Q&A: practice questions with answers

Concept: Loser $35\%$, winner $65\%$; margin $=30\%$ of votes polled.

1. $65\%-35\%=30\%$
2. $30\%$ of total $=15900$
3. Total $=\frac{15900}{0.3}=53000$

Trick: Margin $\%=100-2\times \text{loser}\%$.

Answer: (D) $53,000$

Concept: Simplify a continued fraction from the bottom upward.

1. $2+\frac{1}{3}=\frac{7}{3}$, so $\frac{1}{7/3}=\frac{3}{7}$
2. $2-\frac{3}{7}=\frac{11}{7}$, so $\frac{1}{11/7}=\frac{7}{11}$
3. $1+\frac{7}{11}=\frac{18}{11}$, so $\frac{1}{18/11}=\frac{11}{18}$
4. $1-\frac{11}{18}=\frac{7}{18}$

Trick: Each step: reciprocal of the lower result.

Answer: (A) $\frac{7}{18}$

Concept: Let upstream speed $=u$, downstream $=d$; use $\frac{1}{u}=a$, $\frac{1}{d}=b$ to make the equations linear.

1. $24a+36b=10$ and $36a+24b=12$
2. Adding: $60(a+b)=22\Rightarrow a+b=\frac{11}{30}$; subtracting: $12(a-b)=2\Rightarrow a-b=\frac{1}{6}$
3. $a=\frac{4}{15}\Rightarrow u=\frac{15}{4}$; $b=\frac{1}{10}\Rightarrow d=10$
4. Current $=\frac{d-u}{2}=\frac{10-3.75}{2}=\frac{25}{8}\text{km/h}$

Trick: Swapped coefficients ($24,36$ / $36,24$): add and subtract the equations.

Answer: (D) $\frac{25}{8}\text{km/h}$

Concept: At CI, if money becomes $k$ times in $n$ years, it becomes $k^m$ times in $mn$ years.

1. $3$ times in $5$ years
2. $9=3^2$ times in $2\times 5=10$ years

Trick: Powers multiply the time: $3\to 9\to 27$ in $5,10,15$ years.

Answer: (B) $10$ years

Concept: Solve innermost brackets first; $(-)\times (-)=(+)$.

1. $9-(6-2)=5$; $(-4)\times 5=-20$; $25-(-20)=45$
2. $(-3)\times (-3)=9$; $6+9=15$; $3\times 15=45$
3. $45÷45=1$

Trick: Watch the double negative: $25-(-20)=25+20$.

Answer: (A) $1$

Concept: Bracket first, then division.

1. $\frac{13}{4}\times 80=260$
2. $260-\frac{91}{4}=\frac{1040-91}{4}=\frac{949}{4}$
3. $1898÷\frac{949}{4}=\frac{1898\times 4}{949}=2\times 4=8$

Trick: Notice $1898=2\times 949$.

Answer: (B) $8$

Concept: Average $=\frac{\text{sum}}{5}$; values are in hundreds.

1. Demand: $35+25+30+30+40=160\Rightarrow$ average $32$
2. Production: $30+40+25+40+45=180\Rightarrow$ average $36$
3. Difference $=4$ hundred $=400$

Trick: Difference of averages $=\frac{180-160}{5}=4$ hundred.

Answer: (D) $400$

Concept: $\text{SI}=\frac{P\times R\times T}{100}$.

1. SI $=4900-2500=2400$
2. $2400=\frac{2500\times 12\times T}{100}=300T$
3. $T=8$ years

Trick: Yearly SI $=12\%$ of $2500=300$; $\frac{2400}{300}=8$.

Answer: (C) $8$

Concept: Cone $=\frac{1}{3}\pi r^{2}h$, cylinder $=\pi r^{2}h$; $\pi$ cancels in the ratio.

1. Cone $=\frac{1}{3}\pi \times 25\times 6=50\pi$
2. Cylinder $=\pi \times 6.25\times 3=18.75\pi$
3. Ratio $=\frac{50}{18.75}=\frac{8}{3}$

Trick: $\frac{1}{3}\times \frac{5^2\times 6}{{2.5}^2\times 3}=\frac{1}{3}\times 4\times 2=\frac{8}{3}$.

Answer: (D) $8:3$

Concept: $a^2-b^2=(a+b)(a-b)$ and $a^2-2ab+b^2=(a-b{)}^2$ with $a=3.2$, $b=0.2$.

1. Numerator $=(3.2+0.2)(3.2-0.2)=3.4\times 3$
2. Denominator $=(3.2-0.2{)}^2=9$
3. Value $=\frac{3.4\times 3}{9}=\frac{3.4}{3}=\frac{17}{15}$

Trick: Expression $=\frac{a+b}{a-b}=\frac{3.4}{3.0}$.

Answer: (C) $\frac{17}{15}$

Concept: Find CP and SP of one orange (or of LCM $60$ oranges).

1. CP of 1 $=\frac{50}{10}=5$
2. SP of 1 $=\frac{74}{12}=\frac{37}{6}$
3. Gain $\%=\frac{\frac{37}{6}-5}{5}\times 100=\frac{7}{30}\times 100=23\frac{1}{3}\%$

Trick: For $60$ oranges: CP $=300$, SP $=370$, gain $=\frac{70}{300}=23\frac{1}{3}\%$.

Answer: (A) $23\frac{1}{3}\%$ gain

Concept: Sides are proportional to the sines of opposite angles (sine rule).

1. $x+2x+3x={180}^{\circ }\Rightarrow {30}^{\circ },{60}^{\circ },{90}^{\circ }$
2. $\sin {30}^{\circ }:\sin {60}^{\circ }:\sin {90}^{\circ }=\frac{1}{2}:\frac{\sqrt{3}}{2}:1$
3. $=1:\sqrt{3}:2$

Yaad rakhein: ${30}^{\circ }$-${60}^{\circ }$-${90}^{\circ }$ triangle: sides $1:\sqrt{3}:2$.

Answer: (A) $1:\sqrt{3}:2$

Concept: Inside brackets: $÷$ and $\times$ left to right, then $+$/$-$.

1. $48÷8\times 3=6\times 3=18$; $82+18=100$
2. $100+5\times 6=130$
3. $130-7.5\times 2=130-15=115$

Trick: Do $÷$ before $\times$ when $÷$ comes first (left to right).

Answer: (A) $115$

Concept: Multiply the factors: $0.9\times 1.3\times 0.8$.

1. $0.9\times 1.3=1.17$
2. $1.17\times 0.8=0.936$
3. Net decrease $=1-0.936=0.064=6.4\%=6\frac{2}{5}\%$

Trick: Start with $100$: $100\to 90\to 117\to 93.6$.

Answer: (C) $6\frac{2}{5}\%$

Concept: Write ${\cot }^2\theta =\frac{1}{{\tan }^2\theta }$ and simplify.

1. Let $t={\tan }^2\theta$: numerator $=1-\frac{1}{t}=\frac{t-1}{t}$
2. $\frac{(t-1)/t}{t-1}=\frac{1}{t}$
3. $={\cot }^2\theta$

Trick: Put $\theta ={60}^{\circ }$: $\frac{1-\frac{1}{3}}{3-1}=\frac{1}{3}={\cot }^2{60}^{\circ }$.

Answer: (C) ${\cot }^2\theta$

Concept: Add the three pair rates to get $2(R+S+M)$.

1. $R+S=\frac{1}{2}$, $S+M=\frac{1}{4}$, $R+M=\frac{1}{2}$
2. $2(R+S+M)=\frac{5}{4}\Rightarrow R+S+M=\frac{5}{8}$
3. $R=\frac{5}{8}-\frac{1}{4}=\frac{3}{8}$
4. Ravi alone $=\frac{8}{3}$ days

Trick: Total work $=4$ units: pairs do $2,1,2$; all three $2.5$; Ravi $=1.5$ units/day.

Answer: (B) $\frac{8}{3}$ days

Concept: SP $=$ cost $+$ margin; the discount slab depends on the bill total.

1. Shirt $=900\times 1.09=981$; jacket $=1900\times 1.06=2014$
2. Total $=2995<3000$, so discount $=12\%$
3. Pay $=2995\times 0.88=2635.60$

Trick: Check the slab first: $2995$ is just below $3000$.

Answer: (D) ₹$2,635.60$

Concept: Capacity of hemispherical bowl $=\frac{2}{3}\pi r^3$.

1. $r^3=9^3=729$
2. $V=\frac{2}{3}\times \frac{22}{7}\times 729=\frac{32076}{21}$
3. $\approx 1527.43{\text{cm}}^3$

Trick: $\frac{2}{3}\times 729=486$; $486\times \frac{22}{7}=1527.43$.

Answer: (D) $1527.43{\text{cm}}^3$

Concept: $\frac{\sec A+\tan A}{\sec A-\tan A}=\frac{1+\sin A}{1-\sin A}$; then componendo–dividendo.

1. $2\frac{51}{79}=\frac{209}{79}$
2. $\frac{1+\sin A}{1-\sin A}=\frac{209}{79}$
3. $\sin A=\frac{209-79}{209+79}=\frac{130}{288}=\frac{65}{144}$

Trick: If $\frac{1+x}{1-x}=\frac{p}{q}$, then $x=\frac{p-q}{p+q}$.

Answer: (B) $\frac{65}{144}$

Concept: “of” is done before $÷$ and $\times$.

1. $5\text{ of }4=20$: $20÷20\times 8=8$
2. $24\text{ of }3=72$: $6÷72\times 48=4$
3. $6÷18\times 9=3$
4. $8+4-3=9$

Trick: Resolve every “of” first, then go left to right.

Answer: (C) $9$