Arithmetic Question with Answer for CAT

Overview: It's important to remember that the regular practice of quantitative ability questions is important to accomplish well in this section. Keep practicing regularly to improve your skills and work on your weak areas.

CAT Exam 

The Common Admission Test (CAT) is a prestigious annual MBA entrance exam in India, conducted by the IIMs. Held in November or December, it's a 3-hour computer-based test with sections on Verbal Ability and Reading Comprehension (VARC), Data Interpretation and Logical Reasoning (DILR), and Quantitative Ability (QA). Scoring is scaled with percentiles. Eligibility requires a bachelor's degree with 50% marks (45% for SC/ST/PWD). CAT scores open doors to top business schools.

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Basic for Understanding Arithmetic Questions

Number Systems

Number systems form the backbone of arithmetic. It includes the study of various types of numbers and their properties.

• Natural Numbers (N): Positive integers starting from 1 (1, 2, 3, ...).
• Whole Numbers (W): Natural numbers including zero (0, 1, 2, 3, ...).
• Integers (Z): All positive and negative whole numbers including zero (..., -2, -1, 0, 1, 2, ...).
• Rational Numbers (Q): Numbers that can be expressed as a fraction of two integers where the denominator is not zero (e.g., 1/2, -3/4).
• Irrational Numbers: Numbers that cannot be expressed as a simple fraction (e.g., √2, π).
• Real Numbers (R): All rational and irrational numbers.
• Prime Numbers: Natural numbers greater than 1 with no positive divisors other than 1 and itself (e.g., 2, 3, 5, 7).
• Composite Numbers: Natural numbers greater than 1 that are not prime (e.g., 4, 6, 8, 9).

Important Arithmetic Formula for CAT

1. Number Systems

• Prime Numbers: A number greater than 1 that has no divisors other than 1 and itself.
• LCM (Least Common Multiple): LCM of a and b is the smallest positive integer divisible by both a and b.
• HCF (Highest Common Factor): HCF of a and b is the largest positive integer that divides both a and b without leaving a remainder.

2. Percentages

• Percentage Formula: Percentage = (Part / Whole) * 100
• Percentage Increase/Decrease: Increase/Decrease = [(New Value - Original Value) / Original Value] * 100

3. Profit and Loss

• Profit: Profit = Selling Price (SP) - Cost Price (CP)
• Loss: Loss = CP - SP
• Profit Percentage: Profit % = (Profit / CP) * 100
• Loss Percentage: Loss % = (Loss / CP) * 100

4. Simple and Compound Interest

• Simple Interest (SI): SI = (Principal (P) * Rate (R) * Time (T)) / 100
• Compound Interest (CI): CI = P * (1 + R / 100)^T - P

5. Ratio and Proportion

• Ratio: Ratio of a to b = a / b
• Proportion: If a / b = c / d, then ad = bc

6. Time, Speed, and Distance

• Speed: Speed = Distance / Time
• Distance: Distance = Speed * Time
• Time: Time = Distance / Speed

7. Time and Work

• Work: Work Done = Number of Days * Work Rate
• Work Rate: Work Rate = Work Done / Number of Days

8. Mixtures and Alligations

• Weighted Average: Weighted Average = (Sum of (Weight * Value)) / (Sum of Weights)
• Alligation Rule: Quantity of Cheaper / Quantity of Dearer = (CP of Dearer - Mean Price) / (Mean Price - CP of Cheaper)

9. Pipes and Cisterns

• Filling Time: If a pipe can fill a tank in x hours, then the part filled in 1 hour is 1 / x.
• Emptying Time: If a pipe can empty a tank in y hours, then the part emptied in 1 hour is 1 / y.

10. Arithmetic Progression (AP)

• Nth Term of AP: a_n = a + (n - 1)d
• Sum of First n Terms of AP: S_n = n / 2 [2a + (n - 1)d]

Check: CAT Exam Pattern

Arithmetic for CAT

Basics of Arithmetic for CAT

1. Number Systems and Operations

Understanding number systems is the foundation of arithmetic. It includes:

• Integers: Whole numbers and their opposites (e.g., -3, -2, -1, 0, 1, 2, 3).
• Fractions: Numbers expressed as the quotient of two integers (e.g., 1/2, 3/4).
• Decimals: Numbers expressed in the base-10 system (e.g., 0.5, 1.75).

Operations include addition, subtraction, multiplication, and division of these numbers.

1. Fractions, Decimals, and Percentages

Conversions and calculations involving these forms are crucial:

• Converting Fractions to Decimals: Divide the numerator by the denominator.
• Converting Decimals to Percentages: Multiply by 100.
• Converting Percentages to Fractions: Divide by 100 and simplify if necessary.

1. Ratio and Proportion

Ratios compare two quantities, and proportions state that two ratios are equal:

• Ratio: Ratio of a to b = a / b.
• Proportion: If a / b = c / d, then ad = bc.

Learn more: CAT Exam Day Guidelines

Arithmetic Progression (AP) for CAT

Fundamentals of AP

Arithmetic Progression is a sequence of numbers in which the difference between consecutive terms is constant:

• Nth Term of AP: a_n = a + (n - 1)d, where 'a' is the first term and 'd' is the common difference.
• Sum of First n Terms of AP: S_n = n / 2 [2a + (n - 1)d].

Solving CAT-Level Problems on AP

• Finding Terms: Calculate specific terms in the sequence.
• Summation: Determine the sum of a given number of terms.
• Application Problems: Apply AP concepts to solve real-world problems.

Arithmetic Topics for CAT

Profit and Loss

• Profit: Profit = Selling Price (SP) - Cost Price (CP).
• Loss: Loss = CP - SP.
• Profit Percentage: Profit % = (Profit / CP) * 100.
• Loss Percentage: Loss % = (Loss / CP) * 100.

Simple and Compound Interest

• Simple Interest (SI): SI = (Principal (P) * Rate (R) * Time (T)) / 100.
• Compound Interest (CI): CI = P * (1 + R / 100)^T - P.

Time, Speed, and Distance

• Speed: Speed = Distance / Time.
• Distance: Distance = Speed * Time.
• Time: Time = Distance / Speed.

Time and Work

• Work Done: Work Done = Number of Days * Work Rate.
• Work Rate: Work Rate = Work Done / Number of Days.

Mixtures and Alligations

• Weighted Average: Weighted Average = (Sum of (Weight * Value)) / (Sum of Weights).
• Alligation Rule: Quantity of Cheaper / Quantity of Dearer = (CP of Dearer - Mean Price) / (Mean Price - CP of Cheaper).

Also Check: CAT Registration Dates 2024

Must Solve Arithmetic Questions for CAT 

Question 1: If y is a negative number such that 2y2log35 = 5log23, then y equals:

1. Log2 (1/5)
2. Log2 (1/3)
3. log2 (1/3)
4. log2 (1/5)

Answer: log2(1/5)

Question 2: The number of real-valued solutions of the equation 2x+2−x=2−(x−2)22^x + 2^{-x} = 2 - (x - 2)^22x+2−x=2−(x−2)2

1. 0
2. 1
3. Infinite
4. 2

Answer: 0

Question 3:  A train traveled at one-third of its usual speed, and hence reached 30 minutes late. On the return journey, the train initially traveled at its usual speed for 5 minutes but then stopped for 4 minutes for an emergency. The percentage by which the train must now increase its usual speed so as to reach the destination at the scheduled time is nearest to:

1. 58
2. 67
3. 50
4. 61

Answer: 67

Question 4: The mean of all 4-digit even natural numbers of the form ‘aabb’, where a>0a > 0a>0, is:

1. 5050
2. 4864
3. 5544
4. 4466

Answer: 5544

Question 5: An alloy is prepared by mixing metals A, B, and C in the proportion 3:4:7 by volume. Weights of the same volume of metals A, B, and C are in the ratio 5:2:6. In 130 kg of the alloy, the weight of metal C is:

1. 84
2. 96
3. 70
4. 48

Answer: 84

Question 6: If f (5 + x) = f (5 - x) for every real x, and f (x) = 0 has four distinct real roots, then the sum of these roots is

1. 20
2. 40
3. 0
4. 10

Answer: 20

Read more: Negative Marking In CAT Exam

Also Read : Is CAT Exam Tough

What is Arithmetic Progression?

An Arithmetic Progression (AP) is a sequence of numbers in which each term after the first is obtained by adding a fixed constant to the previous term. This fixed constant is called the common difference (d). In other words, an AP is a sequence of numbers where each term is obtained by adding a fixed constant to the previous term.

Example:

2, 5, 8, 11, 14, ...

In this example, the common difference (d) is 3, and each term is obtained by adding 3 to the previous term.

Key Concepts:

1. Common Difference (d): The fixed constant added to each term to get the next term. It is denoted by the symbol 'd'.
2. First Term (a): The initial term of the AP. It is denoted by the symbol 'a'.
3. Nth Term (an): The nth term of the AP. It is denoted by the symbol 'an'.
4. Sum of n Terms (Sn): The sum of the first n terms of the AP. It is denoted by the symbol 'Sn'.

Formulas:

1. Nth Term (an): an = a + (n-1)d This formula is used to find any term in the AP. It states that the nth term is equal to the first term plus the product of the common difference and the number of terms minus one.
2. Sum of n Terms (Sn): Sn = (n/2)(2a + (n-1)d) This formula is used to find the sum of any number of terms in the AP. It states that the sum of n terms is equal to the product of the number of terms, the first term, and the common difference.

Properties of Arithmetic Progression:

1. The sum of any two terms equidistant from the beginning and end of an AP is equal to the sum of the first and last terms : This property states that if we take any two terms that are equidistant from the beginning and end of an AP, their sum will be equal to the sum of the first and last terms.

2. The sum of an AP is equal to the average of the first and last terms multiplied by the number of terms : This property states that the sum of an AP is equal to the product of the average of the first and last terms and the number of terms.

Practice Questions:

1. The 10th term of an AP is 52, and the 20th term is 92. Find the common difference.

Answer: d = 4

Solution: Let the first term be 'a'. Then, the 10th term is a + 9d = 52, and the 20th term is a + 19d = 92. Subtracting the first equation from the second, we get 10d = 40, which gives d = 4.

2. The sum of the first 15 terms of an AP is 210. If the first term is 2, find the common difference.

Answer: d = 3

Solution: Using the formula for the sum of n terms, we get (15/2)(2(2) + (15-1)d) = 210. Simplifying, we get 15(4 + 14d) = 420, which gives d = 3.

3. In an AP, the 5th term is 11, and the 11th term is 25. Find the 15th term.

Answer: an = 39

Solution: Let the first term be 'a'. Then, the 5th term is a + 4d = 11, and the 11th term is a + 10d = 25. Subtracting the first equation from the second, we get 6d = 14, which gives d = 7/3. Then, the 15th term is a + 14d = 11 + 14(7/3) = 39.

4. The sum of the first 20 terms of an AP is 400. If the first term is 5, find the common difference.

Answer: d = 2

Solution: Using the formula for the sum of n terms, we get (20/2)(2(5) + (20-1)d) = 400. Simplifying, we get 10(10 + 19d) = 400, which gives d = 2.

5. In an AP, the 3rd term is 7, and the 7th term is 15. Find the 10th term.

Answer: an = 21

Solution: Let the first term be 'a'. Then, the 3rd term is a + 2d = 7, and the 7th term is a + 6d = 15. Subtracting the first equation from the second, we get 4d = 8, which gives d = 2. Then, the 10th term is a + 9d = 7 + 9(2) = 21.

Tips and Tricks:

1. Use the formula for the nth term (an) to find any term in the AP.
2. Use the formula for the sum of n terms (Sn) to find the sum of any number of terms.
3. Look for patterns in the AP to find the common difference or first term.
4. Use the properties of AP to simplify complex problems.

Know more : Important Formulas For CAT

CAT-level Questions:

1. The sum of the first 10 terms of an AP is 250. If the first term is 10, what is the common difference?

Answer: d = 5

2. In an AP, the 5th term is 17, and the 9th term is 29. What is the 13th term?

Answer: an = 41

3. The 10th term of an AP is 42, and the 15th term is 62. What is the sum of the first 20 terms?

Answer: Sn = 420Solution: Let the first term be 'a' and the common difference be 'd'. Then, the 10th term is a + 9d = 42, and the 15th term is a + 14d = 62. Subtracting the first equation from the second, we get 5d = 10, which gives d = 2. Then, the sum of the first 20 terms is (20/2)(2a + (20-1)d) = 10(2a + 19d) = 10(2a + 38) = 420.

4. In an AP, the 4th term is 15, and the 8th term is 23. What is the sum of the first 15 terms?

Answer: Sn = 450

5. The sum of the first 12 terms of an AP is 228. If the 6th term is 20, what is the 18th term?

Answer: an = 46

6. The sum of the first n terms of an AP is 15n - 19n^2. If the 10th term is 11, what is the 20th term?

Answer: an = 19

Solution: Let the first term be 'a' and the common difference be 'd'. Then, the 10th term is a + 9d = 11, and the 20th term is a + 19d. From the given sum formula, we have (1/2)n(3a - n + 1) = 15n - 19n^2. Substituting n = 10, we get a = 29/2. Substituting a = 29/2 and n = 20, we get the 20th term as a + 19d = (29/2) + 19d = 19. Solving for d, we get d = -1/2. Substituting d = -1/2 and a = 29/2, we get the 20th term as a + 19d = 29/2 - 19/2 = 19.

Read : CAT Exam Scores

Conclusion : 

Mastering Arithmetic Progression is crucial for acing CAT's arithmetic questions. Understanding key concepts, formulas, and properties helps solve problems efficiently. Practice a variety of questions to develop problem-solving skills and speed. By mastering AP, students can improve their chances of scoring well in the CAT exam and excel in various mathematical disciplines.