Q&A: practice questions with answers

Concept: $\text{Average}=\frac{\text{Sum of observations}}{\text{Number of observations}}$

1. Sum $=40+49+56+76+36=257$ kg.
2. Number of members $=5$.
3. Average $=\frac{257}{5}=51.4$ kg.

Trick: Take 50 as base: deviations $-10,-1,+6,+26,-14$ → total $+7$; $50+\frac{7}{5}=51.4$.

Answer: (C) $51.4$

Concept: Sum of first $n$ even numbers $=n(n+1)$, so their average $=n+1$.

1. Here $n=174$.
2. Sum $=174\times 175$.
3. Average $=\frac{174\times 175}{174}=175$.

Trick: Average of first $n$ even numbers $=n+1$; of first $n$ odd numbers $=n$.

Answer: (A) $175$

Concept: Work per hour of a pipe $=\frac{1}{\text{time}}$; together, add the rates.

1. Combined rate $=\frac{1}{13}+\frac{1}{27}+\frac{1}{19}$.
2. LCM $=13\times 27\times 19=6669$: rate $=\frac{513+247+351}{6669}=\frac{1111}{6669}$.
3. Time $=\frac{6669}{1111}=6\frac{3}{1111}$ hours (since $6\times 1111=6666$).

Trick: The fastest pipe alone takes 13 h, so the answer must be far less; rough rates $0.077+0.037+0.053\approx 0.167$ → about 6 h.

Answer: (C) $6\frac{3}{1111}$

Concept: Cube side $a=\sqrt[3]{V}$; cuboid TSA $=2(lb+bh+hl)$.

1. $a=\sqrt[3]{125}=5$ cm.
2. Joining end to end: $l=10,b=5,h=5$ cm.
3. TSA $=2(10\times 5+5\times 5+5\times 10)=2(125)=250{\text{cm}}^2$.

Trick: Two cubes have $12a^2$ area; joining hides 2 faces → $10a^2=10\times 25=250$.

Answer: (D) $250{\text{cm}}^2$

Concept: Compare what he actually gives (cost) with what he actually receives (selling price).

1. Let CP of the stated 1 kg $=₹100$. Loss $65\%$ ⇒ SP $=₹35$.
2. Weight $70\%$ less ⇒ he gives only $0.3$ kg, costing $₹30$.
3. Profit $\%=\frac{35-30}{30}\times 100=\frac{50}{3}=16.67\%$.

Trick: $\text{Profit}\%=\frac{(100-\text{loss})-(100-\text{less wt})}{100-\text{less wt}}\times 100=\frac{35-30}{30}\times 100$.

Answer: (D) $16.67\%$

Concept: Successive discounts: final price $=\text{MP}\times (1-\frac{d_1}{100})(1-\frac{d_2}{100})$.

1. After $15\%$ trade discount: $8000\times 0.85=6800$.
2. After $5\%$ scheme discount: $6800\times 0.95=6460$.
3. Final price $=₹6,460$.

Trick: Equivalent discount $=15+5-\frac{15\times 5}{100}=19.25\%$; $8000\times 0.8075=6460$.

Answer: (C) ₹6,460

Concept: Profit share $\propto$ capital $\times$ time.

1. Shares: $M+N=2160,M-N=720$ ⇒ $M=1440,N=720$ ⇒ ratio $2:1$.
2. Let N's capital $=x$, M's $=x+69000$: $\frac{5(x+69000)}{3x}=\frac{2}{1}$.
3. $5x+345000=6x$ ⇒ $x=345000$.
4. M's capital $=345000+69000=₹4,14,000$.

Trick: Check: $5\times 414000:3\times 345000=2070000:1035000=2:1$ ✓.

Answer: (B) ₹4,14,000

Concept: $\text{SI}=\frac{P\times R\times T}{100}$, with $T$ in years; count days excluding the deposit day.

1. Days: Feb $29-3=26$ (2024 is leap), March $31$, April $4$ ⇒ $61$ days.
2. 2024 has 366 days, so $T=\frac{61}{366}=\frac{1}{6}$ year.
3. $\text{SI}=\frac{4000\times 7.5\times 1}{100\times 6}=\frac{300}{6}=₹50$.

Trick: Yearly SI $=₹300$; 61 days of 366 is exactly $\frac{1}{6}$ → $₹50$.

Answer: (C) ₹50

Concept: For equal distances at speeds $x$ and $y$: $\text{Average speed}=\frac{2xy}{x+y}$.

1. $x=36,y=45$ km/h.
2. Average $=\frac{2\times 36\times 45}{36+45}=\frac{3240}{81}$.
3. $=40\text{km/h}$.

Trick: Average speed for a round trip is always below the simple mean $40.5$; here $40<40.5$ ✓.

Answer: (D) $40$

Concept: Find each value separately and subtract.

1. $65\%\text{ of }65=\frac{65\times 65}{100}=\frac{4225}{100}=42.25$.
2. $\frac{3}{5}\text{ of }15=9$.
3. Difference $=42.25-9=33.25$.

Trick: $x\%$ of $x=\frac{x^2}{100}$: $\frac{{65}^2}{100}=42.25$.

Answer: (A) $33.25$

Concept: For two numbers, $\text{HCF}\times \text{LCM}=\text{first number}\times \text{second number}$.

1. $\text{Product}=5\times 3190=15950$
2. $\text{Other number}=\frac{15950}{110}=145$
3. Check: $145=5\times 29$ and $110=5\times 22$, HCF $=5$ ✓

Trick: Divide LCM by (one number ÷ HCF): $\frac{3190}{22}=145$.

Answer: (C) 145

Concept: A number is divisible by 11 if (sum of digits at odd places) − (sum of digits at even places) is $0$ or a multiple of $11$.

1. Digits N, 0, 0, M, 4, 9 (places 1 to 6).
2. Odd places: $N+0+4=N+4$; even places: $0+M+9=M+9$.
3. $(N+4)-(M+9)=N-M-5$ must be $0$ or $\pm 11$ → $N-M=5$ (or $-6$).
4. $N-M=5\Rightarrow M-N=-5$.

Trick: Only option (B) gives a difference of 5 between the digits – the others never satisfy $N-M-5=0$ or $\pm 11$.

Answer: (B) $M-N=-5$

Concept: Equal interest for the same time ⇒ $P_1{R}_1=P_2{R}_2$, so the sums are in inverse ratio of the rates.

1. $P_1:P_2=14:9$ (sum at 9% : sum at 14%).
2. Total parts $=23$; one part $=\frac{20700}{23}=900$.
3. Sum at 9% $=14\times 900=12600$ (check: $12600\times 9=113400=8100\times 14$ ✓).

Trick: Same SI, same time → money splits in the reverse ratio of rates.

Answer: (A) ₹12,600

Concept: $\text{Time}=\frac{\text{Distance}}{\text{Speed}}$. Let Pranay's speed be $p$ and Aman's time be $t_A$.

1. $\frac{52}{p}=t_A+7$ … (i)
2. Doubled speed: $\frac{52}{2p}=t_A-6$ … (ii)
3. (i) − (ii): $\frac{52}{p}-\frac{26}{p}=13\Rightarrow \frac{26}{p}=13$
4. $p=2\text{km/h}$ (time 26 h; Aman 19 h, doubled 13 h ✓)

Trick: Doubling speed halves time: half of Pranay's time $=7+6=13$ h, so full time $=26$ h and speed $=\frac{52}{26}=2$.

Answer: (C) $2\text{km/h}$

Concept: Cube: $V=a^3$, total surface area $=6a^2$.

1. $a^3=1728\Rightarrow a=12\text{cm}$
2. $\text{TSA}=6\times {12}^2=6\times 144=864{\text{cm}}^2$

Yaad rakhein: Remember ${12}^3=1728$ – it appears very often.

Answer: (C) 864

Concept: Open brackets by distributing, then combine like terms.

1. $4(\frac{7}{4}{x}^2-27x+19)=7x^2-108x+76$
2. $-7(x^2+9x-14)=-7x^2-63x+98$
3. Add: $(7x^2-7x^2)+(-108x-63x)+(76+98)=-171x+174$

Trick: The $x^2$ terms cancel; check the sign of $x$: both $x$-terms are negative, so the coefficient is $-171$.

Answer: (C) $-171x+174$

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

1. CP of 1 pen $=\frac{150}{12}=12.5$
2. SP of 1 pen $=16$; profit $=3.5$
3. $\text{Profit}\%=\frac{3.5}{12.5}\times 100=28\%$

Trick: Compare per dozen: SP $=16\times 12=192$ vs CP 150 → $\frac{42}{150}=28\%$.

Answer: (A) 28

Concept: Eliminate options by quick rules, then divide.

1. 889897 is odd → not divisible by 52 or 64 (even numbers).
2. Digit sum $=8+8+9+8+9+7=49$ → not divisible by 3, so not by $69=3\times 23$.
3. $889897÷59=15083$ exactly ($59\times 15083=889897$).

Trick: Odd number ⇒ strike out all even divisors first.

Answer: (C) 59

Concept: Make decimal places equal, find LCM of the whole numbers, then put back the decimal places.

1. $2.1=2.10=\frac{210}{100}$, $0.68=\frac{68}{100}$
2. $210=2\times 3\times 5\times 7$, $68=2^2\times 17$
3. $\text{LCM}(210,68)=2^2\times 3\times 5\times 7\times 17=7140$
4. $\text{LCM}=\frac{7140}{100}=71.4$

Trick: LCM of decimals must be a multiple of both: $71.4÷2.1=34$ and $71.4÷0.68=105$ ✓.

Answer: (A) 71.4

Concept: Do division and multiplication before addition/subtraction; watch the signs.

1. $(-60)÷(-6)=10$
2. $(-4)\times 8=-32$
3. $-9-10+(-32)=-51$

Trick: Negative ÷ negative = positive, then subtracting it makes it negative again.

Answer: (B) $-51$