Mixed Again

Simplification

Simplify:
$\frac{3}{4} + \frac{2}{3} - \frac{1}{2}$

Solution:
LCM of 4, 3, 2 = 12
= (9 + 8 − 6)/12 = 11/12

✅ Answer: 11/12

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Square Root and Cube Root

Find the cube root of 27 × 125.

Solution:
27 = 3³, 125 = 5³ → 27×125 = (3×5)³ = 15³
∛(27×125) = 15

✅ Answer: 15

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Surds and Indices

Simplify:
$\sqrt{50} + \sqrt{18}$

Solution:
√50 = √(25×2) = 5√2
√18 = √(9×2) = 3√2
Sum = (5 + 3)√2 = 8√2

✅ Answer: 8√2

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Ratio and Proportion

Divide Nu. 3,600 between Choki and Dema in the ratio 5:7.

Solution:
Total parts = 12
Choki = (5/12)×3600 = Nu. 1,500
Dema = (7/12)×3600 = Nu. 2,100

✅ Answer: Choki = Nu. 1500, Dema = Nu. 2100

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Chain Rule

If 5 workers can build a wall in 12 days, how many days will 8 workers take to build the same wall?

Solution:
More workers → less time
New time = (5×12)/8 = 7.5 days

✅ Answer: 7.5 days

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Pipes and Cistern

A pipe fills a tank in 6 hours, another empties it in 9 hours. If both work together, how long to fill the tank?

Solution:
Net rate = 1/6 − 1/9 = (3−2)/18 = 1/18
Time = 18 hours

✅ Answer: 18 hours

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Boats and Streams

Karma rows upstream 24 km in 6 hours and returns downstream in 3 hours. What is the speed of the stream?

Solution:
Upstream speed = 24/6 = 4 km/h
Downstream = 24/3 = 8 km/h
Speed of stream = (8 − 4)/2 = 2 km/h

✅ Answer: 2 km/h

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Alligation or Mixture

Mix sugar at Nu. 40/kg with sugar at Nu. 60/kg to get 50 kg at Nu. 50/kg. How much of each is needed?

Solution:
Cheaper → 60 − 50 = 10
Costlier → 50 − 40 = 10
Ratio = 1:1 → 25 kg each

✅ Answer: 25 kg of each

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Logarithm

If log₁₀2 = 0.3010, find log₁₀8.

Solution:
log₁₀8 = log₁₀(2³) = 3×log₁₀2 = 3×0.3010 = 0.9030

✅ Answer: 0.9030

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Races and Games

Pema beats Sonam by 20 meters in a 200 m race. When Sonam runs 200 m, how far has Pema run?

Solution:
Pema's speed : Sonam's = 200 : 180 = 10 : 9
When Sonam runs 200 m, Pema = (10/9)×200 = 222.22 m

✅ Answer: ~222 meters

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Stocks and Shares

A 10% stock is bought at 90. Find the yield.

Solution:
Yield = (Dividend / Market Price) × 100 = (10 / 90)×100 = 11.11%

✅ Answer: 11.11%

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Probability

A card is drawn from a deck. What’s the probability it’s a queen or a heart?

Solution:
P(queen) = 4/52
P(heart) = 13/52
P(both queen & heart) = 1/52
So: 4 + 13 − 1 = 16 → 16/52 = 4/13

✅ Answer: 4/13

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True Discount

True discount on Nu. 600 due in 6 months at 10% p.a.?

Solution:
TD = (P × R × T) / (100 + (R×T))
= (600 × 10 × 0.5) / (100 + 5) = 3000 / 105 = Nu. 28.57

✅ Answer: Nu. 28.57

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Banker’s Discount

Banker’s Discount on Nu. 1,200 due in 1 year at 6% p.a.?

Solution:
BD = (P×R×T)/100 = 1200×6×1 /100 = Nu. 72

✅ Answer: Nu. 72

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Odd Man Out and Series

Find the odd one out:
3, 5, 11, 14, 17, 21

Solution:
All except 14 are prime.

✅ Answer: 14 (not prime)