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**Easy Level**

These questions are targeted to improve your knowledge on basic concepts, though easy questions are rare in CAT. These are extremely important for conceptual understanding at the foundation level.

**Example 1:** **The average of the first ten whole numbers is **

(a) 4.5

(b) 5

(c) 5.5

(d) 4

**Ans:** (a)

**Solution:** Required average = (0 + 1 + 2 + … + 9)/10 = 45/10 = 4.5**Example 2:** **The average of the first ten prime numbers is**

(a) 15.5

(b) 12.5

(c) 10

(d) 12.9

**Ans:** (d)

**Solution:** Required average = (2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29)/10

= 129/10 = 12.9

**Example 3:** **There are three fractions A, B and C. If A = 1/5 and B = 1/8 and the average of A, B and C is 1/10. What is the value of C?**

(a) –1/20

(b) – 1/60

(c) –1/30

(d) – 1/40

**Ans:** (d)

**Solution:** 1/5 + 1/8 + C = 3* 1/10

C = –1/40.

**Example 4:** **The average age of 10 men is increased by 3 years when one of them, whose age is 54 years is replaced by a woman. What is the age of the woman?**

(a) 68 years

(b) 82 years

(c) 72 years

(d) 84 years

**Ans:** (d)

**Solution:** The woman’s age would be 10 * 3 = 30 years more than the age of the man she replaces.

► Age of the woman = 54 + 3* 10 = 84 years.

**Example 5:** **The average of 50 numbers is zero. Of them, at the most, how many may be greater than zero?**

**Solution:** If the average of 50 numbers is 0, then at most 49 of them can be greater than 0 and 50th number can be such that its’ negative value equals to the positive value of the first 49 numbers.

**Example 6:** **The average height of 13 people reduces by 2 cm if a person of height 184 cm is replaced by a new person. Find the height of the new person?**

(a) 154 cm

(b) 159 cm

(c) 197 cm

(d) 158 cm

**Ans:** (d)

**Solution:** Average got reduced by 2 cm.

► So, an overall decrease of 26 cm for 13 people.

► Height of new person = 184 – 13 × 2 = 184 – 26 = 158 cm.

**Example 7: The monthly salaries of two persons A and B are in the ratio of 3 : 5 respectively. If both of them received an increment of Rs. 250, then the ratio becomes 2 : 3. What were their respective salaries before the increment?**

(a) Rs. 850 & Rs. 1,275

(b) Rs. 700 & Rs. 1,050

(c) Rs.750 & Rs. 1,250

(d) Rs. 650 & Rs. 975

**Ans:** (c)

**Solution:** Let 3x and 5x be the salaries of A and B respectively.

⇒ 9x + 750 = 10x + 500

⇒ x = 250

Salary of A = 250 × 3 = Rs. 750 and B = 250 × 5 = Rs. 1,250.

Try yourself:The average weight of 3 boys A, B and C is 74 kg. Another boy D joins the group and the average now becomes 70 kg. If another boy E, whose weight is 3 kg more than that of D, replaces A then the average weight of B, C, D and E becomes 75 kg. The weight of A is

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*Answer can only contain numeric values

Try yourself:What is the average of the first seven natural number multiples of 11?

Check

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Try yourself:Ramesh receives the following scores in his math test: 78, 92, 83 and 99. What score does he need in the next test in order to have an average of 90 in the five tests?

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**Medium Level**

Almost 70% of questions in CAT are of Medium based questions. Though the conceptually they seem easier, the trick is to solve the calculations faster & we curated problems for you to help you do problems easier.

**Example 1: With an average speed of 25 km/h, a train reaches its destination in time. If it goes with an average speed of 20 km/h, it is late by 1 hour. The length of the total journey is:**

(a) 90 km

(b) 100 km

(c) 120 km

(d) 80 km

**Ans:** (b)

**Solution:** The train needs to travel 60 minutes extra @ 20 kmph. Hence, it is behind by 20 km.

► The rate of losing distance is 5 kmph. Hence, the train must have travelled for 20/5 = 4 hours@ 25 kmph = 100 km.

► Alternatively, you can also see that 20% drop in speed results in 25% increase in time. Hence, the total time required is 4 hours @ 25 kmph so 100 kilometres.

► Alternatively, solve through options.

**Example 2: ****In hotel Clarks, the rooms are numbered from 101 to 150 on the first floor, 201 to 240 on the second floor and 316 to 355 on the third floor. In the month of May 2018, the room occupancy was 50% on the first floor, 50% on the second floor and 30% on the third floor. If it is also known that the room charges are ₹ 2000, ₹1000 and ₹1500 on each of the floors, then find the average income per room (in ₹) for the month of May 2017.**

(a) 676.92

(b) 880.18

(c) 783.3

(d) 650.7

**Ans:** (a)

**Solution:** The number of rooms is 50 + 40 + 40 = 130 on the three floors respectively.

► Total revenues are: 25 * 2000 + 20 * 1000 + 12 * 1500 = 88000.

► Hence, the required average = 88000/130 = 676.92

**Example 3: ****The average weight of 10 men is decreased by 5 kg when one of them weighing 100 kg is replaced by another person. This new person is again replaced by another person, whose weight is 10 kg lower than the person he replaced. What is the overall change in the average due to this dual change?**

(a) 5 kg

(b) 6 kg

(c) 12 kg

(d) 15 kg

**Ans:** (b)

**Solution:** The weight of the second man is 50 kg and that of the third is 40 kg.

► Hence, the net result is a drop of 60 for 10 people.

► Hence, 6 kg is the drop in the average.

**Example 4: ****The average price of 3 precious diamond-studded platinum thrones is ₹ 97610498312. If their prices are in the ratio 4:7:9. The price of the cheapest is **

(a) 5, 65, 66, 298.972

(b) 5, 85, 66, 29, 8987.2

(c) 58, 56, 62, 889.72

(d) None of these

**Ans:** (b)

**Solution:** The total price of the three stones would be 97610498312 * 3 = 292831494936.

► Since this price is divided into three stones in the ratio of 4: 7: 9, the price of the cheapest one would be = (4 * 2928314936/20) = 58566298987.2

**Example 5:**** The average age of a group of 15 persons is 25 years and 5 months. Two persons, each 40 years old, left the group. What will be the average age of the remaining persons in the group?**

(a) 23.17 years

(b) 24.25 years

(c) 25.35 years

(d) 25 years

**Ans:** (a)

**Solution:** (15 * 305 – 2 * 480)/13 = 278.07 months or 23.17 years.

**Example 6: ****There are 24 students in a class whose average marks in a subject, the maximum marks of which is 100, is 89. If 3 students leave the class, then what is the maximum by which the average could go up?**

(a) 10.7

(b) 10.5

(c) 11.2

(d) 11

**Ans:** (d)

**Solution:** Sum of marks for 24 students = 24 × 89 = 2136

► Average marks of a student cannot increase beyond 100.

► So, total marks for 21 students cannot exceed 2100.

► So, maximum increase in average = 100 – 89 = 11.

**Example 7: ****Ajay started a firm with a capital of Rs. 28,000. After 5 months, Boman joined him and invested Rs. 40,000 in the firm. Chirag was also added as a new partner with an individual investment of Rs. 56,000 after 7 months of commencement. If at the end of the year, the profit of the firm is Rs. 32,000, what is the share of Boman? **

(a) Rs. 12,000

(b) Rs. 8,000

(c) Rs. 14,000

(d) Rs. 10,000

**Ans:** (d)

**Solution:** The ratio in which profit will be shared among Ajay, Boman and Chirag = 28000 × 12 **: **40000 × 7 **: **560000 × 5 = 6 : 5 : 5

Hence, share of Boman = 5 / 16 × 32000 = Rs. 10,000.

Try yourself:A is the average of a series of n consecutive odd numbers starting with x and B is the average of n consecutive odd numbers starting with x + 4. The value of (B – A)

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Try yourself:20 persons went to a hotel for a combined dinner party. 15 of them spent ₹ 90 each on their dinner and the rest spent ₹ 30 more than the average expenditure of all the 20. What was the total money spent (in ₹) by them?

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Try yourself:One-fifth of a certain journey is covered at the rate of 20 km/h, one-fourth at the rate of 50 km/h and the rest at 55 km/h. Find the average speed for the whole journey.

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**Hard Level**

Around 25% of these type questions come in CAT - If your target is above 95%ile, we recommend you to solve these questions as well.

**Example 1: ****A salesman gets a bonus according to the following structure: If he sells articles worth ₹ x then he gets a bonus of ₹(x/10 – 1000). In the month of January, the value of his sales was ₹10000, in February it was ₹12000, from March to November it was ₹30000 for every month and in December it was ₹12000. Apart from this, he also receives a basic salary of ₹3000 per month from his employer. Find his average income per month (in ₹) during the year. **

(a) 4533

(b) 4517

(c) 4532

(d) 4668

**Ans****:** (a)

**Solution:** Replace x with the sales value to calculate the bonus in a month. Bonus = 0 in January, 200 in February 2000 each from March to November and 200 in December.

► Hence, his Total bonus = 0 + 200 + 2000 x 9 + 200 = 18400.

► Salary for the year = 3000 x 12. Total annual income = 36000+18400 = 54400.

► Hence, the average monthly income = 4533.33.

Option (a) is closest and hence is the correct answer.

**Example 2: ****Ramu appears in six different papers in his semester examination, where the maximum marks were 50 for each paper. His marks in these papers are in the proportion of 8 : 9 : 10 : 13 : 14 : 15. Considering his aggregate in all the papers together, he fails to obtain 50% of the total marks. What is the minimum possible additional marks Ramu should get to obtain 50% of the total marks, given that he got integral marks in each paper?**

(a) 81

(b) 57

(c) 12

(d) 18

**Ans:** (b)

**Solution:** Marks obtained should be less than 50 in each paper.__There are three cases possible: __(a) Marks are 8, 9, 10, 13, 14, 15.

(b) Marks are 16, 18, 20, 26, 28, 30.

(c) Marks are 24, 27, 30, 39, 42, 45.

► If we take the case (a), then he required 81 marks more to get 50% of the total.

► If we take the case (c), then he got more than 50% marks.

► In case (b) he required 12 marks more to get 50% marks, which is less than the case (a). So this is the right answer.

Try yourself:A, B and C invested their money in the ratio 3 : 6 : 7. If the total amount invested by them was Rs. 80,000 and the profit earned was 40% of the amount invested. Then what was the sum of the shares of B and C in the profit?

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Try yourself:In the month of January of a certain year, the average daily expenditure of an organisation was ₹ 60. For the first 15 days of the month, the average daily expenditure was ₹ 80 and for the last 17 days, ₹50. Find the amount spent by the organisation on the 15th of the month.

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Try yourself:A man covers 1/4th of his journey by cycle at 40 km/h, the ½ of the remaining by car at 20 km/h, and the rest by walking at 10 km/h. Find his average speed during the whole journey.

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