That step is the key: translating different setups into comparable expressions in the same unit, which in this case is time, simplifies the question and reduces it to an algebra problem.

What This Question Tests 
This question tests algebraic translation in a rate setting, especially the ability to express two different travel plans in terms of time and relate them correctly. It also tests whether you can use a speed ratio effectively, organize a multi-step setup without confusion, and extract a distance ratio from the resulting equation.

Bunuel · Answer

BATCH #2 (10 QUESTIONS)

6. Exponents

6.1. What is the units digit of 4^3^2?

A. 0
B. 2
C. 4
D. 6
E. 8

If exponentiation is written with stacked exponents, the rule is to evaluate from the top down:

a^{m^n} = a^{(m^n)} , not  (a^m)^n , which equals  a^{(mn)}

So:

4^{3^2} = 4^{(3^2)} = 4^9

Now, the units digit of 4 raised to positive integer powers follows a repeating pattern:

4, 6, 4, 6, ...

The cycle length is 2. Therefore, the units digit depends on whether the exponent is odd or even. Since 9 is odd, the units digit of 4^9 is the same as the units digit of 4^1, which is 4.

Answer: C.

Takeaway 
• If exponentiation is written with stacked symbols and no brackets are given, the expression must be evaluated from the top down. That interpretation is essential, because reading the exponent structure incorrectly changes the entire problem.

• Also, units-digit questions like this are fundamentally pattern-recognition questions. The key is to identify the repeating cycle of units digits and then determine where the given exponent falls within that cycle, instead of trying to compute the full power directly.

What This Question Tests 
This question tests understanding of exponent rules, specifically how to interpret stacked exponents, along with number properties involving cyclic patterns in units digits. It checks whether you can simplify expressions correctly before applying patterns.

Bunuel · Answer

6. Exponents

6.2. If  2^{6x}=49  and  3^{\sqrt{y}}=4 , what is the value of  \frac{(2^{(x-1)})^3}{9^{-\sqrt{y}}} ?

A.  \frac{7}{2} 
B. 7
C. 14
D. 21
E. 28

The key point to notice is that  x  and  y  cannot be integers and must be irrational numbers. This is because 49 does not contain 2 in its prime factorization, and 4 does not contain 3. As a result,  x  and  y  cannot be determined directly in any simple way, so the solution must proceed by manipulating the given expressions into a form that matches the target expression.

Start with  2^{6x}=49 .

Since  a^{mn}=(a^m)^n , rewrite  2^{6x}  as  (2^{3x})^2 :

(2^{3x})^2=7^2

Take the square root of both sides:

2^{3x}=7

Next since  (a^m){^n}=a^{mn}  and  a^{-m}=\frac{1}{a^m} , we get:

\frac{(2^{(x-1)})^3}{9^{-\sqrt{y}}}=

2^{(3x-3)}*9^{\sqrt{y}} .

Since  a^{m + n}=a^m*a^n  and  (a^m)^n=a^{mn}=(a^n)^m , we get:

(2^{3x}*2^{-3})* (3^2)^{\sqrt{y}} =

=(2^{3x}*2^{-3})*(3^{\sqrt{y}})^2

Finally, since  2^{3x}=7  and  3^{\sqrt{y}}=4 , then:

2^{3x}*2^{-3}*(3^{\sqrt{y}})^2=

=7*2^{-3}*4^2=

=7*\frac{1}{8}*16=

=7*2=

=14

Answer: C.

Takeaway 
To handle exponent questions well, you need a strong grasp of the basic exponent rules and be comfortable manipulating expressions using those rules. In problems like this one, that skill is essential, because rewriting the given expressions in a useful form is what makes the problem solvable.

In some exponent questions, direct solving for the variables does not lead anywhere useful. In those cases, the correct approach is to work with the structure of the expressions, apply exponent rules carefully, and transform the given information into the exact form needed.

What This Question Tests 
This question tests exponent rules and algebraic manipulation, especially the ability to rewrite powers in equivalent forms and substitute them into a target expression. It also tests whether you can recognize when direct solving is not the right path and instead use the structure of the given equations strategically.

Bunuel · Answer

7. Fractions and Ratios

7.1. At a technology summit, each registered participant belonged to exactly one of five groups: clients, journalists, support staff, speakers, or vendors. Among those registered,  \frac{3}{5}  were clients,  \frac{1}{4}  were journalists, and the remainder were divided equally among the other three groups. On the day of the summit, however, only  \frac{3}{4}  as many clients attended as had originally registered, twice as many journalists attended as had originally registered, only half of the registered support staff showed up, and none of the speakers attended because their session was canceled. What fraction of the actual attendees were clients?

A.  \frac{18}{41} 
B.  \frac{9}{20} 
C.  \frac{12}{25} 
D.  \frac{1}{2} 
E.  \frac{24}{41}

Since the fractions of the total are three-fifths and one-fourth, let us choose some smart number for the total. The LCM of 5 and 4 is 20, but let us take a larger multiple of 20, say 40, to be on the safer side and avoid fractions in later steps.

• Clients =  \frac{3}{5} * 40 = 24 
• Journalists =  \frac{1}{4} * 40 = 10

The remaining number is 40 - 24 - 10 = 6.

This remainder is divided equally among the other three groups, so each of those groups has 6/3 = 2. Thus, the registered counts are:

• Clients = 24
• Journalists = 10
• Support staff = 2
• Speakers = 2
• Vendors = 2

Now compute the actual attendees:

• Clients =  \frac{3}{4} * 24 = 18 
• Journalists =  2 * 10 = 20 
• Support staff =  \frac{1}{2} * 2 = 1 
• Speakers = 0
• Vendors = 2

So the total number of actual attendees is

18 + 20 + 1 + 0 + 2 = 4

Therefore, the fraction of actual attendees who were clients is

\frac{18}{41}

Answer: A.

Takeaway 
To successfully tackle fraction problems like this, you should be comfortable with phrases such as “twice as many,” “half as many,” or “three-fourths as many,” and quickly translate them into numerical changes. The key is to convert each condition into exact counts step by step.

Also, in percentage or fraction problems where the total is not given, choosing a smart number for the total often simplifies the work significantly. Instead of carrying a variable through the entire solution, picking a suitable total can turn the fractions into clean whole-number counts and make the arithmetic much easier to track.

What This Question Tests 
This question tests fraction translation, careful bookkeeping, and the ability to track how category counts change under different conditions. It also tests whether you can choose a smart total when none is given and use it to turn the fractional information into workable counts, while organizing the multi-step arithmetic without losing track of the groups.

Bunuel · Answer

7. Fractions and Ratios

7.2. At a business convention, there are 540 attendees, and 4 out of every 9 attendees are wearing name badges. Of the guests, only 1 out of every 6 is wearing a name badge. Among the attendees wearing name badges, the ratio of staff members to guests is 7:1. If the convention includes only staff members and guests, how many staff members are not wearing name badges?

A. 90
B. 120
C. 150
D. 180
E. 210

The total number of attendees wearing name badges is 540 * 4/9 = 240.

Among those wearing name badges, the ratio of staff members to guests is 7:1. So 7/8 * 240 = 210 staff members and 1/8 * 240 = 30 guests are wearing name badges.

Since only 1 out of every 6 guests is wearing a name badge, the total number of guests is 30 * 6 = 180.

So the total number of staff members is 540 - 180 = 360.

Therefore, the number of staff members not wearing name badges is 360 - 210 = 150.

Answer: C.

Takeaway 
For fraction and ratio problems, it is important to be comfortable with the standard wording these questions use and to translate that wording accurately into mathematical relationships. In particular, you should be able to transform ratios into fractions and fractions into ratios, since many problems require switching between those two forms at different stages of the solution.

What This Question Tests 
This question tests fraction translation, ratio interpretation, and multi-step part-to-whole reasoning. It also tests whether you can connect a fraction of the total, a ratio within that subgroup, and a fraction of one category without mixing up which quantity each piece of information refers to.

Bunuel · Answer

8. Functions and Custom Characters

8.1. For any real number x, f(x) denotes the greatest integer less than or equal to x and g(x) denotes the least integer greater than or equal to x. If f(a) = -3 and g(b) = -6, which of the following must be true?

A. -12 < a + b < -10
B. -11 < a + b < -9 
C. -10 < a + b < -9 
D. -10 < a + b < -8 
E. -9 < a + b < -8

The function f(x) represents the greatest integer less than or equal to x, so it rounds a number down to the nearest integer. For example:

• f(2.7) = 2, because 2 is the largest integer less than or equal to 2.7
• f(5) = 5, because 5 is the largest integer less than or equal to 5
• f(-2.3) = -3, because -3 is the largest integer less than or equal to -2.3
• f(-10) = -10, because -10 is the largest integer less than or equal to -10

The function g(x) represents the least integer greater than or equal to x, so it rounds a number up to the nearest integer. For example:

• g(2.7) = 3, because 3 is the least integer greater than or equal to 2.7
• g(5) = 5, because 5 is the least integer greater than or equal to 5
• g(-2.3) = -2, because -2 is the least integer greater than or equal to -2.3
• g(-10) = -10, because -10 is the least integer greater than or equal to -10

According to the above, f(a) = -3 means that -3 is the greatest integer less than or equal to a. So a must satisfy

-3 ≤ a < -2 (for any number from that range, the largest integer less than or equal that number will be -3)

Similarly, g(b) = -6 means that -6 is the least integer greater than or equal to b. So b must satisfy

-7 < b ≤ -6 (for any number from that range, the least integer greater than or equal to that number will be -6)

Now add the two inequalities:

-10 < a + b < -8

Answer: D.

Takeaway 
You should be comfortable with the greatest-integer and least-integer functions and know how to translate each one quickly into a range. In questions like this, the key step is not the notation itself, but turning that notation into the correct interval and then working from there.
Also, manipulating inequalities is an important tool to have in your GMAT toolkit. Once the correct ranges are established, the rest of the problem becomes an inequality question, so being able to combine and interpret inequalities accurately is essential.

What This Question Tests 
This question tests understanding of the floor and ceiling functions, specifically how to convert each function value into the correct interval. It also tests inequality properties, with the extra layer of combining those inequalities correctly to determine the required range.

Bunuel · Answer

8. Functions and Custom Characters

8.2. For all integers x,   = -2x when x is odd and   = -x^2 when x is even. What is   /   equivalent to?

A.   
B.   
C.   
D.   
E.

Since -6 is even, we use the even-case definition   = -x^2. So:

= -(-6)^2 = -36

Since -3 is odd, we use the odd-case definition   = -2x. So:

= -2(-3) = 6

Therefore:

/   = -36 / 6 = -6

Now notice that the answer choices are also written in brackets, so we need to find which choice is equal to -6.

First, all even inputs can be ruled out. For an even integer x, we have   = -x^2 but -6 is not the square of an integer, so no even input can give a value of -6. Therefore, choices A, B, and E are eliminated, since -10, -6, and 6 are all even.

We are left with choices C and D.

Check C:

= -2(-3) = 6

So C does not work.

Check D:

= -2(3) = -6

So D works.

Answer: D

Takeaway 
In questions involving custom-defined functions, the first step is to apply the definition carefully and compute the value exactly. Just as important, however, is answering the question that is actually being asked. In a GMAT question like this, a common trap is to stop after finding that   /   = -6 and then choose the option that shows that value directly. But that is not what the question asks. It asks which bracket expression is equivalent to that value, so one more step is required.

What This Question Tests 
This question tests function notation, piecewise definitions, and logical case analysis. It also tests whether you can evaluate an expression using a custom rule and then connect that result to the form in which the answer choices are presented, rather than stopping too early.

Bunuel · Answer

9. Inequalities

9.1. Which of the following describes all values of x for which 1/(x - 1) + 1/(x + 7) < 0 ?

A. x < -7
B. -7 < x < -3
C. -3 < x < 1
D. -7 < x < -3 or x > 1
E. x < -7 or -3 < x < 1

First, combine the two fractions:

\(\frac{1}{(x - 1)} + \frac{1}{(x + 7)} = \)

\(=\frac{(x + 7 + x - 1)}{(x - 1)(x + 7)} = \)

\( =\frac{(2x + 6)}{(x - 1)(x + 7)} = \)

\( =\frac{2(x + 3)}{(x - 1)(x + 7)}\)

So we need to solve \(\frac{(x + 3) }{ (x - 1)(x + 7) } < 0 \).

Divide both sides by 2 to get \(\frac{(x + 3)}{(x - 1)(x + 7)} < 0 \)

The transition points are x = -7, x = -3, and x = 1. These are the values of x for which the factors equal 0 and at which the whole expression alternates sign. This gives us four ranges:

x < -7
-7 < x < -3
-3 < x < 1
x > 1

Next, test an extreme value for x. If x is a large enough number, say 100, then the numerator and both denominator factors are positive, so the whole expression is positive. Therefore, when x > 1, the expression is positive.

Now, here is the key pattern: since there are four ranges and the sign changes at each transition point, the signs must alternate. Since the expression is positive in the 4th range, the pattern must be:

-, +, -, +

So the expression is negative in the 1st and 3rd ranges. Thus, the expression is negative for:

x < -7 and -3 < x < 1

So the correct answer is E.

Answer: E.

Takeaway
In inequality questions like this, where an expression is written as a product or quotient of factors and you need to determine when the overall value is positive or negative, the method used above is usually the most efficient one. The key steps are to identify the transition points, split the number line into intervals, and then determine the sign pattern across those intervals. Practicing this method on many similar questions is important, because it helps you avoid common traps and apply the process quickly and confidently.

What This Question Tests
This question tests your ability to solve sign-based inequalities by analyzing how the sign of an expression changes across critical values. It requires you to combine fractions, identify the critical values, and then determine on which intervals the resulting expression is negative. It also tests whether you can use interval logic efficiently instead of relying on slow case-by-case algebra.

Bunuel · Answer

9. Inequalities

9.2. If p < q < r < s < t and pqrst < 0, which of the following must be true?

I. rs > 0
II. qr < 0
III. st > 0

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

Since pqrst < 0, there must be an odd number of negative values among p, q, r, s, and t, so there are 1, 3, or 5 negative numbers.

Also, because p < q < r < s < t, all negative values must come before all positive values.

So the only possible sign patterns are:

So the only possible sign patterns are:

p | q | r | s | t

- | - | - | - | -
- | - | - | + | +
- | + | + | + | +

Now check each statement.

I. rs > 0

This is true in the first and third cases, but false in the second case. So this does not have to be true.

II. qr < 0

In all three cases, q and r have the same sign, so qr > 0 in every case. Therefore, this does not have to be true.

III. st > 0

In the first case, both s and t are negative, so st > 0.
In the second case, both s and t are positive, so st > 0.
In the third case, both s and t are positive, so st > 0.

So this must always be true.

Answer: C.

Takeaway
In inequality questions like this one, it is important to break the given information into all valid cases and be comfortable determining the sign of each variable from the constraints. You also need to distinguish carefully between must be true and could be true. In a must be true question, a statement has to hold for every possible set of values that satisfies the given conditions, not just for one valid case.

What This Question Tests
This question tests inequality properties together with sign analysis on the number line. It also tests your ability to use case analysis efficiently and to evaluate statements in a must be true setting.

Bunuel · Answer

BATCH #3 (10 QUESTIONS)

10. Min-Max Problems

10.1. Elena has a box containing 11 strawberry candies and 16 lemon candies. What is the least number of candies that Elena must randomly select from the box to guarantee that she has selected at least three candies of each flavor?

A. 15
B. 17
C. 18
D. 19
E. 20

To guarantee that Elena has selected at least three candies of each flavor, we must consider the  worst-case scenario .

The most unfavorable case is that Elena first selects  all  16 lemon candies. At that point, she still has no strawberry candies. However, the next 3 candies she selects must then be strawberry candies.

So the least number of candies she must select to guarantee at least three candies of each flavor is 16 + 3 = 19.

Answer: D.

To elaborate further, it is important to recognize that the candies can be selected in many different orders. Ideally, the best-case scenario for meeting the requirement of selecting at least three candies of each flavor would involve Elena selecting 3 strawberry candies and 3 lemon candies right away. However, this specific order of selection is  not guaranteed . What we need to determine is a number of selections that,  regardless of the order in which the candies are selected , will always  ensure  that at least three candies of each flavor are chosen. Therefore,  to guarantee  that at least three strawberry candies and at least three lemon candies are selected under any possible sequence, it is necessary to consider the worst-case scenario, which is the most unfavorable order of selection.

Takeaway 
In guarantee questions like this one, the key is to think in terms of the worst-case scenario. You should ask how long the most unfavorable sequence can continue before the required outcome becomes unavoidable. In a question that asks for the least number needed to guarantee a result, the correct answer must work for every possible order of selection, not just for a favorable one.

What This Question Tests 
This question tests worst-case counting logic and guarantee reasoning. It also tests whether you can distinguish between a result that could happen in some favorable order and a result that must happen in every possible order once enough selections have been made.

Bunuel · Answer

10. Min-Max Problems

10.2. Eleven salespeople at a store each sold a different number of smartphones during a particular week. If the average (arithmetic mean) number sold was 42 and the median was 37, what is the minimum possible number of smartphones sold by the top seller?

A. 49
B. 50
C. 51
D. 52
E. 53

Let the 11 numbers be

a < b < c < d < e < f < g < h < i < j < k

We want the minimum possible value of k. Since the average is 42, the total number of smartphones sold is

11 * 42 = 462

Since the median is 37, the sixth value must be 37. So

a < b < c < d < e < 37 < g < h < i < j < k

To minimize k, we should maximize the first six values.

The greatest possible values for a, b, c, d, and e are

32, 33, 34, 35, and 36

So the greatest possible sum of the first six values is

32 + 33 + 34 + 35 + 36 + 37 = 207

That leaves 462 - 207 = 255 for the last five values:

g + h + i + j + k = 255

Now we want k to be as small as possible. So the five numbers greater than 37 should be as close together as possible. If those five numbers were equal, we’d get g = h = i = j = k = 255/5 = 51. However, since they must be distinct, the least value k can take is when g = 49, h = 50, i = 51, j = 52, and k = 53.

The final distribution that yields the minimum possible value of k is: {32, 33, 34, 35, 36, 37, 49, 50, 52, 52, 53}.

Therefore, the minimum possible number of phones sold by the top-selling associate is 53.

Answer: E.

Takeaway 
A useful general rule in problems  where the total is fixed  is this:

• to maximize one quantity, minimize the others; 
• to minimize one quantity, maximize the others.

Here, since we want the greatest value to be as small as possible, the other values should be made as large as the constraints allow.

What This Question Tests 
This question tests the concept of minimizing and maximizing under constraints. In addition, this optimization question is built on statistical parameters, specifically arithmetic mean and median.

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