To find the boat's speed in still water, use the relationship between distance, time, and speed in upstream/downstream motion. Let \(b\) = boat's speed and \(s\) = stream's speed.

Step 1: Find upstream speed

Upstream, the boat travels 48 km in 8 hours:

This gives us:

Step 2: Find downstream speed

Downstream, the boat travels 48 km in 6 hours:

This gives us:

Step 3: Solve for boat's speed

Add equations (1) and (2):

Verification: Stream speed \(s = 8 - 7 = 1\,\text{km/h}\). Upstream: \(7 - 1 = 6\,\text{km/h} \checkmark\); Downstream: \(7 + 1 = 8\,\text{km/h} \checkmark\)

Answer: The boat's speed in still water is \(7\,\text{km/h}\) (Option C)

Work done by A in 1 hour = 1/20, by B = 1/30. Combined = 1/20 + 1/30 = 5/60 = 1/12. In 8 hours = 8/12 = 2/3. Remaining = 1/3... Wait, recalculating: Combined rate = 1/12, in 8 hours = 8/12 = 2/3 filled, remaining = 1/3. Check: (1/20 + 1/30)×8 = (5/60)×8 = 40/60 = 2/3. Remaining = 1/3. Error in options - closest is 1/15 if different scenario

Work per day: A = 1/15, B = 1/20. Combined = 1/15 + 1/20 = (4+3)/60 = 7/60. Time = 60/7 ≈ 8.57 days

Selling Price = 500 - 20% of 500 = 500 - 100 = ₹400. Profit = 400 - 300 = ₹100. Profit% = (100/300)×100 = 33.33%... Rechecking: 100/300 = 1/3 = 33.33%. Hmm, should be D.

78 = 2×3×13, 104 = 2³×13, 156 = 2²×3×13. HCF = 2×13 = 26

Distance = Train length + Bridge length = 180 + 420 = 600 m. Speed = 600/30 = 20 m/s = 20×3.6 = 72 km/h

Amount = P(1+R/100)² = 10000(1.1)² = 10000×1.21 = ₹12,100. CI = 12100 - 10000 = ₹2100

HCF × LCM = a × b. 6 × 180 = 30 × b. 1080 = 30b. b = 36

264 = 2³×3×11, 396 = 2²×3²×11, 528 = 2⁴×3×11. HCF = 2²×3×11 = 4×3×11 = 132

# Solution: Finding Pairs with Given HCF and LCM

When two numbers share a specific HCF and LCM, they can be expressed as multiples of the HCF, and their product equals the product of HCF and LCM.

Step 1: Express Numbers in Terms of HCF

Since HCF = 18, both numbers must be multiples of 18. Let the two numbers be \(18a\) and \(18b\), where \(a\) and \(b\) are coprime (HCF of \(a\) and \(b\) is 1).

Step 2: Use the HCF-LCM Product Formula

The fundamental property states that for any two numbers, their product equals HCF × LCM.

Step 3: Find Coprime Factor Pairs of 24

We need pairs \((a,b)\) such that \(ab = 24\) and HCF\((a,b) = 1\) (coprime pairs).

Factorizations of 24: \(1 \times 24\), \(2 \times 12\), \(3 \times 8\), \(4 \times 6\)

Checking which are coprime:

- HCF\((1, 24) = 1\) ✓

- HCF\((2, 12) = 2\) ✗

- HCF\((3, 8) = 1\) ✓

- HCF\((4, 6) = 2\) ✗

Step 4: Find the Number Pairs

The coprime pairs are \((1, 24)\) and \((3, 8)\), giving us:

- Numbers: \(18 \times 1 = 18\) and $