Q&A: practice questions with answers

Concept: 'Buy one get one 30% off' means the first item is at full price and the second at a 30% discount.

1. First product: ₹350.
2. Second product: $350\times \frac{70}{100}=245$, i.e. ₹245.
3. Total $=350+245=595$, i.e. ₹595.

Yaad rakhein: Total $=350\times (1+0.7)=350\times 1.7=595$.

Answer: (A) ₹595

Concept: If numbers are in ratio $a:b$ (co-prime) with HCF $h$, the numbers are $ah$ and $bh$, and LCM $=a\times b\times h$.

1. Numbers $=5\times 7=35$ and $6\times 7=42$.
2. LCM $=5\times 6\times 7=210$.
3. Check: $\text{HCF}\times \text{LCM}=7\times 210=1470=35\times 42$ ✓

Yaad rakhein: LCM = product of ratio terms × HCF.

Answer: (B) $210$

Concept: For the same time period, profit is shared in the ratio of investments.

1. Total parts $=58+72+20=150$.
2. 1 part $=\frac{3750}{150}=25$, i.e. ₹25.
3. Y − Z $=72-20=52$ parts $=52\times 25=1300$, i.e. ₹1,300.

Yaad rakhein: Directly: $\frac{52}{150}\times 3750=1300$.

Answer: (A) ₹1,300

Concept: Divisibility by 7 (also 11, 13): make groups of 3 digits from the right and take the alternating sum.

1. Groups: 862 | 026 | 417 → $417-26+862=1253$.
2. $1253=7\times 179$, so 862026417 is divisible by 7.
3. Not by 5 (last digit 7); not by 6 (odd number); for 11, alternating digit sum $=7-1+4-6+2-0+2-6+8=10$, not a multiple of 11.

Yaad rakhein: Direct check: $7\times 123146631=862026417$.

Answer: (C) $7$

Concept: Simplify the continued fraction inside the bracket first; equal terms with opposite signs cancel.

1. $+\frac{2}{5}$ and $-\frac{2}{5}$ cancel.
2. $1+\frac{5}{7}=\frac{12}{7}$.
3. $\frac{1}{\frac{12}{7}}=\frac{7}{12}$.

Yaad rakhein: Spot $+\frac{2}{5}-\frac{2}{5}=0$ at once – only the bracket matters.

Answer: (C) $\frac{7}{12}$

Concept: Use $p^2-q^2=(p+q)(p-q)$ and ${\cot }^{2}A-{\cos }^{2}A={\cot }^{2}A{\cos }^{2}A$.

1. $p+q=2\cot A$, $p-q=2\cos A$ ⇒ $p^2-q^2=4\cot A\cos A$.
2. $pq={\cot }^{2}A-{\cos }^{2}A={\cos }^{2}A(\frac{1}{{\sin }^{2}A}-1)={\cos }^{2}A{\cot }^{2}A$.
3. So $\sqrt{pq}=\cot A\cos A$ ⇒ $p^2-q^2=4\sqrt{pq}$.

Yaad rakhein: Put $A={45}^{\circ }$: $\cot A=1$, $\cos A=\frac{1}{\sqrt{2}}$ ⇒ $p^2-q^2=2\sqrt{2}$, $\sqrt{pq}=\frac{1}{\sqrt{2}}$; ratio 4.

Answer: (A) $4\sqrt{pq}$

Concept: Apply successive discounts one after another on the reduced price.

1. After 10%: $2000\times 0.90=1800$.
2. After 5%: $1800\times 0.95=1710$.
3. Final price = ₹1,710.

Yaad rakhein: Net discount $=10+5-\frac{10\times 5}{100}=14.5\%$; $2000\times 0.855=1710$.

Answer: (A) ₹1,710

Concept: Expand the brackets and combine like terms.

1. $x(5x-9)=5x^2-9x$.
2. $7(x^2-2)=7x^2-14$.
3. Sum $+17$: $5x^2+7x^2-9x-14+17=12x^2-9x+3$.

Yaad rakhein: Check with $x=1$: $1(-4)+7(-1)+17=6$ and $12-9+3=6$ ✓

Answer: (C) $12x^2-9x+3$

Concept: Volume of a cylinder $V=\pi r^{2}h$, so a % change in $r$ acts twice (squared) and the change in $h$ acts once.

1. New radius $=0.73r$ (decrease 27%), new height $=2.65h$ (increase 165%).
2. New volume $=\pi (0.73r{)}^2(2.65h)=0.5329\times 2.65\pi r^{2}h$.
3. $0.5329\times 2.65=1.4122$, i.e. $V$ becomes $1.4122V$.
4. Increase $=41.22\%\approx 41\%$.

Trick: Multiply the factors: $(0.73{)}^2\times 2.65-1\approx 0.41$ → 41%.

Answer: (C) $41\%$

Concept: Volume of a cylinder $=\pi r^{2}h$.

1. $r=5\text{cm}$, $h=21\text{cm}$, $\pi =\frac{22}{7}$.
2. $V=\frac{22}{7}\times 5^2\times 21=22\times 25\times 3$.
3. $V=1650{\text{cm}}^3$.

Trick: When $h$ is a multiple of 7, cancel 7 first: $21÷7=3$, then $22\times 25\times 3$.

Answer: (B) $1650{\text{cm}}^3$

Concept: Sum of numbers $=$ average $\times$ count.

1. Sum of first two $=80\times 2=160$; third $=107$.
2. Sum of last two $=655-160-107=388$.
3. Average of last two $=\frac{388}{2}=194$.

Trick: Subtract the known parts from the total, then divide by how many are left.

Answer: (D) 194

Concept: With a false weight, the trader gives less goods but charges the full (marked-up) price: $\text{Profit}\%=(\frac{\text{SP of goods charged}}{\text{CP of goods given}}-1)\times 100$.

1. Let CP of 1 kg $=$ ₹100. He charges for 1 kg at 29% profit: SP $=$ ₹129.
2. He actually gives only $0.7$ kg, whose CP $=$ ₹70.
3. Profit $=\frac{129-70}{70}\times 100=\frac{5900}{70}=84.29\%$.
4. Nearest whole number $=84\%$.

Trick: Profit factor $=\frac{1.29}{0.70}=1.843$ → about 84%.

Answer: (B) $84\%$

Concept: For a fixed distance, speed $\propto \frac{1}{\text{time}}$.

1. Distance $=86\times 7=602\text{km}$.
2. Required speed $=\frac{602}{14}=43\text{km/h}$.

Trick: Time doubles (7 h to 14 h), so speed halves: $86÷2=43$.

Answer: (A) $43\text{km/h}$

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

1. $P=1200$, $R=3\%$ per month, $T=6$ months.
2. SI $=\frac{1200\times 3\times 6}{100}=216$.

Trick: 1 month interest $=3\%$ of 1200 $=36$; for 6 months $=36\times 6=216$.

Answer: (C) ₹216

Concept: Increasing by 60% means multiplying by $1.6$.

1. $x\times 1.6=3590$.
2. $x=\frac{3590}{1.6}=\frac{35900}{16}=2243.75$.

Trick: Check: $2243.75\times 1.6=3590$ ✓.

Answer: (B) 2243.75

Concept: Area of a trapezium $=\frac{1}{2}\times (\text{sum of parallel sides})\times \text{height}$.

1. $\frac{1}{2}(a+b)\times 24=648$ ⇒ $a+b=54$.
2. $b-a=6$.
3. Adding: $2b=60$ ⇒ $b=30$, $a=24$.

Trick: Sum $=\frac{2\times \text{Area}}{h}$; then sum and difference give both sides directly.

Answer: (B) $30\text{cm}$, $24\text{cm}$

Concept: Average of the first $n$ even numbers $=n+1$.

1. Sum $=n(n+1)$, count $=n$, so average $=n+1$.
2. For $n=172$: average $=173$.

Yaad rakhein: First $n$ even numbers → $n+1$; first $n$ odd numbers → $n$; first $n$ natural numbers → $\frac{n+1}{2}$.

Answer: (D) 173

Concept: $\text{Profit}\%=\frac{\text{SP}-\text{CP}}{\text{CP}}\times 100$ per item.

1. CP of one ruler $=\frac{180}{12}=$ ₹15.
2. SP of one ruler $=$ ₹24, profit $=$ ₹9.
3. Profit% $=\frac{9}{15}\times 100=60\%$.

Trick: Always bring CP and SP to the same quantity (1 dozen = 12).

Answer: (B) 60

Concept: One-day work $=\frac{1}{\text{days}}$; work of a group is the sum of individual one-day works.

1. Isha $=\frac{1}{24}$; I + S + A $=\frac{1}{12}$; I + S $=\frac{1}{20}$.
2. Ashlesha $=\frac{1}{12}-\frac{1}{20}=\frac{5-3}{60}=\frac{1}{30}$.
3. Isha + Ashlesha $=\frac{1}{24}+\frac{1}{30}=\frac{5+4}{120}=\frac{9}{120}=\frac{3}{40}$.
4. Days $=\frac{40}{3}$.

Trick: LCM method: total work $=120$ units; I $=5$, I+S $=6$, I+S+A $=10$ ⇒ A $=4$; I+A $=9$ ⇒ $\frac{120}{9}=\frac{40}{3}$ days.

Answer: (B) $\frac{40}{3}$ days

Concept: Winning margin $=$ (winner% $-$ loser%) of valid votes.

1. Let registered voters $=N$. Votes polled $=0.8N$.
2. Valid votes $=79\%$ of $0.8N=0.632N$.
3. Winner $51\%$, loser $49\%$ ⇒ margin $=2\%$ of $0.632N=0.01264N$.
4. $0.01264N=1817$ ⇒ $N=\frac{1817}{0.01264}=143750$.

Trick: Only one option is a "clean" number; $N\times 0.8\times 0.79\times 0.02$ must be an integer, which fits 143750.

Answer: (A) 143750