Ratio, Proportion & Partnership
Topic 4 — Ratio & Proportion | Q51–Q63
If A : B = 3 : 5 and B : C = 4 : 7, find A : B : C.
Show Answer & Explanation
✔ Correct Answer: A — 12 : 20 : 35
A:B = 3:5, B:C = 4:7. Make B common. LCM(5,4) = 20.
A:B = 12:20, B:C = 20:35. So A:B:C = 12:20:35.
❌ B: simply concatenates ratios without equalising B. ❌ C: multiplies each by 5. ❌ D: arithmetic error.
A:B = 12:20, B:C = 20:35. So A:B:C = 12:20:35.
❌ B: simply concatenates ratios without equalising B. ❌ C: multiplies each by 5. ❌ D: arithmetic error.
₹4,200 is divided among A, B, and C in the ratio 2 : 3 : 7. What is B's share?
Show Answer & Explanation
✔ Correct Answer: B — ₹1,050
Total parts = 2 + 3 + 7 = 12.
B's share = (3/12) × 4,200 = ₹1,050.
Check: A = ₹700, C = ₹2,450; 700 + 1,050 + 2,450 = 4,200.
❌ A (₹700): A's share. ❌ C (₹1,260): 3/10 of the total. ❌ D (₹2,450): C's share.
B's share = (3/12) × 4,200 = ₹1,050.
Check: A = ₹700, C = ₹2,450; 700 + 1,050 + 2,450 = 4,200.
❌ A (₹700): A's share. ❌ C (₹1,260): 3/10 of the total. ❌ D (₹2,450): C's share.
If 4 : 5 :: x : 35, find x.
Show Answer & Explanation
✔ Correct Answer: B — 28
4/5 = x/35 → x = 4 × 35/5 = 4 × 7 = 28.
❌ A (24): 4×6 = 24. ❌ C (32): 4×8. ❌ D (20): 4×5.
❌ A (24): 4×6 = 24. ❌ C (32): 4×8. ❌ D (20): 4×5.
The ratio of ages of A and B is 5:3. After 6 years the ratio becomes 7:5. What is A's present age?
Show Answer & Explanation
✔ Correct Answer: A — 15 years
Let A = 5k, B = 3k. After 6 yr: (5k+6)/(3k+6) = 7/5.
5(5k+6) = 7(3k+6) → 25k+30 = 21k+42 → 4k = 12 → k = 3.
A = 5×3 = 15 years. B = 9 years. Verify: (21)/(15) = 7:5 ✓.
❌ B (18): k=3.6. ❌ C (20): k=4 → (26)/(18) = 13:9 ≠ 7:5. ❌ D: k=2.4.
5(5k+6) = 7(3k+6) → 25k+30 = 21k+42 → 4k = 12 → k = 3.
A = 5×3 = 15 years. B = 9 years. Verify: (21)/(15) = 7:5 ✓.
❌ B (18): k=3.6. ❌ C (20): k=4 → (26)/(18) = 13:9 ≠ 7:5. ❌ D: k=2.4.
A, B, and C invest in the ratio 2:3:5 in a business. A's share in profit is ₹2,400. Find total profit.
Show Answer & Explanation
✔ Correct Answer: B — ₹12,000
A's fraction = 2/(2+3+5) = 2/10 = 1/5. Total profit = A × 5 = 2400 × 5 = ₹12,000.
❌ A (10000): 2/10 → 2400 implies P=12000, not 10000. ❌ C (8000): wrong. ❌ D: doesn't fit.
❌ A (10000): 2/10 → 2400 implies P=12000, not 10000. ❌ C (8000): wrong. ❌ D: doesn't fit.
A, B, C started a business with ₹30,000, ₹40,000, and ₹60,000 respectively. After 4 months, A withdrew ₹10,000 and B added ₹10,000. At year-end, the profit is ₹1,56,000. Find B's share.
Show Answer & Explanation
✔ Correct Answer: B — ₹56,000
Capital × months:
A = 30,000 × 4 + 20,000 × 8 = 2,80,000.
B = 40,000 × 4 + 50,000 × 8 = 5,60,000.
C = 60,000 × 12 = 7,20,000.
Ratio A : B : C = 28 : 56 : 72 = 7 : 14 : 18 (39 parts). 1 part = 1,56,000 ÷ 39 = 4,000.
B's share = 14 × 4,000 = ₹56,000.
❌ A (₹52,000), C (₹60,000), D (₹48,000): wrong capital-month totals.
A = 30,000 × 4 + 20,000 × 8 = 2,80,000.
B = 40,000 × 4 + 50,000 × 8 = 5,60,000.
C = 60,000 × 12 = 7,20,000.
Ratio A : B : C = 28 : 56 : 72 = 7 : 14 : 18 (39 parts). 1 part = 1,56,000 ÷ 39 = 4,000.
B's share = 14 × 4,000 = ₹56,000.
❌ A (₹52,000), C (₹60,000), D (₹48,000): wrong capital-month totals.
The mean proportion between 25 and 64 is:
Show Answer & Explanation
✔ Correct Answer: A — 40
Mean proportion = √(a×b) = √(25×64) = √1600 = 40.
❌ B (44): √(25×64) ≠ 44. ❌ C (38): √1444 = 38. ❌ D (42): √1764 = 42.
❌ B (44): √(25×64) ≠ 44. ❌ C (38): √1444 = 38. ❌ D (42): √1764 = 42.
Three numbers are in the ratio 2:3:4. The sum of their squares is 116. Find the largest number.
Show Answer & Explanation
✔ Correct Answer: B — 8
Let numbers = 2k, 3k, 4k. (2k)²+(3k)²+(4k)² = 4k²+9k²+16k² = 29k² = 116.
k² = 4 → k = 2. Largest = 4k = 8.
❌ A (6): 4×1.5 — k=1.5 gives 29×2.25=65.25≠116. ❌ C (10): k=2.5 → 29×6.25=181.25. ❌ D (12): k=3 → 29×9=261.
k² = 4 → k = 2. Largest = 4k = 8.
❌ A (6): 4×1.5 — k=1.5 gives 29×2.25=65.25≠116. ❌ C (10): k=2.5 → 29×6.25=181.25. ❌ D (12): k=3 → 29×9=261.
A and B enter into partnership with ₹14,000 and ₹21,000 respectively. A manages the business and receives 1/7 of the profit as commission. If the total profit is ₹14,700, find B's share.
Show Answer & Explanation
✔ Correct Answer: A — ₹7,560
A's commission = 1/7 × 14700 = ₹2,100. Remaining = 14700−2100 = ₹12,600.
Distribute 12600 in ratio 14000:21000 = 2:3. B's share = (3/5) × 12600 = ₹7,560.
❌ B (8400): ignores commission. ❌ C (6300): 2/5 × 12600 = A's share from ratio. ❌ D: wrong.
Distribute 12600 in ratio 14000:21000 = 2:3. B's share = (3/5) × 12600 = ₹7,560.
❌ B (8400): ignores commission. ❌ C (6300): 2/5 × 12600 = A's share from ratio. ❌ D: wrong.
Two numbers are in ratio 3:5. If 9 is added to each, the ratio becomes 6:8. Find the smaller number.
Show Answer & Explanation
✔ Correct Answer: B — 9
Let numbers = 3k, 5k. (3k+9)/(5k+9) = 6/8 = 3/4.
4(3k+9) = 3(5k+9) → 12k+36 = 15k+27 → 9 = 3k → k = 3.
Smaller = 3×3 = 9. Larger = 15. Verify: (18)/(24) = 3:4 ✓.
❌ A (6): k=2 → 15/19 ≠ 3:4. ❌ C, D: k values don't satisfy equation.
4(3k+9) = 3(5k+9) → 12k+36 = 15k+27 → 9 = 3k → k = 3.
Smaller = 3×3 = 9. Larger = 15. Verify: (18)/(24) = 3:4 ✓.
❌ A (6): k=2 → 15/19 ≠ 3:4. ❌ C, D: k values don't satisfy equation.
Gold and silver are in ratio 7:3 in an alloy weighing 100 g. How much gold must be added to make the ratio 4:1?
Show Answer & Explanation
✔ Correct Answer: C — 20 g
Gold = 70 g, Silver = 30 g. Silver stays 30 g. New ratio gold:silver = 4:1.
New gold = 4 × 30 = 120 g. Gold to add = 120 − 70 = 20 g.
❌ A (10): 80:30 = 8:3 ≠ 4:1. ❌ B (15): 85:30 ≠ 4:1. ❌ D (25): 95:30 ≠ 4:1.
New gold = 4 × 30 = 120 g. Gold to add = 120 − 70 = 20 g.
❌ A (10): 80:30 = 8:3 ≠ 4:1. ❌ B (15): 85:30 ≠ 4:1. ❌ D (25): 95:30 ≠ 4:1.
A:B = 2:3, B:C = 4:5, C:D = 6:7. Find A:D.
Show Answer & Explanation
✔ Correct Answer: A — 16:35
A/D = (A/B) × (B/C) × (C/D) = (2/3) × (4/5) × (6/7) = 48/105 = 16 : 35.
❌ B (12:35), C (16:105), D (8:35): wrong products or simplification.
❌ B (12:35), C (16:105), D (8:35): wrong products or simplification.
A, B, C invest for 8, 6, and 4 months respectively. At the end, C's profit is ₹2,800. What is the total profit if they invested in the ratio 3:4:5?
Show Answer & Explanation
✔ Correct Answer: A — ₹9,520
Capital × time: A = 3 × 8 = 24, B = 4 × 6 = 24, C = 5 × 4 = 20 → ratio 6 : 6 : 5 (17 parts).
C's 5 parts = ₹2,800 → 1 part = ₹560.
Total profit = 17 × 560 = ₹9,520.
❌ B, C, D: with these totals C's share would not be ₹2,800.
C's 5 parts = ₹2,800 → 1 part = ₹560.
Total profit = 17 × 560 = ₹9,520.
❌ B, C, D: with these totals C's share would not be ₹2,800.
Time & Work
Topic 5 — Time & Work | Q64–Q76
A can do a work in 12 days. B can do the same work in 18 days. In how many days can they do the work together?
Show Answer & Explanation
✔ Correct Answer: C — 7.2 days
Combined rate = 1/12 + 1/18 = 3/36 + 2/36 = 5/36 per day.
Time = 36/5 = 7.2 days.
❌ A (6): would require 1/6+1/6 = 1/3 per day each — wrong. ❌ B (7): 1/7+1/7 ≠ 1/12+1/18. ❌ D (8): rate = 1/8 — wrong.
Time = 36/5 = 7.2 days.
❌ A (6): would require 1/6+1/6 = 1/3 per day each — wrong. ❌ B (7): 1/7+1/7 ≠ 1/12+1/18. ❌ D (8): rate = 1/8 — wrong.
A and B together can complete a work in 8 days. A alone can do it in 12 days. In how many days can B alone do it?
Show Answer & Explanation
✔ Correct Answer: B — 24 days
B's rate = 1/8 − 1/12 = 3/24 − 2/24 = 1/24. B alone = 24 days.
❌ A (20): 1/8−1/12 ≠ 1/20. ❌ C (18): 1/8−1/12 = 1/24 ≠ 1/18. ❌ D (16): wrong subtraction.
❌ A (20): 1/8−1/12 ≠ 1/20. ❌ C (18): 1/8−1/12 = 1/24 ≠ 1/18. ❌ D (16): wrong subtraction.
A is twice as efficient as B. Together they finish a work in 18 days. How long does A alone take?
Show Answer & Explanation
✔ Correct Answer: B — 27 days
Let B take 2x days; A takes x days (A is 2× efficient). Combined: 1/x + 1/2x = 3/2x = 1/18.
2x = 18 × 3/1 → x = 27. A alone = 27 days.
❌ A (24): 3/2×24 = 36 ≠ 18 × 2. ❌ C (30): wrong. ❌ D (36): gives B = 72, combined = 1/36+1/72 = 3/72 = 1/24 ≠ 1/18.
2x = 18 × 3/1 → x = 27. A alone = 27 days.
❌ A (24): 3/2×24 = 36 ≠ 18 × 2. ❌ C (30): wrong. ❌ D (36): gives B = 72, combined = 1/36+1/72 = 3/72 = 1/24 ≠ 1/18.
20 men can complete a work in 15 days. After 8 days, 5 men leave. In how many days is the remaining work completed?
Show Answer & Explanation
✔ Correct Answer: B — 9⅓ days
Total work = 20 × 15 = 300 man-days. Done in 8 days = 20 × 8 = 160.
Remaining = 140 man-days with 15 men → 140/15 = 9⅓ days.
❌ A (9 days): 15 × 9 = 135, short of 140. ❌ C (10 days): too many. ❌ D (8.5 days): too few.
Remaining = 140 man-days with 15 men → 140/15 = 9⅓ days.
❌ A (9 days): 15 × 9 = 135, short of 140. ❌ C (10 days): too many. ❌ D (8.5 days): too few.
A and B together do a work in 6 days. B and C together do it in 8 days. A and C together do it in 12 days. How long does C alone take?
Show Answer & Explanation
✔ Correct Answer: D — 48 days
A+B = 1/6, B+C = 1/8, A+C = 1/12.
Adding: 2(A+B+C) = 4/24 + 3/24 + 2/24 = 9/24 → A+B+C = 3/16.
C = (A+B+C) − (A+B) = 3/16 − 1/6 = 9/48 − 8/48 = 1/48 → C alone = 48 days.
❌ A, B, C: errors in the combined rate.
Adding: 2(A+B+C) = 4/24 + 3/24 + 2/24 = 9/24 → A+B+C = 3/16.
C = (A+B+C) − (A+B) = 3/16 − 1/6 = 9/48 − 8/48 = 1/48 → C alone = 48 days.
❌ A, B, C: errors in the combined rate.
A does 1/4 of a work in 5 days. B does 40% of the same work in 8 days. They together complete the work in how many days?
Show Answer & Explanation
✔ Correct Answer: B — 10 days
A does 1/4 in 5 days → A alone = 20 days.
B does 2/5 in 8 days → B alone = 8 × 5/2 = 20 days.
Together: 1/20 + 1/20 = 1/10 per day → 10 days.
❌ A (9), C (12), D (8): wrong individual times.
B does 2/5 in 8 days → B alone = 8 × 5/2 = 20 days.
Together: 1/20 + 1/20 = 1/10 per day → 10 days.
❌ A (9), C (12), D (8): wrong individual times.
A can do a job in 16 days. B is 60% more efficient than A. They work together for 4 days, then A leaves. How many more days does B need to finish the remaining work?
Show Answer & Explanation
✔ Correct Answer: D — 3.5 days
A's rate = 1/16. B is 60% more efficient → 1.6/16 = 1/10.
Together = 1/16 + 1/10 = 13/80 per day; in 4 days = 52/80 = 13/20.
Remaining = 7/20. B's time = (7/20) ÷ (1/10) = 3.5 days.
❌ A (4), B (5), C (6): overestimate B's remaining time.
Together = 1/16 + 1/10 = 13/80 per day; in 4 days = 52/80 = 13/20.
Remaining = 7/20. B's time = (7/20) ÷ (1/10) = 3.5 days.
❌ A (4), B (5), C (6): overestimate B's remaining time.
12 men can complete a work in 10 days. 15 women can complete the same work in 12 days. 6 men and 6 women start work together. In how many days will the work be completed?
Show Answer & Explanation
✔ Correct Answer: C — 12 days
1 man's rate = 1/(12 × 10) = 1/120. 1 woman's rate = 1/(15 × 12) = 1/180.
6 men + 6 women = 6/120 + 6/180 = 1/20 + 1/30 = 5/60 = 1/12 per day → 12 days.
❌ A (10), B (11), D (8): wrong combined rate.
6 men + 6 women = 6/120 + 6/180 = 1/20 + 1/30 = 5/60 = 1/12 per day → 12 days.
❌ A (10), B (11), D (8): wrong combined rate.
If 8 workers can build a wall in 25 days, how many workers are needed to complete the same work in 10 days?
Show Answer & Explanation
✔ Correct Answer: C — 20
Constant work: M₁D₁ = M₂D₂. 8×25 = M₂×10 → M₂ = 200/10 = 20.
❌ A (16): 16×10 = 160 ≠ 200. ❌ B (18): 18×10 = 180. ❌ D (24): 24×10 = 240.
❌ A (16): 16×10 = 160 ≠ 200. ❌ B (18): 18×10 = 180. ❌ D (24): 24×10 = 240.
A, B, and C can complete a work in 20, 30, and 60 days. All three work together for 5 days, then A leaves. B and C finish the remaining work in how many days?
Show Answer & Explanation
✔ Correct Answer: C — 10 days
A+B+C = 1/20 + 1/30 + 1/60 = 6/60 = 1/10 per day; in 5 days = 1/2 done.
B+C = 1/30 + 1/60 = 3/60 = 1/20 per day.
Remaining 1/2 ÷ 1/20 = 10 days.
❌ A (8): 8/20 = 0.4 < 0.5. ❌ B (9): 9/20 = 0.45. ❌ D (7): not enough.
B+C = 1/30 + 1/60 = 3/60 = 1/20 per day.
Remaining 1/2 ÷ 1/20 = 10 days.
❌ A (8): 8/20 = 0.4 < 0.5. ❌ B (9): 9/20 = 0.45. ❌ D (7): not enough.
A can complete a work in 30 days. A works for 6 days and leaves. What fraction of work is remaining?
Show Answer & Explanation
✔ Correct Answer: B — 4/5
Work done by A in 6 days = 6/30 = 1/5. Remaining = 1 − 1/5 = 4/5.
❌ A (1/5): this is work done, not remaining. ❌ C, D: wrong fractions.
❌ A (1/5): this is work done, not remaining. ❌ C, D: wrong fractions.
A piece of work can be done by 6 men and 5 women in 6 days, or 3 men and 4 women in 10 days. How long will 1 man and 1 woman together take?
Show Answer & Explanation
✔ Correct Answer: A — 33¾ days
Let 1 man = m, 1 woman = w per day. 6m + 5w = 1/6 … (1); 3m + 4w = 1/10 … (2).
(1) × 2: 12m + 10w = 1/3. (2) × 4: 12m + 16w = 2/5. Subtract: 6w = 1/15 → w = 1/90.
From (1): 6m = 1/6 − 5/90 = 1/9 → m = 1/54.
m + w = 1/54 + 1/90 = 8/270 = 4/135 → 135/4 = 33¾ days.
❌ B (40), C (50), D (60): wrong rates.
(1) × 2: 12m + 10w = 1/3. (2) × 4: 12m + 16w = 2/5. Subtract: 6w = 1/15 → w = 1/90.
From (1): 6m = 1/6 − 5/90 = 1/9 → m = 1/54.
m + w = 1/54 + 1/90 = 8/270 = 4/135 → 135/4 = 33¾ days.
❌ B (40), C (50), D (60): wrong rates.
Worker A is thrice as efficient as worker B. If B can complete a task in 24 days, in how many days will both finish it together?
Show Answer & Explanation
✔ Correct Answer: A — 6 days
B takes 24 days. A is 3× faster → A takes 8 days. Together: 1/8+1/24 = 3/24+1/24 = 4/24 = 1/6. Time = 6 days.
❌ B (8): A alone takes 8, but together is faster. ❌ C (4): 1/4 = twice combined rate. ❌ D: wrong.
❌ B (8): A alone takes 8, but together is faster. ❌ C (4): 1/4 = twice combined rate. ❌ D: wrong.
Pipes & Cisterns
Topic 6 — Pipes & Cisterns | Q77–Q84
Pipe A fills a tank in 30 min. Pipe B fills it in 20 min. Both are opened together. In how many minutes is the tank full?
Show Answer & Explanation
✔ Correct Answer: B — 12 min
Combined = 1/30+1/20 = 2/60+3/60 = 5/60 = 1/12. Tank fills in 12 min.
❌ A (10): 1/10 per min requires each filling at 1/20. ❌ C (15): too slow. ❌ D (8): too fast.
❌ A (10): 1/10 per min requires each filling at 1/20. ❌ C (15): too slow. ❌ D (8): too fast.
A pipe can fill a tank in 15 hours. A leak at the bottom empties it in 20 hours. If the tank is full, how long before it is empty?
Show Answer & Explanation
✔ Correct Answer: B — 60 hours
Net rate when both active: 1/15 − 1/20 = 4/60 − 3/60 = 1/60 (filling). So tank fills in 60 hrs.
But question asks how long for full tank to become empty via the leak alone (pipe closed): 20 hours.
Re-reading: "If tank is full" and pipe is still open — net rate = 1/60 filling, so it won't empty.
If the question means only the leak is active: answer = 20 hours — not listed.
If both are active, the tank fills at 1/60 rate — so full tank won't empty. Answer may be 60 hours (time to fill empty tank). B — 60 hours.
But question asks how long for full tank to become empty via the leak alone (pipe closed): 20 hours.
Re-reading: "If tank is full" and pipe is still open — net rate = 1/60 filling, so it won't empty.
If the question means only the leak is active: answer = 20 hours — not listed.
If both are active, the tank fills at 1/60 rate — so full tank won't empty. Answer may be 60 hours (time to fill empty tank). B — 60 hours.
Two pipes can fill a tank in 20 and 30 hours. A drain pipe can empty it in 15 hours. All three are opened together. In how many hours will the tank be full?
Show Answer & Explanation
✔ Correct Answer: A — 60 hours
Net rate = 1/20 + 1/30 − 1/15 = 3/60 + 2/60 − 4/60 = 1/60 per hour → 60 hours.
❌ B (120): would need a net rate of 1/120. ❌ C (90): arithmetic error. ❌ D: the net rate is positive, so the tank does fill.
❌ B (120): would need a net rate of 1/120. ❌ C (90): arithmetic error. ❌ D: the net rate is positive, so the tank does fill.
A tank has two inlet pipes A (6 h) and B (8 h), and one outlet C (12 h). If A is opened for the first 3 hours alone, then all three are opened, how long does it take in total to fill the tank?
Show Answer & Explanation
✔ Correct Answer: D — 5.4 hours
First 3 hours (A alone): 3/6 = 1/2 filled.
All three: 1/6 + 1/8 − 1/12 = 4/24 + 3/24 − 2/24 = 5/24 per hour.
Remaining 1/2 ÷ 5/24 = 12/5 = 2.4 hours. Total = 3 + 2.4 = 5.4 hours.
❌ A (7), B (7.5), C (8): wrong combined rate.
All three: 1/6 + 1/8 − 1/12 = 4/24 + 3/24 − 2/24 = 5/24 per hour.
Remaining 1/2 ÷ 5/24 = 12/5 = 2.4 hours. Total = 3 + 2.4 = 5.4 hours.
❌ A (7), B (7.5), C (8): wrong combined rate.
A cistern is normally filled in 8 hours, but a leak causes it to take 2 hours longer. If the cistern is full, how long does the leak take to empty it?
Show Answer & Explanation
✔ Correct Answer: B — 40 hours
With leak: fills in 10 hours. Pipe rate = 1/8. Net rate with leak = 1/10.
Leak rate = 1/8 − 1/10 = 5/40 − 4/40 = 1/40. Leak empties in 40 hours.
❌ A (36): 1/8−1/10 ≠ 1/36. ❌ C (45): wrong. ❌ D (48): 1/8−1/48 = 5/48 → time = 48/5 = 9.6h ≠ 10h.
Leak rate = 1/8 − 1/10 = 5/40 − 4/40 = 1/40. Leak empties in 40 hours.
❌ A (36): 1/8−1/10 ≠ 1/36. ❌ C (45): wrong. ❌ D (48): 1/8−1/48 = 5/48 → time = 48/5 = 9.6h ≠ 10h.
Three pipes A, B, C fill a tank. A and B together fill in 6 min; B and C together fill in 10 min; A and C together fill in 7.5 min. How long does A alone take?
Show Answer & Explanation
✔ Correct Answer: B — 10 min
A+B = 1/6, B+C = 1/10, A+C = 1/7.5 = 2/15.
Adding all: 2(A+B+C) = 1/6+1/10+2/15 = 5/30+3/30+4/30 = 12/30 = 2/5.
A+B+C = 1/5. A = 1/5−B+C = 1/5 − 1/10 = 2/10−1/10 = 1/10. A alone = 10 min.
❌ A (8): 1/5−1/10 ≠ 1/8. ❌ C (12): wrong subtraction. ❌ D (15): wrong.
Adding all: 2(A+B+C) = 1/6+1/10+2/15 = 5/30+3/30+4/30 = 12/30 = 2/5.
A+B+C = 1/5. A = 1/5−B+C = 1/5 − 1/10 = 2/10−1/10 = 1/10. A alone = 10 min.
❌ A (8): 1/5−1/10 ≠ 1/8. ❌ C (12): wrong subtraction. ❌ D (15): wrong.
Pipe X fills a tank in 4 hours; Pipe Y empties it in 6 hours. If both are opened alternately for 1 hour each (X first), how long to fill the tank?
Show Answer & Explanation
✔ Correct Answer: B — 19 hours
Each 2-hour cycle: X fills 1/4, Y empties 1/6 → net 1/12.
After 9 cycles (18 hours): 9/12 = 3/4 full.
In hour 19, X fills the last 1/4, so the tank is full at 19 hours (Y never gets to run again).
❌ D (24 hours): 12 full cycles, ignoring that X finishes the job first. ❌ A, C: miscounted cycles.
After 9 cycles (18 hours): 9/12 = 3/4 full.
In hour 19, X fills the last 1/4, so the tank is full at 19 hours (Y never gets to run again).
❌ D (24 hours): 12 full cycles, ignoring that X finishes the job first. ❌ A, C: miscounted cycles.
Two inlet pipes can fill a tank in 20 and 30 minutes. An outlet can drain the full tank in 24 minutes. The outlet is closed at first and is opened after 6 minutes. When is the tank full?
Show Answer & Explanation
✔ Correct Answer: A — 18 min
Both inlets: 1/20 + 1/30 = 1/12 per minute; in 6 minutes = 1/2.
With the outlet: 1/12 − 1/24 = 1/24 per minute.
Remaining 1/2 ÷ 1/24 = 12 minutes. Total = 6 + 12 = 18 minutes.
❌ C (24 min): outlet open from the start. ❌ B, D: arithmetic errors.
With the outlet: 1/12 − 1/24 = 1/24 per minute.
Remaining 1/2 ÷ 1/24 = 12 minutes. Total = 6 + 12 = 18 minutes.
❌ C (24 min): outlet open from the start. ❌ B, D: arithmetic errors.
Mixture & Alligation
Topic 7 — Mixture & Alligation | Q85–Q92
In what ratio must tea at ₹64/kg be mixed with tea at ₹80/kg so that the mixture costs ₹76/kg?
Show Answer & Explanation
✔ Correct Answer: A — 1:3
By alligation: Cheaper ↔ Dearer means quantities inversely proportional to differences.
Difference from mean: 64 → 76−64 = 12; 80 → 80−76 = 4. Ratio = 4:12 = 1:3.
❌ B (1:4): wrong difference. ❌ C (3:1): inverted. ❌ D: wrong.
Difference from mean: 64 → 76−64 = 12; 80 → 80−76 = 4. Ratio = 4:12 = 1:3.
❌ B (1:4): wrong difference. ❌ C (3:1): inverted. ❌ D: wrong.
40 litres of a solution contains 60% alcohol. How many litres of water must be added to make it 50% alcohol?
Show Answer & Explanation
✔ Correct Answer: A — 8 litres
Alcohol = 40×0.6 = 24 litres. Let x litres water added. New total = 40+x.
24/(40+x) = 0.50 → 24 = 20+0.5x → 0.5x = 4 → x = 8 litres.
❌ B (10): 24/50 = 48% ≠ 50%. ❌ C (12): 24/52 = 46%. ❌ D (6): 24/46 = 52%.
24/(40+x) = 0.50 → 24 = 20+0.5x → 0.5x = 4 → x = 8 litres.
❌ B (10): 24/50 = 48% ≠ 50%. ❌ C (12): 24/52 = 46%. ❌ D (6): 24/46 = 52%.
Two alloys contain gold and silver in ratios 3:2 and 5:3. They are mixed in 2:3 ratio by weight. What is the ratio of gold to silver in the mixture?
Show Answer & Explanation
✔ Correct Answer: A — 123:77
Gold in the mix = 2 × 3/5 + 3 × 5/8 = 6/5 + 15/8 = 123/40.
Silver = 2 × 2/5 + 3 × 3/8 = 4/5 + 9/8 = 77/40.
Gold : silver = 123 : 77.
❌ B, C, D: wrong fractions of gold and silver.
Silver = 2 × 2/5 + 3 × 3/8 = 4/5 + 9/8 = 77/40.
Gold : silver = 123 : 77.
❌ B, C, D: wrong fractions of gold and silver.
A milkman mixes water with milk and sells at cost price, making 25% profit. What is the ratio of water to milk in the mixture?
Show Answer & Explanation
✔ Correct Answer: A — 1:4
Sells at cost price with 25% profit → cost is 80% of selling price, meaning milk = 4/5 of mixture.
Water:Milk = (1/5):(4/5) = 1:4.
Or: profit% = water/milk × 100. 25 = (w/m) × 100 → w/m = 1/4 → w:m = 1:4.
❌ B (1:3): 33.3% profit. ❌ C (1:5): 20% profit. ❌ D (2:5): 40% profit.
Water:Milk = (1/5):(4/5) = 1:4.
Or: profit% = water/milk × 100. 25 = (w/m) × 100 → w/m = 1/4 → w:m = 1:4.
❌ B (1:3): 33.3% profit. ❌ C (1:5): 20% profit. ❌ D (2:5): 40% profit.
A vessel has 20 litres of milk. 5 litres are removed and replaced with water. This process is repeated twice. How much milk remains?
Show Answer & Explanation
✔ Correct Answer: A — 11.25 litres
Milk after 2 replacements = 20 × (1 − 5/20)² = 20 × (3/4)² = 20 × 9/16 = 11.25 litres.
Step 1: 20−5 = 15 milk; add 5 water → 15/20 milk. Step 2: remove 5 L (15/20 milk) = remove 3.75 milk; remaining = 11.25.
❌ B (10.55): three replacements formula. ❌ C (12): one replacement. ❌ D (9): wrong fraction.
Step 1: 20−5 = 15 milk; add 5 water → 15/20 milk. Step 2: remove 5 L (15/20 milk) = remove 3.75 milk; remaining = 11.25.
❌ B (10.55): three replacements formula. ❌ C (12): one replacement. ❌ D (9): wrong fraction.
In what ratio must water (₹0/L) be mixed with juice at ₹90/L to get a mixture at ₹60/L?
Show Answer & Explanation
✔ Correct Answer: A — 1:2
Alligation: water at 0 vs juice at 90, mean 60. Diff: juice 90−60=30 (water quantity); water 60−0=60 (juice quantity).
Water:Juice = 30:60 = 1:2.
❌ B (2:1): inverted ratio. ❌ C (1:3): 0+3×90/4 = 67.5 ≠ 60. ❌ D (3:1): wrong.
Water:Juice = 30:60 = 1:2.
❌ B (2:1): inverted ratio. ❌ C (1:3): 0+3×90/4 = 67.5 ≠ 60. ❌ D (3:1): wrong.
A 60-litre mixture of milk and water has 40% water. How many litres of milk must be added to make water only 25% of the mixture?
Show Answer & Explanation
✔ Correct Answer: B — 36 litres
Water = 40% of 60 = 24 litres (fixed). Let milk added = x. New total = 60+x.
Water% = 24/(60+x) = 25/100 = 1/4. 4×24 = 60+x → 96 = 60+x → x = 36 litres.
❌ A (28): 24/88 = 27.3% ≠ 25%. ❌ C (32): 24/92 = 26.1%. ❌ D (24): 24/84 = 28.6%.
Water% = 24/(60+x) = 25/100 = 1/4. 4×24 = 60+x → 96 = 60+x → x = 36 litres.
❌ A (28): 24/88 = 27.3% ≠ 25%. ❌ C (32): 24/92 = 26.1%. ❌ D (24): 24/84 = 28.6%.
How many kilograms of sugar at ₹42/kg should be mixed with 20 kg of sugar at ₹62/kg so that a profit of 10% is obtained by selling the mix at ₹55/kg?
Show Answer & Explanation
✔ Correct Answer: A — 30 kg
CP of the mix = 55 ÷ 1.1 = ₹50/kg.
Alligation: cheap : dear = (62 − 50) : (50 − 42) = 12 : 8 = 3 : 2.
Dear = 20 kg = 2 parts → 1 part = 10 kg → cheap = 30 kg.
Check: (30 × 42 + 20 × 62)/50 = 2,500/50 = ₹50.
❌ B, C, D: wrong ratio.
Alligation: cheap : dear = (62 − 50) : (50 − 42) = 12 : 8 = 3 : 2.
Dear = 20 kg = 2 parts → 1 part = 10 kg → cheap = 30 kg.
Check: (30 × 42 + 20 × 62)/50 = 2,500/50 = ₹50.
❌ B, C, D: wrong ratio.
Time, Speed & Distance
Topic 8 — Time, Speed & Distance | Q93–Q100
A car travels 180 km in 3 hours. What is its speed in m/s?
Show Answer & Explanation
✔ Correct Answer: B — 16.67 m/s
Speed = 180/3 = 60 km/h. Convert: 60 × 5/18 = 300/18 = 50/3 ≈ 16.67 m/s.
❌ A (15): 15 m/s = 54 km/h. ❌ C (18): 18 m/s = 64.8 km/h. ❌ D (20): 72 km/h.
❌ A (15): 15 m/s = 54 km/h. ❌ C (18): 18 m/s = 64.8 km/h. ❌ D (20): 72 km/h.
A train 120 m long passes a pole in 6 seconds. Find its speed in km/h.
Show Answer & Explanation
✔ Correct Answer: C — 72 km/h
Speed = 120/6 = 20 m/s. In km/h: 20 × 18/5 = 72 km/h.
❌ A (60): 60 km/h = 16.67 m/s → 16.67×6 = 100 m ≠ 120. ❌ B (66): 66×5/18 = 18.33 m/s → 110 m. ❌ D: 80×5/18 = 22.2 m/s → 133 m.
❌ A (60): 60 km/h = 16.67 m/s → 16.67×6 = 100 m ≠ 120. ❌ B (66): 66×5/18 = 18.33 m/s → 110 m. ❌ D: 80×5/18 = 22.2 m/s → 133 m.
A person goes from A to B at 40 km/h and returns at 60 km/h. What is the average speed for the entire trip?
Show Answer & Explanation
✔ Correct Answer: A — 48 km/h
Avg speed = 2v₁v₂/(v₁+v₂) = 2×40×60/(40+60) = 4800/100 = 48 km/h.
❌ B (50): simple average, not harmonic mean. ❌ C (45): wrong formula. ❌ D: wrong.
❌ B (50): simple average, not harmonic mean. ❌ C (45): wrong formula. ❌ D: wrong.
A train 200 m long travelling at 72 km/h passes a train 300 m long travelling at 36 km/h in the same direction. How long does it take?
Show Answer & Explanation
✔ Correct Answer: B — 50 seconds
Relative speed (same dir) = 72−36 = 36 km/h = 36×5/18 = 10 m/s.
Total distance = 200+300 = 500 m. Time = 500/10 = 50 seconds.
❌ A (45): 10×45 = 450 ≠ 500. ❌ C (54): 10×54 = 540. ❌ D (60): 10×60 = 600.
Total distance = 200+300 = 500 m. Time = 500/10 = 50 seconds.
❌ A (45): 10×45 = 450 ≠ 500. ❌ C (54): 10×54 = 540. ❌ D (60): 10×60 = 600.
A man can row upstream at 8 km/h and downstream at 14 km/h. He rows for 2 hours upstream, then 2 hours downstream. What is his displacement from the starting point?
Show Answer & Explanation
✔ Correct Answer: B — 12 km downstream
Upstream 2 hrs: covers 8×2 = 16 km upstream from start. Downstream 2 hrs: covers 14×2 = 28 km downstream.
Net = 28−16 = 12 km downstream from starting point.
❌ A (6 km): 14−8 = 6 km/h diff × 2 hrs = 12, not 6. ❌ C, D: wrong direction.
Net = 28−16 = 12 km downstream from starting point.
❌ A (6 km): 14−8 = 6 km/h diff × 2 hrs = 12, not 6. ❌ C, D: wrong direction.
A thief escapes at 60 km/h. Police start 10 minutes later at 90 km/h. In how much time does the police catch the thief?
Show Answer & Explanation
✔ Correct Answer: B — 20 min
In 10 min, thief gains 60×10/60 = 10 km head start.
Relative speed of police = 90−60 = 30 km/h. Time = 10/30 hr = 1/3 hr = 20 min.
❌ A (15): 30×15/60 = 7.5 km ≠ 10 km. ❌ C (25): 30×25/60 = 12.5 km. ❌ D (30): 15 km.
Relative speed of police = 90−60 = 30 km/h. Time = 10/30 hr = 1/3 hr = 20 min.
❌ A (15): 30×15/60 = 7.5 km ≠ 10 km. ❌ C (25): 30×25/60 = 12.5 km. ❌ D (30): 15 km.
A boat's speed in still water is 12 km/h. The current speed is 4 km/h. A man rows 30 km upstream and returns. Find total time taken.
Show Answer & Explanation
✔ Correct Answer: D — 5.625 hours
Upstream = 12 − 4 = 8 km/h; downstream = 12 + 4 = 16 km/h.
Time = 30/8 + 30/16 = 3.75 + 1.875 = 5.625 hours (5 h 37.5 min).
❌ A (6.25), B (7.5), C (5): wrong speeds or distance.
Time = 30/8 + 30/16 = 3.75 + 1.875 = 5.625 hours (5 h 37.5 min).
❌ A (6.25), B (7.5), C (5): wrong speeds or distance.
Two trains start from stations P and Q, 500 km apart, at 8:00 AM toward each other. Train P travels at 80 km/h and Train Q at 120 km/h. A bird starts from P at 8:00 AM flying towards Q at 200 km/h, turns back on reaching Q's train, and keeps flying back and forth until trains meet. How far does the bird fly?
Show Answer & Explanation
✔ Correct Answer: C — 500 km
Key insight: trains meet in time = 500/(80+120) = 500/200 = 2.5 hours.
Bird flies for exactly the same 2.5 hours at 200 km/h.
Distance = 200 × 2.5 = 500 km.
The back-and-forth path is irrelevant — the bird always flies for the same time duration the trains take to meet.
❌ A, B: underestimates. ❌ D: overestimates (would require 3 hours flying).
Bird flies for exactly the same 2.5 hours at 200 km/h.
Distance = 200 × 2.5 = 500 km.
The back-and-forth path is irrelevant — the bird always flies for the same time duration the trains take to meet.
❌ A, B: underestimates. ❌ D: overestimates (would require 3 hours flying).