Q&A: practice questions with answers

Concept: Find each quantity separately and subtract.

1. $65\%\text{ of }65=\frac{65\times 65}{100}=\frac{4225}{100}=42.25$
2. $\frac{2}{5}\text{ of }55=22$
3. $\text{Difference}=42.25-22=20.25$

Trick: ${65}^2=4225$ (numbers ending in 5: $6\times 7=42$, append 25).

Answer: (D) $20.25$

Concept: Apply the divisibility tests for 10, 11, 3 and 8.

1. Last digit is 9, so not divisible by 10.
2. Last three digits 059 are not divisible by 8.
3. For 11: (sum of odd-place digits from right) $-$ (even-place) $=(9+0+7+8+6)-(5+9+6+4)=30-24=6$, not a multiple of 11.
4. Digit sum $=6+4+8+6+7+9+0+5+9=54$, divisible by 3 (and by 9) ✔

Trick: Digit sum divisible by 3 ⇒ number divisible by 3.

Answer: (C) 3

Concept: $a^m÷a^n=a^{m-n}$ and $a^m\times a^n=a^{m+n}$; work left to right.

1. ${57}^{-9}÷{57}^4={57}^{-9-4}={57}^{-13}$
2. ${57}^{-13}\times {57}^{-10}={57}^{-13-10}={57}^{-23}$

Trick: Add the exponents with signs: $-9-4+(-10)=-23$.

Answer: (C) ${57}^{-23}$

Concept: Surface area of a sphere $=4\pi r^2$, so area $\propto r^2$.

1. Before: $S_1=4\pi r^2$
2. After: $S_2=4\pi (5r{)}^2=25\times 4\pi r^2$
3. $S_2:S_1=25:1$

Trick: Radius $\times k$ ⇒ area $\times k^2$, volume $\times k^3$.

Answer: (C) $25:1$

Concept: If $a:b=b:x$, then $x$ is the third proportional of $a$ and $b$: $x=\frac{b^2}{a}$.

1. $60:30=30:x$
2. $x=\frac{{30}^2}{60}=\frac{900}{60}=15$

Trick: Third proportional $=\frac{(\text{second}{)}^2}{\text{first}}$.

Answer: (C) $15$

Concept: Sum of the first $n$ even numbers $=n(n+1)$.

1. $\text{Sum}=156\times 157$
2. $\text{Average}=\frac{156\times 157}{156}=157$

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

Answer: (D) $157$

Concept: Take CP = 100 units and follow the mark-up and discount.

1. List price $=125$ units; $\text{SP}=125\times \frac{84}{100}=105$ units.
2. Profit $=105-100=5$ units $=$ ₹64 ⇒ 1 unit $=$ ₹$12.8$.
3. List price $=125\times 12.8=$ ₹$1600$

Trick: Net effect $=25-16-\frac{25\times 16}{100}=5\%$ profit; $\text{CP}=\frac{64}{5\%}=$ ₹$1280$, list $=1280\times 1.25$.

Answer: (B) ₹$1,600$

Concept: Let the numbers be $x$ and $x+2$.

1. $x^2+(x+2{)}^2=2452\Rightarrow 2x^2+4x+4=2452\Rightarrow x^2+2x-1224=0$
2. $(x+36)(x-34)=0\Rightarrow x=34$
3. Numbers 34 and 36: ${34}^2+{36}^2=1156+1296=2452$ ✔; sum $=70$

Trick: Half of 2452 is 1226 ≈ ${35}^2=1225$ ⇒ numbers around 35, i.e. 34 and 36.

Answer: (A) $70$

Concept: Same base: $(a^m{)}^n=a^{mn}$; compare exponents.

1. $x^z=({132}^{0.14}{)}^z={132}^{0.14z}$
2. $y^2=({132}^{0.26}{)}^2={132}^{0.52}$
3. Equate powers: $0.14z=0.52\Rightarrow z=\frac{0.52}{0.14}=\frac{26}{7}\approx 3.71$

Trick: Base is the same, so just divide the exponents: $\frac{2\times 0.26}{0.14}$.

Answer: (D) $3.71$

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

1. $M+N=3519$, $M-N=207\Rightarrow M=1863,N=1656$
2. Ratio $=\frac{1863}{1656}=\frac{9}{8}$
3. Let N's capital $=c$: $\frac{6(c+49000)}{7c}=\frac{9}{8}\Rightarrow 48c+2352000=63c$
4. $c=156800$, so M's capital $=156800+49000=205800$

Trick: Get the profit ratio first ($9:8$), then plug capitals into $\frac{6(c+49000)}{7c}$.

Answer: (C) ₹2,05,800

Concept: $1-2{\sin }^{2}A=\cos 2A$ and $2{\cos }^{2}A-1=\cos 2A$.

1. Numerator $=\sin A(1-2{\sin }^{2}A)=\sin A\cos 2A$
2. Denominator $=\cos A(2{\cos }^{2}A-1)=\cos A\cos 2A$
3. Fraction $=\frac{\sin A}{\cos A}=\tan A$
4. $K\times \tan A=1\Rightarrow K=\cot A$

Trick: Put $A={30}^{\circ }$: fraction $=\frac{1/2-1/4}{3\sqrt{3}/4-\sqrt{3}/2}=\frac{1}{\sqrt{3}}$, so $K=\sqrt{3}=\cot {30}^{\circ }$.

Answer: (A) $\cot A$

Concept: Work on the remaining part step by step.

1. After debt: $100\%-33\%=67\%$ of salary remains
2. Spends $25\%$ of it, saves $75\%$: $0.75\times 0.67=0.5025$ of salary
3. $0.5025S=402\Rightarrow S=\frac{402}{0.5025}=800$

Trick: Check options: $800\times 0.67=536$, $536\times 0.75=402$ ✓.

Answer: (D) $800$

Concept: Same distance: time $\propto \frac{1}{\text{speed}}$.

1. Distance $=100\times 14=1400\text{km}$
2. Disha's time $=\frac{1400}{35}=40$ hours

Trick: $t_2=t_1\times \frac{s_1}{s_2}=14\times \frac{100}{35}=40$.

Answer: (C) 40 hours

Concept: Difference of ratio parts gives one part's value.

1. Largest − smallest $=13-3=10$ parts $=36846$
2. 1 part $=3684.6$
3. Total $=13+3+9=25$ parts $=25\times 3684.6=92115$

Trick: Total $=36846\times \frac{25}{10}=36846\times 2.5$.

Answer: (D) 92,115

Concept: Test each option.

1. Ends in 3 → not divisible by 5; odd → not by 6 or 12
2. Rule of 7: $26753-2\times 3=26747$; $2674-14=2660$; $266-0=266=7\times 38$ ✓
3. Direct: $267533=7\times 38219$

Trick: Odd number kills 6 and 12 at once; last digit kills 5 — only 7 is left.

Answer: (B) 7

Concept: All sides equal + one right angle + parallelogram ⇒ square.

1. Opposite sides parallel and opposite angles equal → parallelogram
2. All sides equal → rhombus
3. One angle ${90}^{\circ }$ → every angle ${90}^{\circ }$ → square

Trick: Rhombus + right angle = square; rectangle needs only opposite sides equal.

Answer: (B) square

Concept: Apply discounts one after another.

1. After 15%: $12000\times 0.85=10200$
2. After 10%: $10200\times 0.9=9180$

Trick: Net discount $=15+10-\frac{15\times 10}{100}=23.5\%$; $12000\times 0.765=9180$.

Answer: (C) ₹9,180

Concept: Average $=\frac{\text{sum}}{\text{number}}$.

1. Sum $=40+49+56+74+37=256$
2. Average $=\frac{256}{5}=51.2$

Trick: Take 50 as base: deviations $-10,-1,+6,+24,-13=+6$; $50+\frac{6}{5}=51.2$.

Answer: (D) $51.2$

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

1. CP per pen $=\frac{180}{12}=15$
2. SP $=22$, profit $=7$
3. $\text{Profit}\%=\frac{7}{15}\times 100=46.67\%$

Trick: $\frac{7}{15}=\frac{1}{3}+\frac{2}{15}=33.33+13.33=46.67$.

Answer: (A) $46.67$

Concept: Total work = LCM of days; find units done.

1. Total $=\text{LCM}(16,22)=176$ units; A $=11$, B $=8$ units/day
2. Together in 2 days: $2\times 19=38$ units
3. Left $=176-38=138$ units $=\frac{138}{176}=\frac{69}{88}$

Trick: Done $=2(\frac{1}{16}+\frac{1}{22})=\frac{19}{88}$, left $=1-\frac{19}{88}$.

Answer: (B) $\frac{69}{88}$