Govt. Exams
Entrance Exams
After 30% discount: 1500 × 0.7 = ₹1050. After additional 10%: 1050 × 0.9 = ₹945
Filling rate = 1/10, Emptying rate = 1/15. Net rate = 1/10 - 1/15 = 3/30 - 2/30 = 1/30. Time = 30 hours
Let numbers be 4k and 5k. HCF = k = 8. So numbers are 32 and 40
When selling multiple articles at different profit/loss percentages with equal cost prices, we calculate the overall result by comparing total selling price to total cost price.
Step 1: Calculate selling price of first article (20% profit)
Cost price of first article = ₹1000
Selling price with 20% profit:
Step 2: Calculate selling price of second article (20% loss)
Cost price of second article = ₹1000
Selling price with 20% loss:
Step 3: Find total cost price and total selling price
Step 4: Calculate overall profit or loss
Since the total selling price equals the total cost price, there is no profit and no loss.
Answer: No profit, no loss (Option A)
Amount = P(1 + r/100)^t = 8000(1.1)^2 = 8000 × 1.21 = ₹9680. CI = 9680 - 8000 = ₹1680
96 = 2^5 × 3, 144 = 2^4 × 3^2, 192 = 2^6 × 3. HCF = 2^4 × 3 = 16 × 3 = 48
Downstream speed = 48/3 = 16 km/h. Upstream speed = 24/4 = 6 km/h. Speed in still water = (16+6)/2 = 11 km/h. (Closest option A is 10, likely ₹typo in question setup)
When a train crosses a platform, it must travel a distance equal to its own length plus the platform length. We'll use the relationship between speed, distance, and time.
Step 1: Convert speed to m/s
The train travels at \(60\,\text{km/h}\). Convert to metres per second:
Step 2: Calculate total distance traveled
The train crosses the platform in 30 seconds. Using \(\text{Distance} = \text{Speed} \times \text{Time}\):
Step 3: Apply the crossing condition
When a train crosses a platform, the total distance it travels equals the train's length plus the platform's length:
Step 4: Solve for train length
Answer: The length of the train is \(200\,\text{m}\) (Option A)
# 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 $
264 = 2³×3×11, 396 = 2²×3²×11, 528 = 2⁴×3×11. HCF = 2²×3×11 = 4×3×11 = 132
