Q&A: practice questions with answers

Concept: Weighted average $=\sum (\text{price}\times \text{weight})$ with weights as fractions.

1. $0.55\times 156=85.8$
2. $0.45\times 220=99$
3. Weighted average $=85.8+99=184.8$

Trick: $156+0.45\times (220-156)=156+28.8=184.8$.

Answer: (B) ₹184.80

Concept: Successive % growth works like compound interest: $P{(1+\frac{r}{100})}^n$.

1. Factor $={1.95}^2=3.8025$
2. Population 1950 $=348000\times 3.8025$
3. $=1323270$ → 13,23,270

Trick: Last digits: $348000\times 3.8025$ ends in …270 since $3.8025\times 1000=3802.5$ and $348\times 3802.5=1323270$.

Answer: (B) 13,23,270

Concept: Profit share ∝ capital × time (sum over each period).

1. Take capitals 30, 40, 100. X: $30\times 12=360$
2. Y: $40\times 6+20\times 6=360$
3. Z withdraws 40% after 3 months: $100\times 3+60\times 9=840$
4. Ratio $360:360:840=3:3:7$; Z $=\frac{7}{13}\times 23400=12600$

Trick: '9 months prior to completion' = after 3 months.

Answer: (A) 12,600

Concept: Solid hemisphere TSA = curved $2\pi r^2$ + flat base $\pi r^2=3\pi r^2$.

1. $r=16\text{cm}$, $r^2=256$
2. TSA $=3\pi \times 256=768\pi {\text{cm}}^2$

Trick: Solid hemisphere → $3\pi r^2$; hollow (curved only) → $2\pi r^2$.

Answer: (D) $768\pi {\text{cm}}^2$

Concept: Take total work = LCM(3, 9, 27) = 27 units; efficiencies 9, 3, 1 units/day.

1. Let total days $=n$. Nick works 1 day → 9 units.
2. Kevin works $n-2$ days → $3(n-2)$; Joe works $n$ days → $n$.
3. $9+3(n-2)+n=27\Rightarrow 4n=24\Rightarrow n=6$

Trick: All three for $n$ days $=13n$; subtract absences: $13n-9(n-1)-3\times 2=27\Rightarrow n=6$.

Answer: (D) 6

Concept: Mode = the value that occurs most often.

1. Counts: 4 → 5 times, 5 → 4 times, 7 → 3 times, 8 → 2 times; 2 and 9 once.
2. Highest frequency is for 4.

Trick: Tally quickly – mark each value once as you pass.

Answer: (D) 4

Concept: Write the numbers as $2k,7k,13k$; since 2, 7, 13 are pairwise co-prime, LCM $=2\times 7\times 13\times k$.

1. LCM $=182k=3094$
2. $k=\frac{3094}{182}=17$
3. HCF $=k=17$

Trick: Co-prime ratio terms → HCF = the common multiplier $k$.

Answer: (A) 17

Concept: Let one item cost $x$; the other costs $4800-x$. Write the total selling price as an equation.

1. $1.34x+0.90(4800-x)=5442$
2. $0.44x+4320=5442\Rightarrow 0.44x=1122\Rightarrow x=2550$
3. Other item $=4800-2550=2250$
4. Difference $=2550-2250=300$

Trick: If all were sold at 10% loss, SP would be ₹4320. Extra ₹1122 comes from $34\%+10\%=44\%$ of item 1, so item 1 $=\frac{1122}{0.44}=2550$.

Answer: (B) ₹300

Concept: Half-yearly compounding: rate per half-year $=\frac{1.1}{2}=0.55\%$. The January deposit earns for 2 half-years, the July deposit for 1 half-year.

1. Jan deposit: $7745[(1.0055{)}^2-1]=7745\times 0.01103025=85.43$
2. July deposit: $7745\times 0.0055=42.60$
3. Total interest $=85.43+42.60=128.03$

Trick: CI for 2 periods at $r\%$ $=2r+\frac{r^2}{100}=1.1+0.003025=1.103025\%$; add $0.55\%$ for the second deposit: $1.653025\%$ of 7745 $\approx 128.03$.

Answer: (B) ₹128.03

Concept: Divide each option by 41 and check the remainder.

1. $9430÷41=230$ exactly, since $41\times 230=9430$
2. $9879=41\times 240+39$; $9249=41\times 225+24$; $10113=41\times 246+27$ – not divisible

Trick: $9430=943\times 10$ and $943=41\times 23$ (since $41\times 23=943$).

Answer: (A) 9430

Concept: Use $(x-y{)}^2=(x+y{)}^2-4xy$.

1. $(x-y{)}^2={15}^2-4\times 36=225-144=81$
2. $x-y=\sqrt{81}=9$ (positive since $x>y$)
3. Check: $x=12,y=3$ ⇒ $12+3=15,12\times 3=36$

Trick: Find two numbers with product 36 and sum 15 directly: 12 and 3.

Answer: (B) 9

Concept: Crossing distance = sum of lengths; use relative speed (difference for same direction, sum for opposite).

1. Same direction: $78-62=16\text{km/h}=\frac{40}{9}\text{m/s}$; time $=4.5\times 60=270\text{s}$
2. $L_P+120=\frac{40}{9}\times 270=1200\Rightarrow L_P=1080\text{m}$
3. Opposite: $78+81=159\text{km/h}=159\times \frac{5}{18}=\frac{265}{6}\text{m/s}$
4. Time $=\frac{1080+245}{265/6}=\frac{1325\times 6}{265}=30\text{s}$

Trick: $1325=5\times 265$, so time $=5\times 6=30$ s.

Answer: (C) 30

Concept: Average $=\frac{\text{sum}}{\text{count}}$.

1. Sum $=8+12+10+14+6=50$
2. Average $=\frac{50}{5}=10$

Trick: Pair the deviations around 10: $-2,+2,0,+4,-4$ cancel out, so the average is 10.

Answer: (D) 10

Concept: Take total work = LCM(5, 55, 66) = 330 units.

1. Efficiencies: A $=\frac{330}{5}=66$, B $=\frac{330}{55}=6$, C $=\frac{330}{66}=5$ units/day
2. 4 days together: $(66+6+5)\times 4=308$ units
3. Left: $330-308=22$ units by B + C at 11/day ⇒ $\frac{22}{11}=2$ days
4. Total $=4+2=6$ days

Trick: Units method avoids fractions – always use LCM as total work.

Answer: (D) 6

Concept: Average speed $=\frac{\text{total distance}}{\text{total time}}$. Let first part $=d$, second part $=2d$.

1. Onward time $=\frac{d}{88}+\frac{2d}{16}=\frac{d}{88}+\frac{11d}{88}=\frac{12d}{88}=\frac{3d}{22}$
2. Total distance $=6d$; total time $=\frac{6d}{33}=\frac{2d}{11}=\frac{4d}{22}$
3. Return time $=\frac{4d}{22}-\frac{3d}{22}=\frac{d}{22}$
4. Return speed $=\frac{3d}{d/22}=66\text{km/h}$

Trick: Take $d=88$: onward time 12 h, total time 16 h, return time 4 h for 264 km ⇒ 66 km/h.

Answer: (C) 66

Concept: The largest circle inside a rectangle touches the two longer sides, so its diameter = the shorter side.

1. Diameter $=112\text{m}$
2. Circumference $=\pi d=\frac{22}{7}\times 112=22\times 16=352\text{m}$

Trick: Diameter = smaller side of the rectangle; the longer side (281) is irrelevant.

Answer: (A) 352

Concept: Take square roots of numerator and denominator separately.

1. $\sqrt{0.04}=0.2$ and $\sqrt{0.0004}=0.02$
2. $\frac{0.2}{0.02}=10$

Trick: $\sqrt{\frac{0.04}{0.0004}}=\sqrt{100}=10$ in one step.

Answer: (C) 10

Concept: Let pen $=p$, pencil $=c$ and solve the two equations.

1. $8p+2c=96$ and $4p+7c=72$
2. Double the second: $8p+14c=144$; subtract: $12c=48\Rightarrow c=4$
3. $8p=96-8=88\Rightarrow p=11$
4. $2p+8c=22+32=54$

Trick: Make the pen coefficients equal (8) and subtract to get pencils first.

Answer: (C) ₹54

Concept: Sum of interior angles of an $n$-sided polygon $=(n-2)\times {180}^{\circ }$.

1. $(n-2)\times 180=1800$
2. $n-2=10\Rightarrow n=12$

Trick: Divide the sum by 180 and add 2: $\frac{1800}{180}+2=12$.

Answer: (C) 12

Concept: Solve innermost brackets first, then multiplication, then addition, then division.

1. $2-6=-4$
2. $7\times (-4)=-28$
3. $30+(-28)=2$
4. $160÷2=80$

Trick: Watch the sign: $7\times (-4)$ is negative, so the curly bracket becomes just 2.

Answer: (A) 80