Q&A: practice questions with answers

Concept: The expression has the form $a^2-2ab+b^2=(a-b{)}^2$.

1. Here $a=75$, $b=25$.
2. Value $=(75-25{)}^2={50}^2$.
3. ${50}^2=2500$.

Trick: Spot the middle term $-2\times 75\times 25$ — it signals $(a-b{)}^2$, no need to multiply each term.

Answer: (D) 2500

Concept: Work done = rate $\times$ time; take the tank as 1 (or LCM of 15 and 20 $=60$ units).

1. Tank $=60$ units; A fills $4$ units/min, B fills $3$ units/min.
2. In 4 min together: $4\times (4+3)=28$ units.
3. Remaining $60-28=32$ units by B alone: $\frac{32}{3}=10\frac{2}{3}$ min $=10$ min 40 s.
4. Total $=4+10\frac{2}{3}=14\frac{2}{3}$ min $=14$ min 40 s.

Trick: LCM-unit method avoids fractions: $\frac{2}{3}$ min $=40$ s.

Answer: (C) 14 min 40 sec

Concept: To cross a bridge, a train covers (its length + bridge length).

1. Distance $=100+150=250$ m.
2. Speed $=36\times \frac{5}{18}=10$ m/s.
3. Time $=\frac{250}{10}=25$ s.

Yaad rakhein: km/h to m/s: multiply by $\frac{5}{18}$ (36 km/h $=10$ m/s).

Answer: (B) 25

Concept: Area $=\pi r^2$, so area changes with the square of the radius.

1. New radius $=\frac{r}{2}$.
2. New area $=\pi {\left(\frac{r}{2}\right)}^2=\frac{1}{4}\pi r^2$.
3. Reduction $=1-\frac{1}{4}=\frac{3}{4}=75\%$.

Trick: Net % change $=a+b+\frac{ab}{100}=-50-50+\frac{2500}{100}=-75\%$.

Answer: (C) 75%

Concept: For a pure recurring part: $0.\overline{ab}=\frac{ab}{99}$.

1. $1.\overline{24}=1+\frac{24}{99}$.
2. $=\frac{99+24}{99}=\frac{123}{99}$.
3. Divide by 3: $\frac{41}{33}$.

Trick: Always reduce — $\frac{124}{99}$ and $\frac{123}{90}$ are traps for wrong rules.

Answer: (D) $\frac{41}{33}$

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

1. $ab=0.06\times 24=1.44$.
2. $\sqrt{1.44}=1.2$.

Trick: $\sqrt{1.44}=1.2$ because ${12}^2=144$ (two decimal places in 1.44 → one in the root).

Answer: (D) 1.2

Concept: Exterior angle at a vertex $={180}^{\circ }-$ interior angle $=$ sum of the two opposite interior angles.

1. $\angle C={180}^{\circ }-{70}^{\circ }-{70}^{\circ }={40}^{\circ }$.
2. Exterior $\angle A={180}^{\circ }-{70}^{\circ }={110}^{\circ }$.
3. Check: $\angle B+\angle C={70}^{\circ }+{40}^{\circ }={110}^{\circ }$.

Yaad rakhein: Interior and exterior angles at the same vertex form a linear pair (${180}^{\circ }$).

Answer: (A) ${110}^{\circ }$

Concept: Volume is conserved on recasting: $\frac{4}{3}\pi R^3=27\times \frac{4}{3}\pi r^3$; surface area $=4\pi r^2$.

1. ${12}^3=27r^3\Rightarrow r^3=64\Rightarrow r=4$ cm.
2. Original area $=4\pi (12{)}^2=576\pi$.
3. 8 small spheres: $8\times 4\pi (4{)}^2=512\pi$.
4. Ratio $=576:512=9:8$.

Trick: $27=3^3$ spheres → each radius $=\frac{R}{3}$; ratio $=R^2:8r^2=144:128$.

Answer: (B) 9 : 8

Concept: Square root of a decimal has half as many decimal places.

1. $0.0009=\frac{9}{10000}$.
2. $\sqrt{0.0009}=\frac{3}{100}=0.03$.

Trick: Check by squaring: ${0.03}^2=0.0009$.

Answer: (A) 0.03

Concept: Let age at marriage $=x$; present age $=x+6$.

1. $x+6=\frac{5}{4}x\Rightarrow \frac{x}{4}=6\Rightarrow x=24$.
2. Namya now $=30$ years.
3. Son $=\frac{1}{5}\times 30=6$ years.

Trick: $1\frac{1}{4}$ times means the extra $\frac{1}{4}$ part $=6$ years, so $x=24$.

Answer: (C) 6 years

Concept: $a$ is what % of $b$ $=\frac{a}{b}\times 100$.

1. $\frac{3}{20}÷\frac{12}{25}=\frac{3}{20}\times \frac{25}{12}=\frac{75}{240}=\frac{5}{16}$.
2. $\frac{5}{16}\times 100=31.25\%$.

Yaad rakhein: $\frac{1}{16}=6.25\%$, so $\frac{5}{16}=31.25\%$.

Answer: (A) 31.25%

Concept: Leak rate $=$ (filling rate without leak) $-$ (net rate with leak).

1. Without leak: 7 h; with leak: 8 h.
2. Leak rate $=\frac{1}{7}-\frac{1}{8}=\frac{1}{56}$.
3. Leak empties the full cistern in $56$ hours.

Trick: Leak time $=\frac{a\times b}{b-a}=\frac{7\times 8}{8-7}=56$ h.

Answer: (A) 56 hours

Concept: $\text{Discount}\%=\frac{\text{MP}-\text{SP}}{\text{MP}}\times 100$.

1. Discount $=80-68=$ ₹12.
2. Rate $=\frac{12}{80}\times 100=15\%$.

Yaad rakhein: Discount % is always on the marked price, not the selling price.

Answer: (D) 15%

Concept: Equal profit % and loss % means equal profit and loss amounts, so CP is the average of the two SPs.

1. $\text{CP}=\frac{1920+1500}{2}=$ ₹1710.
2. For 10% profit: $\text{SP}=1710\times \frac{110}{100}=$ ₹1881.

Trick: Same CP, equal % gain and loss → $\text{CP}=\frac{{\text{SP}}_1+{\text{SP}}_2}{2}$.

Answer: (B) ₹1,881

Concept: BODMAS: division and multiplication first, then addition/subtraction from left to right.

1. $36÷4=9$ and $2\times 5=10$.
2. $4-9+10=5$.

Yaad rakhein: Never do $4-36$ first — that breaks the order of operations.

Answer: (D) 5

Concept: $\text{CSA}=2\pi rh$, $V=\pi r^{2}h$.

1. $2\times \frac{22}{7}\times r\times 7=484\Rightarrow 44r=484\Rightarrow r=11$ cm.
2. $V=\frac{22}{7}\times {11}^2\times 7=22\times 121=2662$ cm${}^3$.

Trick: $V=\frac{\text{CSA}\times r}{2}=\frac{484\times 11}{2}=2662$.

Answer: (A) 2662

Concept: List the primes after 20, then average $=\frac{\text{sum}}{\text{count}}$.

1. First five primes above 20: $23,29,31,37,41$.
2. Sum $=161$.
3. Average $=\frac{161}{5}=32.2$.

Trick: Skip 21, 25, 27, 33, 35, 39 — all composite.

Answer: (B) 32.2

Concept: $\text{SI}=\frac{P\times R\times T}{100}$, with $R$ and $T$ in the same time unit.

1. Rate is per month, time $=9$ months.
2. $\text{SI}=\frac{600\times 7\times 9}{100}=6\times 63=$ ₹378.

Yaad rakhein: Match units: monthly rate → time in months.

Answer: (D) ₹378

Concept: Share of A $=\frac{a}{a+b}\times \text{total}$.

1. Total parts $=4+5=9$.
2. A $=\frac{4}{9}\times 63=28$.

Trick: 1 part $=\frac{63}{9}=7$, so A $=4\times 7=28$.

Answer: (A) 28

Concept: Convert averages to totals; B is common to both pairs.

1. $A+B+C=3\times 45=135$.
2. $A+B=80$, $B+C=86$.
3. $B=(A+B)+(B+C)-(A+B+C)=80+86-135=31$ kg.

Trick: Middle common term $=$ sum of pair totals $-$ grand total.

Answer: (B) 31 kg