Alligation and Mixture, Numerical Problems, Numerical Inequalities

In what ratio must rice costing ₹ 60 per kg be mixed with rice costing ₹ 75 per kg so that the mixture costs ₹ 65 per kg?

1. a1 : 2
2. b4 : 5
3. c2 : 1
4. d3 : 2

8 kg of tea costing ₹ 300 per kg is mixed with 12 kg of tea costing ₹ 250 per kg. The mean price of the mixture is:

1. a₹ 270 per kg
2. b₹ 275 per kg
3. c₹ 280 per kg
4. d₹ 260 per kg

Wheat costing ₹ 20 per kg is mixed with wheat costing ₹ 30 per kg. Match the price of the mixture (List I) with the ratio cheaper : dearer in which the two are mixed (List II):

1. a(i)-R, (ii)-S, (iii)-P, (iv)-Q
2. b(i)-S, (ii)-R, (iii)-Q, (iv)-P
3. c(i)-R, (ii)-S, (iii)-Q, (iv)-P
4. d(i)-S, (ii)-R, (iii)-P, (iv)-Q

Assertion (A): Sugar costing ₹ 40 per kg mixed with sugar costing ₹ 60 per kg in the ratio 1 : 1 gives a mixture worth ₹ 50 per kg.

Reason (R): The mean price of a mixture of two varieties always lies between the prices of the two varieties.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

How many litres of water must be added to 60 litres of milk costing ₹ 50 per litre so that the mixture is worth ₹ 40 per litre? (Take water as free of cost.)

1. a12 litres
2. b240 litres
3. c15 litres
4. d20 litres

Rice of variety A costing ₹ 40 per kg is mixed with variety B costing ₹ 55 per kg. The mixture is sold at ₹ 52.80 per kg at a profit of 10%. In what ratio were A and B mixed?

1. a8 : 7
2. b7 : 8
3. c11 : 64
4. d1 : 1

Assertion (A): If a boat goes downstream at \(u\) km/h and upstream at \(v\) km/h, then the speed of the stream is \(\dfrac{u - v}{2}\) km/h.

Reason (R): If the speed of the boat in still water is \(b\) km/h and the speed of the stream is \(s\) km/h, then \(u = b + s\) and \(v = b - s\).

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

A man rows 30 km downstream in 2 hours and 18 km upstream in 2 hours. His speed in still water and the speed of the stream are, respectively:

1. a15 km/h and 9 km/h
2. b3 km/h and 12 km/h
3. c12 km/h and 6 km/h
4. d12 km/h and 3 km/h

1. I. 'A beats B by 20 m in a 200 m race' means that when A finishes, B has covered 180 m.
2. II. For a boat in a flowing stream, its upstream speed is greater than its downstream speed.
3. III. The speed of a boat in still water is half the sum of its downstream and upstream speeds.

Which of the statements above about races and about boats and streams is/are correct?

1. aOnly I
2. bOnly II and III
3. cOnly I and III
4. dI, II and III

A motorboat whose speed in still water is 18 km/h takes 1 hour more to go 24 km upstream than to return 24 km downstream. The speed of the stream is:

1. a6 km/h
2. b4 km/h
3. c3 km/h
4. d9 km/h

1. I. If a pipe can fill a tank in \(x\) hours, it fills \(\dfrac{1}{x}\) of the tank in one hour.
2. II. If two pipes can fill a tank in \(a\) hours and \(b\) hours, together they fill it in \(\dfrac{a + b}{2}\) hours.
3. III. If an inlet pipe fills a tank in 4 hours and an outlet pipe empties it in 6 hours, then with both open the empty tank fills in 10 hours.

Which of the statements above about pipes and cisterns is/are correct?

1. aOnly I and III
2. bOnly I
3. cOnly II
4. dOnly I and II

A pipe can fill a tank in 6 hours. Because of a leak at the bottom, it takes 8 hours to fill the tank. The leak alone can empty the full tank in:

1. a2 hours
2. b14 hours
3. c48 hours
4. d24 hours

Match each pair of pipes (List I) with the time they take together to fill the empty tank (List II):

1. a(i)-S, (ii)-R, (iii)-P, (iv)-Q
2. b(i)-R, (ii)-S, (iii)-Q, (iv)-P
3. c(i)-R, (ii)-S, (iii)-P, (iv)-Q
4. d(i)-R, (ii)-P, (iii)-S, (iv)-Q

Pipes A and B can fill a tank in 10 hours and 15 hours respectively. Both are opened together on the empty tank, and A is closed after 3 hours. How many more hours does B take to fill the tank?

1. a7.5 hours
2. b12 hours
3. c10.5 hours
4. d6 hours

In a 1 km race, A beats B by 40 m or by 8 seconds. The time A takes to finish the race is:

1. a200 seconds
2. b192 seconds
3. c208 seconds
4. d160 seconds

On sports day, Riya beats Sara by 20 m in a 200 m race, and in another 200 m race Sara beats Tara by 20 m. If all three run at the same speeds as before, by how many metres will Riya beat Tara in a 200 m race?

1. a40 m
2. b36 m
3. c20 m
4. d38 m

1. I. \(2^{10} > 10^{3}\)
2. II. \(\dfrac{11}{13} < \dfrac{13}{15}\)
3. III. \(3^{4} < 4^{3}\)

Which of the numerical inequalities above is/are correct?

1. aOnly I and II
2. bOnly II
3. cOnly I and III
4. dI, II and III

Assertion (A): If \(-3x + 7 \ge 1\), then \(x \ge 2\).

Reason (R): Multiplying or dividing both sides of an inequality by a negative number reverses the inequality sign.

1. aBoth A and R are true, and R is the correct explanation for A
2. bBoth A and R are true, but R is NOT the correct explanation for A
3. cA is true, but R is false
4. dA is false, but R is true

Which of the following numbers is the greatest?

1. a\(2^{300}\)
2. b\(3^{200}\)
3. c\(7^{100}\)
4. d\(6^{100}\)

A taxi charges a fixed fee of AED 12 plus AED 2.50 per km. Rahul can spend at most AED 62 on the ride. What is the greatest distance he can travel?

1. a24.8 km
2. b29.6 km
3. c20 km
4. d19 km