Q&A: practice questions with answers

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

1. LHS: $855÷15=57$, $32\times 5=160$, so LHS $=57-k+160=217-k$.
2. RHS: $1152÷16=72$, $72\times 111=7992$, $7992÷37=216$.
3. $217-k=216\Rightarrow k=1$.

Trick: On the RHS, $111÷37=3$ directly, so RHS $=72\times 3=216$.

Answer: (D) $1$

Concept: Every fifth vote invalid means invalid $=20\%$ of the total. The 2500 margin among the remaining (valid) votes equals $10\%$ of those votes.

1. Valid votes $=\frac{2500}{0.10}=25000$, so total $=\frac{25000}{0.8}=31250$ and invalid $=6250$.
2. Valid split: $A+B=25000$, $A-B=2500\Rightarrow A=13750,B=11250$.
3. Invalid votes: B gets $90\%=5625$, A gets $625$. New totals: $A=14375$, $B=16875$.
4. B wins by $2500$, i.e. $\frac{2500}{31250}\times 100=8\%$ of the votes polled.

Trick: B gains a net $80\%$ of invalid votes $=5000$; A's lead of 2500 turns into B's lead of 2500.

Answer: (C) B would have won by 8%

Concept: Solve the bracket first.

1. $1002÷20.04=\frac{100200}{2004}=50$.
2. $5055-50=5005$.

Trick: $20.04\times 50=1002$ – spot the multiple quickly.

Answer: (D) $5005$

Concept: Population after $n$ years $=P{(1+\frac{R}{100})}^n$.

1. $16000\times {\left(\frac{21}{20}\right)}^2=16000\times \frac{441}{400}$.
2. $=40\times 441=17640$.

Trick: Two years at $5\%$ = $10.25\%$ effective; $10.25\%$ of 16000 $=1640$.

Answer: (D) $17640$

Concept: Required $\%=\frac{\text{Medicine}}{\text{MBA}}\times 100$ for city A.

1. City A: Medicine $=1100$, MBA $=1200$.
2. $\frac{1100}{1200}\times 100=91.67\%\approx 92\%$.

Trick: $\frac{11}{12}=1-\frac{1}{12}\approx 1-0.083=0.917$.

Answer: (C) 92%

Concept: Gain $\%=\frac{\text{True weight}-\text{False weight}}{\text{False weight}}\times 100$.

1. He gives 950 g but charges for 1000 g: gain $=50$ g on 950 g.
2. Gain $\%=\frac{50}{950}\times 100=\frac{100}{19}=5\frac{5}{19}\%$.

Trick: Divide the error by the weight actually given (950), not by 1000.

Answer: (D) $5\frac{5}{19}\%$

Concept: BODMAS: 'of', then $÷$ and $\times$, then addition.

1. $15\%$ of $1.5=0.225$; $0.15÷0.1=1.5$.
2. $0.01$ of $1.1=0.011$, $\times 0.5=0.0055$; $2.5\times 25=62.5$.
3. Sum $=0.225+1.5+0.0055+62.5=64.2305$.

Trick: The big term 62.5 plus about 1.73 already points to 64.23.

Answer: (D) $64.2305$

Concept: Express $\sin x$ in terms of $\cos x$ and substitute.

1. $\cos x=(\sqrt{2}-1)\sin x\Rightarrow \sin x=\frac{\cos x}{\sqrt{2}-1}=(\sqrt{2}+1)\cos x$.
2. $\sin x-\cos x=(\sqrt{2}+1)\cos x-\cos x=\sqrt{2}\cos x$.

Trick: Standard pair: if $\sin x+\cos x=\sqrt{2}\sin x$ then $\sin x-\cos x=\sqrt{2}\cos x$ (square and add both: sum $=2$).

Answer: (D) $\sqrt{2}\cos x$

Concept: Surface area $\propto r^2$.

1. New radius $=2.25r$.
2. New area $=(2.25{)}^2=5.0625$ times.
3. Increase $=(5.0625-1)\times 100=406.25\%$.

Trick: Use $a+b+\frac{ab}{100}=125+125+156.25=406.25\%$.

Answer: (C) 406.25%

Concept: Solve innermost bracket first (BODMAS).

1. $5\frac{1}{3}+3\frac{1}{3}-2\frac{1}{3}=6\frac{1}{3}=\frac{19}{3}$.
2. $3.3\times \frac{19}{3}=20.9$; $2\frac{3}{4}+20.9=23.65$.
3. $3\frac{2}{5}\times 23.65=3.4\times 23.65=80.41$.

Trick: Convert mixed fractions to decimals ($3.4$, $2.75$) for quick multiplication.

Answer: (B) $80.41$

Concept: Third side lies strictly between the difference and the sum of the other two sides.

1. $22-12<m<22+12$.
2. $10<m<34$ (equality would make a straight line, not a triangle).

Yaad rakhein: Both ends are always strict (open) – reject any option with $\le$.

Answer: (C) $10<m<34$

Concept: TSA $=2\pi rh+2\pi r^2$ = CSA + two circular ends.

1. CSA $=2200{\text{cm}}^2$ (given).
2. $2\pi r^2=2\times \frac{22}{7}\times 49=308{\text{cm}}^2$.
3. TSA $=2200+308=2508{\text{cm}}^2$.

Trick: No need to find $h$ – just add the two ends.

Answer: (A) $2508{\text{cm}}^2$

Concept: 'of' is done before $÷$; brackets innermost first.

1. $32$ of $2=64$; $64-4=60$.
2. $60÷(4\times 15)=60÷60=1$.
3. $34-1=33$; $66÷33=2$.

Trick: Treat 'of' as multiplication with higher priority than $÷$.

Answer: (B) $2$

Concept: Count = total × percentage, then take the ratio.

1. Females: A $=0.35\times 700=245$, D $=0.55\times 800=440$; sum $=685$.
2. Males: B $=0.7\times 550=385$, C $=0.6\times 600=360$; sum $=745$.
3. $\frac{685}{745}\times 100=91.95\%$.

Trick: $685$ is just below $745$ – answer must be near $92\%$.

Answer: (C) 91.95%

Concept: Women = population × (women's share of the ratio).

1. City A: $2340\times \frac{2}{3}=1560$.
2. City B: $900\times \frac{1}{4}=225$.
3. Total $=1560+225=1785$ hundred.

Trick: Divide by the ratio sum first: $2340÷3=780$, $900÷4=225$.

Answer: (D) $1785$

Concept: Compare money actually received with cost of goods actually given.

1. SP per 'kg' $=38\times 0.88=₹33.44$; for 10 'kg' he gets ₹334.40.
2. Actual sugar given $=10\times 0.9=9\text{kg}$, cost $=9\times 30=₹270$.
3. Profit $\%=\frac{334.40-270}{270}\times 100=\frac{64.4}{270}\times 100=23.85\%$.

Trick: Per 900 g: CP $=27$, SP $=33.44$ → $\frac{6.44}{27}\times 100$.

Answer: (A) 23.85%

Concept: $d\propto t\Rightarrow d=kt$.

1. $72=k\times 2\Rightarrow k=36$.
2. So $d=36t$.

Trick: The constant is just the speed $\frac{72}{2}=36\text{km/h}$.

Answer: (D) $d=36t$

Concept: Volume of the hollow sphere = volume of the cone.

1. Shell: $\frac{4}{3}\pi (5^3-3^3)=\frac{4}{3}\pi \times 98$.
2. Cone: $\frac{1}{3}\pi \times 5^2\times h$.
3. $\frac{1}{3}\pi \times 25h=\frac{4}{3}\pi \times 98\Rightarrow h=\frac{392}{25}=15.68\text{cm}$.

Trick: Cancel $\frac{1}{3}\pi$: $25h=4\times 98$.

Answer: (D) $15.68\text{cm}$

Concept: Sector area $=\frac{\theta }{{360}^{\circ }}\times \pi r^2$.

1. $\pi r^2=\frac{22}{7}\times 196=616$.
2. $\frac{\theta }{360}\times 616=231\Rightarrow \theta =\frac{231\times 360}{616}={135}^{\circ }$.

Trick: $\frac{231}{616}=\frac{3}{8}$, and $\frac{3}{8}\times 360=135$.

Answer: (B) ${135}^{\circ }$

Concept: If a sum of squares is 0, each square is 0.

1. $a=18$, $b=12$, $c=6$.
2. $a+b+c=36$, $(36{)}^{\frac{1}{2}}=\pm 6$ as given in options.

Trick: Square-sum $=0$ ⇒ every bracket $=0$.

Answer: (D) $\pm 6$