## Quantitative Reasoning: Introduction

Quantitative reasoning is the part of Logical Reasoning that requires us to think critically to understand the logic of the problem and then use the mathematical and statistical skills to solve that problem.

Let’s see some examples.

Quantitative Reasoning Question 1: On her walk through the park, Hamsa collected 50 colored leaves, all either maple or oak. She sorted them by category when she got home and found the following

(i) The number of red leaves with spots is even and positive.

(ii) The number of red oak leaves without any spot equals the number of red maple leaves without spots.

(iii) All non-red oak leaves have spots and there are five times as many of them as there are red-spotted oak leaves.

(iv) There are no spotted maple leaves that are not red.

(v) There are exactly 6 red spotted maple leaves.

(vi) There are exactly 22 maple leaves that are neither spotted nor red.

How many oak leaves did she collect?

A. 22

B. 17

C. 25

D. 18

Solution:

There are two types of leaves – Oak and Maple. Further, both the types are divided into two categories – red and non-red, which can be spotted and non-spotted.

There are no spotted maple leaves that are not red. There are exactly 6 red spotted maple leaves. There are exactly 22 maple leaves that are neither spotted nor red. The number of red oak leaves without any spot equals the number of red maple leaves without spots. All non-red oak leaves have spots and there are five times as many of them as there are red-spotted oak leaves.

Total number of leaves = 50

Therefore, y + 5y + 6 + x + x + 22 = 50

⇒ 2x + 6y = 22

⇒ x + 3y = 11

The number of red leaves with spots is even and positive.

This implies ‘y’ is even and greater than 0. Therefore, y can be 2 only.

x + 6 = 11 ⇒ x = 5

The total number of oak leaves = y + 5y + x = 6y + x = 6×2 + 5 = 17

Therefore, B is the correct option.

Quantitative Reasoning Question 2: Three travelers are sitting around a fire and are about to eat a meal. One of them has five small loaves of bread, the second has three small loaves of bread. The third has no food but has eight coins. He offers to pay for some bread. They agree to share the eight loaves equally among the three travelers and the third traveler will pay eight coins for his share of the eight loaves. All loaves were of the same size. The second traveler (who had three loaves) suggests that he be paid three coins and the first traveler be paid five coins. The first traveler says that he should get more than five coins. How much should the first traveler get?

A. 5

B. 7

C. 1

D. None of these

Solution:

There are a total of 8 loaves of bread. Therefore, each traveler would get 8/3 loaves of bread.

So, the third traveler paid 8 coins for 8/3 loaves of bread.

Cost of 8/3 loaves of bread = 8 coins

This implies cost of 1 loaf of bread = 3 coins.

The first traveler gave (5 – 8/3) = 7/3 loaves of bread which are equivalent to 7 coins.

Therefore, B is the correct option.

Quantitative Reasoning Question 3: I have a total of \$1000. Item A costs \$110, item B costs \$90, item C costs \$70, item D costs \$40, and item E costs \$45. For every item D that I purchase, I must also buy two of item B. For every item A, I must buy one of item C. For every item E, I must also buy two of item D and one of item B. For every item purchased, I earn 1000 points and for every dollar not spent I earn a penalty of 1500 points. My objective is to maximize the points I earn. What is the number of items that I must purchase to maximize my points?

A. 13

B. 14

C. 15

D. 16

Solution:

For every item purchased, I earn 1000 points and for every dollar not spent I earn a penalty of 1500 points.

To maximize the points I earn, I would have to buy the maximum number of items possible with the least amount of money left.

> For every item D that I purchase, I must also buy two of item B.

Cost of 1 item of D and 2 items of B = \$220

The effective cost of 1 item = \$73.33

Total points earned = 3000

> For every item A, I must buy one of item C.

Cost of 1 item of A and 1 item of C = \$180

The effective cost of 1 item = \$90

Total points earned = 2000

> For every item E, I must also buy two of item D and one of item B.

Also, for every item of D that I purchase, I must buy two items of B. Therefore, I would have to buy a total of 5 items of B for every item of E that I purchase.

Cost of 1 item of E, 2 items of D, and 5 items of B = \$575

The effective cost of 1 item = \$71.875

Total points earned = 8000

> Item B costs \$90 and Item C costs \$70.

Case 1:

Buying 14 items of C  would cost us \$980 and we would be left with \$20.

Case 2:

Buying 1 item of E, 2 items of D and 5 items of B would cost us \$575. We would be left with \$425.

The best way to spend this money is by buying 6 items of C that would cost \$420 and I would be left with \$5. The total number of items purchased is 14.

Case 3:

Buying 13 items of C would cost us \$910 and we would be left with \$90. With this money, we can buy 1 item of B and we would be left with no money.

Clearly, Case 3 is the most efficient way of spending money. The total number of items purchased is 14. Hence, B is the correct option.

Quantitative Reasoning Question 4: Read the following information carefully and answer the questions based on it.

A cuboid of dimensions (6cm × 4cm × 1cm) is painted black on both the surfaces of dimensions (4cm × 1cm), green on the surfaces of dimensions (1cm × 6cm), and red on the surfaces of dimensions (6cm × 4cm). Now, the block is divided into various smaller cubes of sides 1cm each. The smaller cubes so obtained are separated.

1. How many cubes will have at least 3 sides painted?

A. 16

B. 12

C. 10

D. 8

2. How many cubes will be formed?

A. 6

B. 12

C. 16

D. 24

3. If cubes having black, as well as green color, are removed, then how many cubes will be left?

A. 4

B. 8

C. 16

D. 20

4. How many cubes will have 4 colored sides and 2 sides without any color?

A. 8

B. 4

C. 16

D. 10

Solution:

1. The cubes along the boundary will have at least 3 sides painted. The number of such cubes would be 16. So, A is the correct option.

2. Number of cubes formed = (Volume of the cuboid)/(Volume of a cube) = 24cm³/1cm³ = 24

Therefore, D is the correct option.

3. Only the four cubes that are cut from the corners will have both black as well as green color. The number of remaining cubes would be 20. So, D is the correct option.

4.  Cubes that are cut from the corners will have 4 colored sides and 2 sides without any color. Such cubes would be 4 in number. So, B is the correct option.

Let’s practice what we have learned. Read the following information and answer the questions.

Quantitative Reasoning Question 5: Two ants start climbing a slippery wall together, from the bottom of the wall. Ant A climbs at a rate of 3 inches per minute. Ant B climbs at a rate of 4 inches per minute. However, owing to the fact that the wall is slippery, ant A slips back 1 inch for every 2 inches it climbs and ant B slips back 1.5 inches for every 2 inches it climbs. Besides this, ant A takes a rest of 1 minute after every 2 minutes, and ant B takes a rest of 1 minute after every 3 minutes. (Assume that both ant A and ant B slip continuously while climbing.)

1. The two ants met each other at _______ inch.

2. If the widest gap achieved between the two ants, within the first 10 minutes is inches, then what is the value of N?

3. If ant B does not have any periods of rest, then how many times do the ants meet in the first 10 minutes?

4. When the ant A reach a height of 12 inches on the wall, the ant B is _______ inches behind ant A.

> Ant A climbs at a rate of 3 inches per minute and slips back 1 inch for every 2 inches it climbs. It takes a rest of 1 minute after every 2 minutes.

Ant A climbs a net 2 inches in the first minute, 1 inch in the second minute, and then rest for 1 minute. It continues to climb in this manner.

> Ant B climbs at a rate of 4 inches per minute and slips back 1.5 inches for every 2 inches it climbs. It takes a rest of 1 minute after every 3 minutes.

Ant B climbs a net 1 inch every minute and takes a rest of 1 minute after every 3 minutes.

The table below shows the distance climbed by each ant.

1. The two ants meet each other after reaching a height of 3 inches.

2. The widest gap between the two ants within the first 10 minutes is 3 inches.

3. If ant B does not take any rest break, the table would be as follows:

Both the ants meet 3 times in the first 10 minutes.

4. When the ant A reaches a height of 12 inches on the wall, the ant B is 3 inches behind ant A.