Q&A: practice questions with answers

Concept: Solve the innermost bracket first; ÷ and × go left to right.

1. $\frac{18-14}{4}=\frac{4}{4}=1$
2. $8÷2\times 1=4$
3. $19\times 3=57$
4. $57\times 4=228$

Trick: The fraction part equals 1, so the answer is just $19\times 3\times 4=19\times 12=228$.

Answer: (C) 228

Concept: Inside the bracket do ÷ and × before +.

1. $\frac{15}{5}=3$
2. $\frac{16}{1}\times (9-8)=16$
3. Bracket = $3+16=19$
4. $12÷4=3\to 3\times 19=57$

Trick: Final answer must be a multiple of 3 (since $12÷4=3$ is a factor) – among the options only 57 and 69 qualify; $3\times 19=57$.

Answer: (A) 57

Concept: Mean proportional of a and b = $\sqrt{ab}$.

1. $12+6\sqrt{2}=6(2+\sqrt{2})$ and $8-4\sqrt{2}=4(2-\sqrt{2})$
2. $ab=24\times (2+\sqrt{2})(2-\sqrt{2})=24\times (4-2)=48$
3. $\sqrt{48}=\sqrt{16\times 3}=4\sqrt{3}$

Trick: Spot $(a+b)(a-b)=a^2-b^2$ whenever surd signs are opposite – the √ terms vanish.

Answer: (A) $4\sqrt{3}$

Concept: Convert percentages into actual numbers before combining groups.

1. Females = $360000-225000=135000$
2. Total literates = 35% of 360000 = 126000
3. Literate males = 17% of 225000 = 38250
4. Literate females = $126000-38250=87750\to \frac{87750}{135000}\times 100=65\%$

Trick: Alligation on %: males 17, females f, overall 35 with weights 225:135 = 5:3 → $5(35-17)=3(f-35)\to f=65$.

Answer: (D) 65%

Concept: Total = average × count; the newcomer's value = new total − old total.

1. 57 students: $57\times 13=741$
2. 58 persons: $58\times 14=812$
3. Teacher = $812-741=71$ years

Trick: Newcomer = new average + (old count × rise) = $14+57\times 1=71$.

Answer: (D) 71

Concept: Two discounts a% and b% equal one discount of $(a+b-\frac{ab}{100})\%$.

1. 10% and 15% → $10+15-1.5=23.5\%$
2. Single 25% vs 23.5% → gap = 1.5%
3. 1.5% of ₹600 = ₹9

Trick: Directly: difference = $(25-23.5)\%$ of bill. Never subtract the rates as $25-25=0$.

Answer: (D) ₹9

Concept: $\text{Speed (km/h)}=\text{distance in km}÷\text{time in hours}$.

1. Distance = 30 km 250 m = 30.25 km
2. Time = 25 min = $\frac{5}{12}$ h
3. $\text{Speed}=30.25÷\frac{5}{12}=30.25\times \frac{12}{5}=72.6$ km/h

Trick: Per-minute speed × 60: $\frac{30.25}{25}=1.21$ km/min → $1.21\times 60=72.6$.

Answer: (C) 72.6

Concept: Fewest rows ⇒ most trees per row ⇒ use the HCF.

1. $\text{HCF}(33,55,88)=11$
2. Rows: $\frac{33}{11}+\frac{55}{11}+\frac{88}{11}=3+5+8$
3. Total = 16 rows

Trick: Shortcut: total trees ÷ HCF = $176÷11=16$.

Answer: (D) 16

Concept: If two sums grow to equal amounts, the one invested longer starts smaller by the growth factor.

1. Chetan's money grows 10 years, Vipul's 9 years; factor per year = $1.08=\frac{27}{25}$.
2. $C\times {\left(\frac{27}{25}\right)}^{10}=V\times {\left(\frac{27}{25}\right)}^9\Rightarrow C\times \frac{27}{25}=V$ ⇒ C : V = 25 : 27
3. Total 52 parts = ₹23400 ⇒ 1 part = ₹450
4. Vipul = $27\times 450=$ ₹12150

Trick: Year gap = 1 ⇒ ratio is simply 100 : 108 = 25 : 27; the younger child (longer wait) gets the smaller share.

Answer: (B) 12150

Concept: Downstream = $x+4$, upstream = $x-4$; total time = 3 h.

1. $\frac{20}{x+4}+\frac{14}{x-4}=3$
2. Try $x=12$: $\frac{20}{16}+\frac{14}{8}=1.25+1.75=3$ ✔
3. So boat speed = 12 km/h.

Trick: Option testing is fastest: x must exceed 4 and make both denominators divide nicely – 12 fits at once.

Answer: (A) 12

Concept: For 2 years, CI − SI = interest on one year's SI.

1. One year's SI = $\frac{44}{2}=$ ₹22
2. CI − SI = 2 ⇒ rate = $\frac{2}{22}\times 100=\frac{100}{11}\%$
3. $P=\frac{\text{SI}\times 100}{R\times T}=\frac{44\times 100\times 11}{100\times 2}=$ ₹242

Trick: $P={\left(\frac{\text{SI}}{2}\right)}^2÷(\text{CI}-\text{SI})={22}^2÷2=484÷2=242$.

Answer: (D) 242

Concept: For equal distances at speeds x and y, average speed = $\frac{2xy}{x+y}$.

1. $x=24$, $y=36$
2. $2\times 24\times 36=1728$; $24+36=60$
3. $\frac{1728}{60}=28.8$ km/hr

Trick: Average must be below the simple mean 30 – only 28.8, 27.9, 26.3 qualify; formula picks 28.8.

Answer: (B) $28.8\mathrm{km}/\mathrm{hr}$

Concept: $\text{Profit}\%=(\text{error}÷\text{true weight given})\times 100$.

1. For 1000 g claimed, he gives $1000-390=610$ g.
2. Gain = 390 g on 610 g.
3. $\text{Profit}=\frac{390}{610}\times 100=\frac{3900}{61}=63\frac{57}{61}\%$

Trick: $\text{Profit}\%=\frac{x}{100-x}\times 100$ with $x=39\to \frac{3900}{61}$.

Answer: (C) $63\frac{57}{61}\%$

Concept: If $p\%$ of $x=N$, then $x=\frac{N\times 100}{p}$.

1. 2% of $x=240$
2. $x=\frac{240\times 100}{2}$
3. $x=12000$

Trick: 2% → 240 means 1% → 120, so 100% → 12000.

Answer: (A) 12000

Concept: Write each ratio as a fraction/decimal and compare.

1. $\frac{2}{3}\approx 0.67$, $\frac{5}{7}\approx 0.71$
2. $\frac{3}{4}=0.75$, $\frac{5}{9}\approx 0.56$
3. Smallest = $\frac{5}{9}$

Trick: All except 5:9 are more than half by a wide margin; $\frac{5}{9}$ is only slightly over 0.5.

Answer: (A) 5 : 9

Concept: $\text{Hemisphere volume}=\frac{2}{3}\pi r^3$.

1. $r=6\sqrt{3}\Rightarrow r^3=216\times 3\sqrt{3}=648\sqrt{3}$
2. $V=\frac{2}{3}\times 648\sqrt{3}\pi$
3. $V=432\sqrt{3}\pi {\text{cm}}^3$

Trick: $(\sqrt{3}{)}^3=3\sqrt{3}$ – remember this to avoid errors; then $\frac{2}{3}$ of $648=432$.

Answer: (C) $432\sqrt{3}\pi {\mathrm{cm}}^3$

Concept: $\text{Profit}=\text{profit}\%\times \text{cost price}$.

1. Profit = 20% of ₹315
2. = $\frac{315}{5}$
3. = ₹63

Trick: $20\%=\frac{1}{5}$ → just divide CP by 5.

Answer: (B) 63

Concept: Equal rate and time ⇒ the interest gap is earned by the principal gap alone.

1. Principal gap: $10200-5400=$ ₹4800
2. $360=\frac{4800\times R\times 3}{100}=144R$
3. $R=\frac{360}{144}=2.5\%$

Trick: Interest per year on ₹4800 = $\frac{360}{3}=120\to \frac{120}{4800}=2.5\%$.

Answer: (C) 2.5

Concept: Take total work = LCM; find work done per cycle of days.

1. Total = $\text{LCM}(12,20,15)=60$ units
2. Per day: A = 5, B = 3, C = 4
3. One 3-day cycle (C, B, A) = 12 units
4. $\frac{60}{12}=5$ cycles exactly ⇒ $5\times 3=15$ days

Trick: If total ÷ cycle-work is a whole number, $\text{days}=\text{cycles}\times \text{cycle length}$ – no leftover check needed.

Answer: (B) 15

Concept: For two numbers, $\text{LCM}\times \text{HCF}=$ product of the numbers.

1. $154\times 7=77\times x$
2. $x=\frac{154\times 7}{77}$
3. $x=2\times 7=14$

Trick: $\text{Other number}=(\text{LCM}÷\text{given number})\times \text{HCF}=(154÷77)\times 7=14$.

Answer: (D) 14