Test: Word Problems - 1 - GMAT MCQ

Let the nos. be 5x and 3x.
Then, 5x - 3x = 10
⇒ 2x = 10, x = 5
Required product = 5x * 3x
⇒ 5 * 5 * 3 * 5 = 375

Let the number be x, 3x, 4x and 7x. Then, x + 3x + 4x + 7x = 75
⇒ 15x = 75 , x = 5
Required value = 7 * 5 = 35

Solution:

Statement 1: It took a total of 5 hours to complete the entire distance.

- Let the total distance be D miles.
- Let the time taken to cover the first part be 2x hours and the time taken to cover the remaining part be 3x hours.
- Using the formula Distance = Speed x Time, we can write the equations:
- Distance covered in the first part = 30 * 2x = 60x
- Distance covered in the remaining part = 50 * 3x = 150x
- Total distance = 60x + 150x = 210x

- We are given that the total time taken is 5 hours, so:
- 2x + 3x = 5
- 5x = 5
- x = 1

- Therefore, the total distance D = 210x = 210 * 1 = 210 miles.

Statement 2: 2/7th of the entire distance was covered during the first part.

- Let the total distance be D miles.
- According to the statement, the distance covered in the first part = 2/7 * D
- Therefore, the distance covered in the remaining part = D - 2/7 * D = 5/7 * D

- However, this information alone does not provide us with enough information to determine the total distance that Charles travelled.

Conclusion: Statement (1) alone is sufficient to answer the question, but statement (2) alone is not sufficient. Therefore, the correct answer is A.

Step 1: Question statement and Inferences

Let the two dogs meet at a distance of x miles from the Starting position of Dog 1.

So, Dog 1 covers a distance of x miles at a speed of 12 mph to reach the meeting point.

In the same time, Dog 2 covers a distance of (2-x) miles at a speed of 8 mph to reach the meeting point

Step 2: Finding required values

We know that

Since the time taken by Dog 1 to reach the meeting point is equal to the time taken by Dog 2 to reach the meeting point, we can write:

The dogs will meet 6/5 miles from Dog 1’s starting position.

Answer: Option (B)

Step 1: Question statement and Inferences

Riding 10 miles at her new speed took Sam 10 minutes less than riding the same distance at 12 miles per hour. What was this new speed?

Step 2: Finding required values

Start by finding the time it took Sam to reach the top of the hill.

Her uphill speed, u = 12 miles per hour

Distance travelled = 10 miles

Now considering the downhill journey,

Step 3: Calculating the final answer

Answer: Option (B)

• ⇒ 1/10 % of 50
• = 1/10 x 1/100 x 50
• = 0.05

• 1. we cannot move to any conclusion if we look at the condition number 1 i.e x = 5.
• 2.  Now, x% of y = 2
• (x/100)y = 2
• xy = 200....Equation 1 
•  
• Now if we have to calculate y% of x
• (y/100)x
• yx/100
• now putting the value of yx from 1 in equation 2
• 200/100 = 2
• Hence, Statement 2 is sufficient.

• Let the number be 100
• 10% of 100 is 10 = 110 (new Number)
• 50% of 110 is 55 = 110-55 = 55 (New Number)
• 100% of 55 is 55= 55+55= 110 (Final Number)
• Final Number- Initial Number =110-100=10
• And 10 is 10% of 100

Given:

• Let Machine A take a minutes to copy x pages
  So, A’s rate, R_A = x / a pages per minute
  This rate is constant
• So, Machine B take a + 6 minutes to copy x pages
  So, B’s rate, R_B = x / a + b pages per minute
  This rate is constant
• Time taken by A and B together to copy 7x pages) = 20 minutes

To Find: Time taken by A alone to copy 2x pages

Approach:
1. Time taken by A alone to copy 2x pages
= 2x / A’s rate = 2x / x / a = 2a minutes

2. We’re given the combined output of A and B and the time taken for this output. Using this data, we can find the combined rate of A and B.
Then, we’ll use the equation (Combined Rate of A and B) = (Rate of A) + (Rate of B) to get an equation in terms of a.

Working out:

• When A and B operate simultaneously, 7x pages take 20 minutes
  • So, Combined Rate = 7x / 20 pages per minute
• Therefore, 7x / 20 = x / a + x / a + 6
  7 / 20 = 1 / a + 1 / a + 6
  7a ( a + 6) = 20(a + 6 + a)
  7a² + 42a = 40a + 120
  7a² + 2a - 120 = 0
  7a² - 28a + 30a - 120 = 0
  7a(a - 4) + 30(a - 4) = 0
  (7a + 30)(a - 4) =0
  So, a = -30 / 7 or a = 4
• The negative root is rejected since time cannot be negative.
• So, a = 4 minutes
• Therefore, Required Time = 2a = 2*4 = 8 minutes

Looking at the answer choices, we see that the correct answer is Option B

Given:

• Number of emails answered by Peter in 1 hour = 25
• Number of emails answered by Mark in 1 hour = 25 + 40% of 25
• Time taken by John to answer 25 emails =
  (1 - 1 / 6) = 5 / 6 hours

To Find: Percentage by which Peter should increase his speed so that all 3 can answer 1 / 6th more emails than they currently do?

Approach:
1. (Percentage by which Peter should increase his speed) = Extra number of emails that he needs to answer per hour / Current number of emails answered per hour * 100

2. So, to find the percentage by which Peter should increase his speed, we should first find the extra number of emails that he needs to answer.
3. For finding the extra number of emails that he needs to answer, we need to find the total number of emails currently answered by Peter, John and Mark together in 1 hour and number of increased emails they need to answer together.

• We are given the number of emails answered by Peter and Mark in 1 hour.
• As we know that John can answer 25 email in 5/6  hours, we can find the number of emails answered by John in 1 hour

Working out:

1. Number of emails answered by Mark in 1 hour = 25 + 40% of 25 = 25 + 10 = 35
2. Number of emails answered by John in 1 hour = 
   25 * 6 / 5 = 30
3.  So, the number of emails currently being answered by Peter, mark and John combined = 25 + 35 + 30 = 90
4. Number of emails they should answer in case of 1/6 increase = 90 + 1 / 6 * 90 = 105  emails
5. As there are 105 – 90 = 15 more emails that need to answered, these should be answered by Peter.
6. So, Peter should answer 15 more emails per hour
   • Hence, Peter’s speed should increase by 
     15 / 25 * 100 = 60%

So, Peter should increase his speed by 60%

Given:

• One side of the triangle is parallel to x – axis
• The x- and y – coordinates of each vertex are integers.
• For each vertex of the triangle,
  • -3 ≤ x < 7
  • 2 < y ≤ 7

To find: Number of different triangles that may be specified.

Approach:

1. The objective here consists of the following tasks:
   • Task 1 – Choose the x-coordinates of the two vertices on the line parallel to x-axis
   • Task 2 – Chose x-coordinate of the 3^rd point
   • Task 3 - Choose y-coordinate of the line parallel to x-axis
   • Task 4 – Choose y-coordinate of the third point
     
     Since each of these tasks must be performed in order to construct this triangle, the answer will be obtained by multiplying the number of ways of doing each task.
      
2. Using the given inequalities and the fact that only integer values of x and y are allowed, we’ll find the number of ways of doing each task.

Working Out:

• Choosing the x-coordinates of the 3 points
  • Finding number of ways to do Task 1:
    • -3 ≤ x < 7
    • That is, -3 ≤ x ≤ 6
    • Therefore, number of possible values of x is 10
    • So, ways to choose x-coordinates of the 2 vertices on the line parallel to x-axis =

• Finding number of ways to do Task 2
  • Choices for third vertex’ x-coordinate = 10
    • (If the x-coordinate of the third vertex is the same as the x-coordinates of either of the two vertices that form the line parallel to x-axis, we get a right triangle)

• Choosing the y-coordinates of the 3 points
  • Finding number of ways to do Task 3:
    • 2 < y ≤ 3
    • That is 3 ≤ y ≤ 7
    • So, number of possible values of y is 5
    • So, (Number of choices for y-coordinate of the line parallel to x-axis) = 5
• Finding number of ways to do Task 4:
  • One of the 5 possible values of y is used up in Task 3
  • So, (Number of choices available for third vertex’ y-coordinate) = 4
• Getting to the answer
  • So, total ways in which the triangle can be formed = 45*10*5*4 = 45*200  = 9000

Looking at the answer choices, we see that the correct answer is Option B

Step I: Define Event 

The event here is that the person was born in a leap year between 1990 and 2010, inclusive. We know that a leap year is a year that is divisible by 4. 

Step II: Find n, the number of ways in which all outcomes can occur

Since the boundary years 1990 and 2010 are included as well, the total number of years to be considered is 2010 – 1990 + 1 = 21.

(Note: Suppose the years given were 1990-1992 inclusive, then how many years would that be? 3 years: 1990, 1991 and 1992. But 1992-1990 is equal to only 2. Therefore, when the boundary years are included, the correct formula to count the number of years will be: Final Year – Initial Year + 1)

The person may be born in any of these years.

Thus, n = ²¹C₁ = 21

Step III: Find x, the number of ways in which the event can occur

To calculate x, we need to find the leap years between 1990 and 2010, inclusive.

We know that a number is divisible by 4 if its last two digits are divisible by 4. Using this rule, we can determine that the leap years will be:

1992,  1996, 2000, 2004 and 2008

Thus, there are 5 leap years. The person being born in any of these 5 years will be a favorable outcome.

Therefore, x = 5

Step IV: Calculate probability 

P = x/n = 5/21     

Answer: Option (D)

Step I: Define the Event 

We are given a set of 20 numbers. A single number is selected from this set. The event will be the selection of a number that is divisible by 5

Step II: Find n, the number of ways in which all outcomes can occur

As we know there are 20 numbers in the set. Any of these numbers may be selected.

Thus, n = ²⁰C₁ = 20     

Step III: Find x, the number of ways in which the event can occur

To find x, we should list the numbers between 1 and 20 that are divisible by 5. These numbers are:

5, 10, 15 and 20.

Thus, there are 4 numbers between 1 and 20 that are divisible by 5.

The event will occur if any of these 4 numbers is selected.

Thus, x =  ⁴C₁ = 4

Step IV: Probability = x/n 

P = x/n = 4/20 = 1/5   

So, the probability of the selected number being divisible by 5 is 1/5.   

Answer: Option (C)  

Interest = Rs. (81-72) = Rs. 9
Let the time be t years.
9 = (72×25×t)/(4×100)
t = (9×400)/(72×25)=2 years

► Cost Price(CP) of Type 1 material is Rs. 15 per kg
► Cost Price(CP) of Type 2 material is Rs. 20 per kg
Type 1 and Type 2 are mixed in the ratio of 2 : 3

Therefore, rice per kg of the mixed variety of material = Rs.18

Steps 1 & 2: Understand Question and Draw Inferences

We are given that there are 150 students in a school. Out of these 150 students, 40 have opted only for French as an extra subject.

There are 30 students who haven’t opted for any of German or French as the extra subject.

Let’s say:

X = Number of students who opted for both the subjects

Y = Number of students who opted only for German as the extra subject

So, the above information can be represented as follows:   

The sum of the numbers in the four zones of the Venn diagram will be equal to the total number of students.

Thus,

40 + X + Y + 30 = 150

We get: 

X + Y = 80      ……….. (1)

The question asks us to find the value of Y.

Step 3: Analyze Statement 1

The total number of students opting for at least one subject out of German and French is 120 

This means,

40 + X + Y = 120

Or, X + Y = 80

As shown in equation (1), we already have this information. However, since we don’t have the value of X, we can’t find the value of Y.

Hence, Statement 1 is not sufficient to answer the question: What is the value of Y?

Step 4: Analyze Statement 2

The number of students who opted for both German and French is one-third of the number of students who opted for neither of the two subjects  

Per this statement,

   X=1/3(30)=10

By plugging in the value of X in Equation (1), we get,

Y = 70

Hence, Statement 2 alone is sufficient to answer the question: What is the value of Y?  

Step 5: Analyze Both Statements Together (if needed)

Since Statement 2 alone is sufficient to answer the question, we don’t need to perform this step.

Answer: Option (B)  

Steps 1 & 2: Understand Question and Draw Inferences

We are given that 500 GMAT aspirants registered on e-GMAT website on a single day. These registrations came in three sections: Quant, Verbal, and Others. 

Let’s say:

X = the number of students who registered in only Verbal courses

Y = the number of students who registered in only Quant courses

Z = the number of students who registered in both Quant and Verbal

P = the number of students who registered for neither of the Verbal or Quant courses.

We are told that

The total number of students who registered in Verbal courses = 50% of 500 =

Thus, X + Z = 250          ……………… (1)

Also given:

The total number of students who registered in Quant courses = 200

Thus, Y + Z = 200          ……………… (2)

Now, the sum of the numbers in the four zones of the Venn diagram will be equal to the total number of students.

That is,

X + Y + Z + P = 500      ………. (3)

We need to find the value of P.

Since we have three equations in four unknowns, we just need one more equation in these unknowns to find the value of P.

Step 3: Analyze Statement 1

30% of the aspirants registered only for Quant courses

Per this statement,

Y = 30% of 500 =  ………. (4)

Substituting Equations (1) and (4) in Equation (3):

250 + 150 +P = 500

P = 100

Hence, Statement 1 alone is sufficient to answer the question: What is the value of P?  

(Note: We have shown the actual calculation for P here just to illustrate the calculation. In the exam, once you determine that Statement 1 provides enough information for you to solve the question, you can move to Step 4)

Step 4: Analyze Statement 2

10% of the aspirants registered for both Quant and Verbal courses   

Per this statement,

Z = 10% of 500 = 

 By substituting this value of Z in Equations 1 and 2 respectively, we’ll be able to find the values of X and Y. And by substituting the values of X, Y and Z in Equation 3, we will be able to determine the value of P.

Thus, Statement 2 alone is sufficient to find the value of P.

Step 5: Analyze Both Statements Together (if needed)

Since Statement 1 and 2 alone are sufficient to answer the question, we don’t need this step.

Answer: Option (D)  

A loss of 25% means a cost price of 100 corresponding to a selling price of 75. CP as a percentage of the SP would then be 133.33%

Alternatively,

Cost price = 1025
Loss% = 25%
SP = 1025 - (1025 * 25/100) = 768.75
⇒ 768.75 * (x/100) = 1025
⇒ 768.75x = 102500
⇒ x = 102500/768.75 = 133.33%

So option D is correct

Change in total weight of 10 students = difference in weight of the student who joined and the student

=> weigth of first student who left = 30 + (10×2) = 50

weight of the student who joined last = 50 – 10 = 40...
Thus change in average weight = (40 – 30)/10 = 1...
 

Initial Milk = 2/5*250*3/5 = 60 L
Water = 2/5*250*2/5 = 40 L
Rest of Tank =150 L
Pipes are opened then can fill rest of tank in 108/25 hours
H/W = constant
then (108/25)/12/x = (108/25)/18(150-x)
X = 90 = Milk, Water = 60
Final ratio = 3:2