If HCF(a, b) = 15 and LCM(a, b) = 450, and a = 75, find b.

The correct answer is A: 90. HCF × LCM = a × b. 15 × 450 = 75 × b. 6750 = 75b. b = 90

Which of the following numbers is divisible by both 6 and 8?

The correct answer is A: 48. For divisibility by both 6 and 8, the number must be divisible by LCM(6,8) = 24. Check: 48 ÷ 24 = 2 ✓. Option A is correct.

A manufacturer sells goods to a wholesaler at 20% profit. The wholesaler sells to a retailer at 25% profit. The retailer sells to the customer at 30% profit. If the customer pays ₹1,950, what is the manufacturer's cost price?

The correct answer is A: ₹1,000. Let manufacturer's CP = x. Wholesaler's CP = 1.2x. Retailer's CP = 1.2x × 1.25 = 1.5x. Customer pays = 1.5x × 1.3 = 1.95x = 1,950. So x = 1,000.

A boat travels 90 km upstream and 110 km downstream. The upstream journey takes 6 hours and downstream takes 5 hours. What is the speed of the boat in still water?

The correct answer is A: 18.5 km/h. # Solution: Speed of Boat in Still Water **Step 1: Define variables** Let b = speed of boat in still water (km/h) Let c = speed of current (km/h) **Step 2: Set up equations using Speed = Distance ÷ Time** For upstream (boat moves against current): - Speed upstream = b - c - 90 ÷ (b - c) = 6 - Theref

How many numbers between 50 and 150 are divisible by 7?

The correct answer is A: 14. To find how many numbers between 50 and 150 are divisible by 7, we identify the smallest and largest multiples of 7 in this range, then count them using the arithmetic sequence formula. **Step 1: Find the smallest multiple of 7 greater than 50** Divide 50 by 7: $$50 \div 7 = 7.14...$$ The next whole

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Q.221 Medium Numbers

A sells goods to B at 20% profit. B sells to C at 15% profit. C's cost price is Rs. 690. What is A's cost price?

ARs. 500

BRs. 550

CRs. 575

DRs. 600

Correct Answer: A. Rs. 500

Explanation:

C's CP = B's SP = 690. B's CP = 690/1.15 = 600. A's SP = B's CP = 600. A's CP = 600/1.2 = 500

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Q.222 Medium Numbers

A and B together can do work in 8 days. A alone can do it in 12 days. In how many days can B alone do it?

A20 days

B22 days

C24 days

D28 days

Correct Answer: C. 24 days

Explanation:

B's rate = 1/8 - 1/12 = (3-2)/24 = 1/24. B takes 24 days

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Q.223 Medium Numbers

A product's price is reduced by 25% then increased by 25%. Net change is?

A0% (no change)

B6.25% decrease

C6.25% increase

D25% decrease

Correct Answer: B. 6.25% decrease

Explanation:

Let price = 100. After -25% = 75. After +25% of 75 = 75 × 1.25 = 93.75. Net decrease = 6.25%

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Q.224 Medium Numbers

A principal becomes Rs. 1331 in 3 years at 10% p.a. compound interest. What was principal?

ARs. 1000

BRs. 1050

CRs. 1100

DRs. 1200

Correct Answer: A. Rs. 1000

Explanation:

A = P(1.1)^3 = 1.331P = 1331. P = 1000

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Q.225 Medium Numbers

A's income is 25% more than B's income. B's income is what percentage of A's income?

A75%

B80%

C85%

D90%

Correct Answer: B. 80%

Explanation:

If B = 100, then A = 125. B/A = 100/125 = 80%

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Q.226 Medium Numbers

A boat's speed in still water is 15 km/h. The speed of the current is 3 km/h. What is the time taken to travel 108 km downstream and return?

A13.5 hours

B14 hours

C15 hours

D16 hours

Correct Answer: C. 15 hours

Explanation:

When traveling downstream and upstream, the boat's effective speed changes due to the current's aid or resistance.

Step 1: Calculate Downstream Speed

Downstream, the current aids the boat's motion, so we add the current speed to the boat's speed in still water.

\text{Downstream speed} = \text{Boat speed} + \text{Current speed} = 15 + 3 = 18 \text{ km/h}

Step 2: Calculate Time for Downstream Journey

Using the formula: Time = Distance ÷ Speed, we find the time to travel 108 km downstream.

\text{Time (downstream)} = \frac{108}{18} = 6 \text{ hours}

Step 3: Calculate Upstream Speed

Upstream, the current opposes the boat's motion, so we subtract the current speed from the boat's speed in still water.

\text{Upstream speed} = \text{Boat speed} - \text{Current speed} = 15 - 3 = 12 \text{ km/h}

Step 4: Calculate Time for Upstream Journey

The boat must return the same 108 km against the current.

\text{Time (upstream)} = \frac{108}{12} = 9 \text{ hours}

Step 5: Calculate Total Time

Add the downstream and upstream times to get the total journey time.

\text{Total time} = 6 + 9 = 15 \text{ hours}

The answer is 15 hours.

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Q.227 Medium Numbers

A can complete a work in 10 days. B can complete the same work in 15 days. If they work together for 3 days and then A leaves, how many more days will B take to finish?

A5 days

B6 days

C7.5 days

D8 days

Correct Answer: C. 7.5 days

Explanation:

To solve this problem, we use the concept of work rates: if A completes work in 10 days, A's rate is $\frac{1}{10}$ of the work per day, and similarly for B.

Step 1: Find individual work rates

A completes the work in 10 days, so A's rate = $\frac{1}{10}$ work/day

B completes the work in 15 days, so B's rate = $\frac{1}{15}$ work/day

Step 2: Calculate work done together in 3 days

Combined rate when working together:

A + B = \frac{1}{10} + \frac{1}{15} = \frac{3}{30} + \frac{2}{30} = \frac{5}{30} = \frac{1}{6}

Work completed in 3 days:

W_{\text{completed}} = 3 \times \frac{1}{6} = \frac{3}{6} = \frac{1}{2}

Step 3: Find remaining work

W_{\text{remaining}} = 1 - \frac{1}{2} = \frac{1}{2}

Step 4: Calculate days B needs to finish alone

B works alone at rate $\frac{1}{15}$ work/day. For remaining work $\frac{1}{2}$:

\text{Days needed} = \frac{\text{Remaining work}}{B\text{'s rate}} = \frac{\frac{1}{2}}{\frac{1}{15}} = \frac{1}{2} \times \frac{15}{1} = \frac{15}{2} = 7.5

Answer: B will take 7.5 more days to finish the work (Option C)

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Q.228 Medium Numbers

A merchant buys goods at Rs. 75 per unit and sells at Rs. 100 per unit. His profit per unit and overall profit percentage are?

ARs. 25, 33.33%

BRs. 25, 25%

CRs. 20, 30%

DRs. 30, 40%

Correct Answer: A. Rs. 25, 33.33%

Explanation:

Profit per unit = 100 - 75 = 25. Profit% = (25/75) × 100 = 33.33%

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Q.229 Medium Numbers

Two trains of lengths 150m and 250m are moving towards each other at speeds of 45 km/h and 35 km/h. How long will they take to completely pass each other?

A18 seconds

B20 seconds

C24 seconds

D30 seconds

Correct Answer: A. 18 seconds

Explanation:

When two trains move towards each other, their relative speed is the sum of their individual speeds, and they must cover the combined length of both trains to completely pass each other.

Step 1: Find the relative speed

Since the trains are moving towards each other, we add their speeds:

\text{Relative speed} = 45 + 35 = 80\,\text{km/h}

Step 2: Convert relative speed to m/s

To work with the lengths given in meters, convert km/h to m/s by multiplying by $\frac{5}{18}$:

\text{Relative speed} = 80 \times \frac{5}{18} = \frac{400}{18} = \frac{200}{9}\,\text{m/s}

Step 3: Find the total distance to be covered

For the trains to completely pass each other, the total distance covered equals the sum of their lengths:

\text{Total distance} = 150 + 250 = 400\,\text{m}

Step 4: Calculate time using distance = speed × time

\text{Time} = \frac{\text{Distance}}{\text{Relative speed}} = \frac{400}{\frac{200}{9}} = 400 \times \frac{9}{200} = \frac{3600}{200} = 18\,\text{seconds}

Answer: The trains will take $18$ seconds to completely pass each other. (Option A)

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Q.230 Medium Numbers

A's salary is 20% less than B's. B's salary is what percentage more than A's?

A20%

B25%

C30%

D33.33%

Correct Answer: B. 25%

Explanation:

If B = 100, A = 80. Increase needed = 20. Percentage = (20/80) × 100 = 25%

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