Q&A: practice questions with answers

Concept: The gap between the two cabs stays fixed; the man closes this gap with relative speed (sum of speeds, as they move towards each other).

1. Gap $=54\times \frac{29}{60}=26.1$ km
2. Relative speed $=\frac{26.1}{12/60}=130.5\text{km/h}$
3. Man's speed $=130.5-54=76.5\text{km/h}$

Trick: $v=54\times \frac{29-12}{12}=54\times \frac{17}{12}=76.5$

Answer: (C) $76.5$

Concept: Let CP $=C$, SP $=S$; profit $=S-C$.

1. New SP $=2S$, new profit $=2S-C=10(S-C)$
2. $2S-C=10S-10C\Rightarrow 9C=8S\Rightarrow S=\frac{9}{8}C$
3. Profit $\%=\frac{S-C}{C}\times 100=\frac{1}{8}\times 100=12.5\%$

Trick: Take $C=8$: $2S-8=10(S-8)\Rightarrow S=9$ → profit 1 on 8 = $12.5\%$.

Answer: (C) $12.5$

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

1. Numbers $=17k$ and $6k$; product $=102k^2=102\Rightarrow k=1$
2. Numbers 17 and 6 are co-prime: HCF $=1$, LCM $=102$
3. $\frac{1}{102}+\frac{1}{1}=\frac{103}{102}$

Trick: Co-prime ratio terms → HCF $=k$, LCM $=17\times 6\times k$.

Answer: (C) $\frac{103}{102}$

Concept: Convert each ratio to a decimal.

1. $\frac{41}{64}\approx 0.64$, $\frac{50}{59}\approx 0.85$
2. $\frac{26}{90}\approx 0.29$, $\frac{40}{70}\approx 0.57$
3. Greatest $=\frac{50}{59}$

Trick: Only 50 : 59 has the first term close to the second → largest.

Answer: (B) $50:59$

Concept: Angles on a straight line add to ${180}^{\circ }$; angles of a triangle add to ${180}^{\circ }$.

1. $\angle BEF={180}^{\circ }-\angle PEB={180}^{\circ }-{59}^{\circ }={121}^{\circ }$
2. $\angle GEF={121}^{\circ }-{50}^{\circ }={71}^{\circ }$
3. In $△EGF$: $\angle EGF={180}^{\circ }-{71}^{\circ }-{33}^{\circ }={76}^{\circ }$

Trick: Draw the figure first: $\angle EGF={180}^{\circ }-({121}^{\circ }-{50}^{\circ })-{33}^{\circ }={76}^{\circ }$.

Answer: (A) ${76}^{\circ }$

Concept: Divisibility by 7: double the last digit and subtract it from the rest; the result must be divisible by 7.

1. 875: $87-2\times 5=77=7\times 11$ ✓ (indeed $875=7\times 125$)
2. 894: $89-8=81$ ✗; 87: $8-14=-6$ ✗; 687: $68-14=54$ ✗

Trick: $875=7\times 125$ – multiples of 125 with 7 are easy to spot.

Answer: (C) 875

Concept: Upstream speed $=b-s$, downstream speed $=b+s$.

1. Upstream $=\frac{10.2}{51/60}=12\text{km/h}$
2. $b:s=7:6$ → upstream $=1$ part $=12$, so $b=84$, $s=72$, downstream $=156\text{km/h}$
3. Time $=\frac{17.7}{12}+\frac{66.3}{156}=1.475+0.425=1.9$ hours

Trick: Upstream = $7-6=1$ part, downstream = $7+6=13$ parts → downstream $=13\times 12$.

Answer: (C) 1.9

Concept: Net effect of $+35\%$ and $-20\%$: $35-20-\frac{35\times 20}{100}=8\%$.

1. MP $=1300\times 1.35=1755$
2. SP $=1755\times 0.8=1404$
3. Profit $=104$; $\frac{104}{1300}\times 100=8\%$

Trick: Successive change: $a+b+\frac{ab}{100}=35-20-7=8\%$ – CP not even needed.

Answer: (D) $8\%$

Concept: Doubling means $SI=P$.

1. $P=\frac{P\times R\times 5}{100}$
2. $R=\frac{100}{5}=20\%$

Trick: Sum becomes $n$ times in $T$ years → $R=\frac{(n-1)\times 100}{T}$.

Answer: (B) $20\%$

Concept: Open brackets from the innermost: ( ), { }, [ ].

1. $9+2=11$; $5+8-11=2$
2. $7+7-2=12$
3. $\frac{8}{4}\times 12=24$
4. $144÷24=6$

Trick: Innermost first, then move outward one bracket at a time.

Answer: (C) 6

Concept: $\tan \theta =\frac{\text{height of rod}}{\text{length of shadow}}$

1. $\tan \theta =\frac{2}{\sqrt{12}}=\frac{2}{2\sqrt{3}}=\frac{1}{\sqrt{3}}$
2. $\theta ={30}^{\circ }$

Trick: Shadow $=\sqrt{3}\times$ height → ${30}^{\circ }$; shadow = height → ${45}^{\circ }$.

Answer: (D) ${30}^{\circ }$

Concept: $x^3-\frac{1}{x^3}=(x-\frac{1}{x}{)}^3+3(x-\frac{1}{x})$

1. $={10}^3+3\times 10$
2. $=1000+30=1030$

Trick: If $x-\frac{1}{x}=a$, then $x^3-\frac{1}{x^3}=a^3+3a$.

Answer: (B) 1030

Concept: Savings $=100\%-60\%=40\%$ of income; one-fourth of a year = 3 months.

1. Monthly saving $=40\%\times 8400=3360$
2. For 3 months: $3\times 3360=10080$

Trick: $8400\times 3\times 0.4=10080$ in one line.

Answer: (A) ₹10,080

Concept: Evaluate the roots first, then multiply.

1. $\sqrt[3]{1331}=11$ (since ${11}^3=1331$) and $\sqrt[4]{10000}=10$ (since ${10}^4=10000$).
2. Expression $=\frac{11}{10}\times \frac{5}{33}\times 30$.
3. $=\frac{11\times 5\times 30}{10\times 33}=\frac{1650}{330}=5$.

Trick: Cancel early: $\frac{11}{33}=\frac{1}{3}$ and $\frac{30}{10}=3$, so the answer is just $5$.

Answer: (C) $5$

Concept: Half-yearly compounding: rate halves, periods double.

1. Rate per half-year $=\frac{8}{2}=4\%$, number of periods $=2$.
2. $A=90625\times {(1+\frac{4}{100})}^2=90625\times \frac{26}{25}\times \frac{26}{25}$.
3. $90625÷625=145$, so $A=145\times 676=98020$.

Trick: $(1.04{)}^2=1.0816$; $90625\times 1.0816=98020$.

Answer: (A) ₹$98,020$

Concept: One-day work: (together) − (Robin) = Varun.

1. Robin + Varun per day $=\frac{1}{6}$; Robin per day $=\frac{1}{42}$.
2. Varun per day $=\frac{1}{6}-\frac{1}{42}=\frac{7-1}{42}=\frac{6}{42}=\frac{1}{7}$.
3. Varun alone takes $7$ days.

Trick: LCM method: total work $=42$ units; together $7$/day, Robin $1$/day → Varun $6$/day → $\frac{42}{6}=7$ days.

Answer: (A) $7$ days

Concept: Average of groups = sum of (share × group average).

1. Children $=100-20-50=30\%$.
2. Average $=0.2\times 45+0.5\times 50+0.3\times 50$.
3. $=9+25+15=49\text{kg}$.

Trick: 80% of people weigh 50 and 20% weigh 5 less → average $=50-0.2\times 5=49$.

Answer: (D) $49$

Concept: Make the common term (red) equal in both ratios.

1. Red : Green $=7:20$ and Pink : Red $=15:12$, i.e. Red : Pink $=12:15=4:5$.
2. LCM of red values $7$ and $4$ is $28$: R : G $=28:80$, R : P $=28:35$.
3. Green : Pink $=80:35=16:7$.

Trick: $\frac{G}{P}=\frac{G}{R}\times \frac{R}{P}=\frac{20}{7}\times \frac{12}{15}=\frac{16}{7}$.

Answer: (C) $16:7$

Concept: Profit is shared in the ratio of investments (same time).

1. Profit left after charity $=69\%$ of $76700=52923$.
2. Ratio $56000:52000:69000=56:52:69$; total $=177$ parts.
3. One part $=\frac{52923}{177}=299$.
4. Suresh $=69\times 299=20631$.

Trick: Check divisibility early: $52923÷177=299$ exactly confirms the setup.

Answer: (A) ₹$20,631$

Concept: Range = highest value − lowest value.

1. Highest mark $=80$, lowest mark $=45$.
2. Range $=80-45=35$.

Trick: Only the two extremes matter for range — ignore the middle values.

Answer: (D) $35$