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# CAT Linear Equations Questions PDF [Most Important]

Linear Equations is a key topic in the CAT Quants section. Make sure you are aware of all the Important Concepts in CAT Quant (Linear Equation). One must not miss out on the questions on Linear Equations in the QA section. Linear Equations fall under the category of Algebra in the CAT Quants; You can also check out these CAT Linear Equation questions from the CAT Previous year papers. This post will look into some important Linear Equation questions for CAT. These are good sources of practice for CAT 2022 preparation. If you want to practice these questions, you can download these Important Linear Equation Questions for CAT (with detailed answers) PDF and the video solutions below, which are completely Free.

Question 1: Suppose you have a currency, named Miso, in three denominations: 1 Miso, 10 Misos and 50 Misos. In how many ways can you pay a bill of 107 Misos?

a) 17

b) 16

c) 18

d) 15

e) 19

Solution:

If two 50 Misos are used, the 107 can be paid in only 1 way.

If one 50 Miso is used, the number of ways of paying 107 is 6 – zero 10 Miso, one 10 Miso and so on till five 10 Misos.

If no 50 Miso is used, the number of ways of paying 107 is 11 – zero 10 Miso, one 10 Miso and so on till ten 10 Misos.

So, the total number of ways is 18

Question 2: A confused bank teller transposed the rupees and paise when he cashed a cheque for Shailaja, giving her rupees instead of paise and paise instead of rupees. After buying a toffee for 50 paise, Shailaja noticed that she was left with exactly three times as much as the amount on the cheque. Which of the following is a valid statement about the cheque amount?

a) Over Rupees 13 but less than Rupees 14

b) Over Rupees 7 but less than Rupees 8

c) Over Rupees 22 but less than Rupees 23

d) Over Rupees 18 but less than Rupees 19

e) Over Rupees 4 but less than Rupees 5

Solution:

Let the value of cheque be x Rs and y ps and the amount she received is y Rs and x ps.

After 50 ps is deducted she has the amount which is 3 times the amount on cheque,

So 100y+x-50=3(100x+y)    (After converting the amount in paise)

y= (299x+50)/97 = 3x+ (8x+50)/97

Now both x and y are integers, so from options we put x=18, (8x+50)/97 = 194/97 =2 which is an integer. Hence, D is the answer.

Question 3: The price of Darjeeling tea (in rupees per kilogram) is 100 + 0.10n, on the nth day of 2007 (n=1, 2, …, 100), and then remains constant. On the other hand, the price of Ooty tea (in rupees per kilogram) is 89 + 0.15n, on the nth day of 2007 (n = 1, 2, …, 365). On which date in 2007 will the prices of these two varieties of tea be equal?

a) May 21

b) April 11

c) May 20

d) April 10

e) June 30

Solution:

Price of Darjeeling tea on 100th day= 100+(0.1*100)=110
Price of Ooty tea on nth day= 89+0.15n
Let us assume that the price of both varieties of tea would become equal on nth day where n<=100
So
89+0.15n=100+0.1n
n=220 which does not satisfy the condition of n<=100
So the price of two varieties would become equal after 100th day.
89+0.15n=110
n=140
140th day of 2007 is May 20 (Jan=31,Feb=28,March=31,April=30,May=20)

Instructions

DIRECTIONS for the following two questions: Answer the questions on the basis of the information given below.

A certain perfume is available at a duty-free shop at the Bangkok international airport. It is priced in the Thai currency Baht but other currencies are also acceptable. In particular, the shop accepts Euro and US Dollar at the following rates of exchange:

US Dollar 1 = 41 Bahts

Euro 1= 46 Bahts

The perfume is priced at 520 Bahts per bottle. After one bottle is purchased, subsequent bottles are available at a discount of 30%. Three friends S, R and M together purchase three bottles of the perfume, agreeing to share the cost equally. R pays 2 Euros. M pays 4 Euros and 27 Thai Bahts and S pays the remaining amount in US Dollars.

Question 4: How much does R owe to S in Thai Baht?

a) 428

b) 416

c) 334

d) 324

Solution:

Total to be paid = 1248 Baht

Each has to pay 1248/3 = 416 Baht

R paid 92 Baht

M paid 184+27 = 211 Baht

So, R owes S 416 – 92 = 324 Baht

Question 5: How much does M owe to S in US Dollars?

a) 3

b) 4

c) 5

d) 6

Solution:

Total to be paid = 1248 Baht

Each has to pay 1248/3 = 416 Baht

R paid 92 Baht

M paid 184+27 = 211 Baht

So, R owes S 416 – 92 = 324 Baht

M owes S 416-211 Baht = 205 Baht = 5 US Dollars

Question 6: Which one of the following conditions must p, q and r satisfy so that the following system of linear simultaneous equations has at least one solution, such that p + q + r $\neq$ 0?
x+ 2y – 3z = p
2x + 6y – 11z = q
x – 2y + 7z = r

a) 5p -2q – r = 0

b) 5p + 2q + r = 0

c) 5p + 2q – r = 0

d) 5p – 2q + r = 0

Solution:

Substitute value of p,q,r in the options only option A satisfies .

5(x+2y-3z)-2(2x+6y-11z)-(x-2y+7z) = 5x+10y-15z-4x-12y+22z-x+2y-7z  = 0

Question 7: A leather factory produces two kinds of bags, standard and deluxe. The profit margin is Rs. 20 on a standard bag and Rs. 30 on a deluxe bag. Every bag must be processed on machine A and on Machine B. The processing times per bag on the two machines are as follows:

The total time available on machine A is 700 hours and on machine B is 1250 hours. Among the following production plans, which one meets the machine availability constraints and maximizes the profit?

a) Standard 75 bags, Deluxe 80 bags

b) Standard 100 bags, Deluxe 60 bags

c) Standard 50 bags, Deluxe 100 bags

d) Standard 60 bags, Deluxe 90 bags

Solution:

.Let x be no. of standard bags and y be no. of deluxe bags. According to given conditions we have 2 equations 4x+5y<=700 and 6x+10y<=1250. Here option A satisfies both the equations.

Question 8: A test has 50 questions. A student scores 1 mark for a correct answer, -1/3 for a wrong answer, and -1/6 for not attempting a question. If the net score of a student is 32, the number of questions answered wrongly by that student cannot be less than

a) 6

b) 12

c) 3

d) 9

Solution:

Let the number of questions answered correctly be x and the number of questions answered wrongly be y.

So, number of questions left unattempted = (50-x-y)

So, x – y/3 – (50-x-y)/6 = 32

=> 6x – 2y – 50 + x + y = 192 => 7x – y = 242 => y = 7x – 242

If x = 35, y = 3

If x = 36, y = 10

So, min. value of y is 3.

The number of wrongly answered questions cannot be less than 3.

Instructions

Directions for the following two questions: Answer the questions on the basis of the information given below.

In an examination, there are 100 questions divided into three groups A, B and C such that each group contains at least one question. Each question in group A carries 1 mark, each question in group B carries 2 marks and each question in group C carries 3 marks. It is known that the questions in group A together carry at least 60% of the total marks.

Question 9: If group B contains 23 questions, then how many questions are there in group C?

a) 1

b) 2

c) 3

d) Cannot be determined

Solution:

Group B contains 23 questions => Marks of group B = 46
Let the number of questions in A be x and in C be 77-x.
Marks of group A = x
So, x/(x+46+3*77-3x) >= 60%
=> 5x >= 3(277-2x)
=> 11x >= 831
=> x >= 75.54
=> x = 76 (min)
So, the possible number of questions in group C = 1.

Question 10: If group C contains 8 questions and group B carries at least 20% of the total marks, which of the following best describes the number of questions in group B?

a) 11 or 12

b) 12 or 13

c) 13 or 14

d) 14 or 15

Solution:

Let the number of questions in group B be x.
So, number of questions in group A = 92-x
Marks of group B = 2x
2x/(92-x+2x+24) >= 20%
=> 10x >= 116+x
=> 9x >= 116
=> x >= 12.88
From the options, x can be 13 or 14

Instructions

An airline has a certain free luggage allowance and charges for excess luggage at a fixed rate per kg. Two passengers, Raja and Praja have 60 kg of luggage between them, and are charged Rs 1200 and Rs 2400 respectively for excess luggage. Had the entire luggage belonged to one of them, the excess luggage charge would have been Rs 5400.

Question 11: What is the weight of Praja’s luggage?

a) 20 kg

b) 25 kg

c) 30 kg

d) 35 kg

e) 40 kg

Solution:

Let the limit be x and the rate of charge be k per kg.
Let the excess luggage with Raja be R kg.
So, excess luggage with Praja = 2R kg
Now, excess luggage with Raja + excess luggage with Praja = 60 – 2x
So, 3R = 60 – 2x => R = 20 – 2x/3 which was charged 1200 Also, if one person had the entire luggage, excess luggage would have been 60 – x, which would have been charged 5400.
So the charge for the excess of (20-$\frac{2x}{3}$) = k(20-$\frac{2x}{3}$) = 1200  ….(1)
Also, the charge for the excess of 60-x = k(60-x) = 5400 …..(2)
Dividing (1) by (2), we get
=>$\frac{\left(60-2x\right)}{3\times\ \left(60-x\right)}=\frac{1200}{5400}$
Solving this, x = 15 kg
So, Praja’s luggage = 35 kg

Question 12: What is the free luggage allowance?

a) 10 kg

b) 15 kg

c) 20 kg

d) 25kg

e) 30kg