Q&A: practice questions with answers

Concept: Single equivalent discount of $a\%$ and $b\%$ $=a+b-\frac{ab}{100}$.

1. $a=65,b=24$
2. $65+24-\frac{65\times 24}{100}=89-15.6=73.4\%$
3. Check: $100\to 35\to 35\times 0.76=26.6$, so discount $=73.4\%$.

Trick: Multiply the remaining parts: $0.35\times 0.76=0.266$ → discount $=1-0.266=73.4\%$.

Answer: (D) $73.4\%$

Concept: Write the percentage as a fraction and form the ratio.

1. $\frac{38}{100}A=\frac{2}{7}B$
2. $\frac{A}{B}=\frac{2}{7}\times \frac{100}{38}=\frac{200}{266}=\frac{100}{133}$
3. So $A:B=100:133$.

Trick: Cross-check with option: $38\%$ of 100 $=38$ and $\frac{2}{7}\times 133=38$ ✓.

Answer: (C) $100:133$

Concept: If SP of $x$ articles = CP of $y$ articles, then gain/loss $\%=\frac{y-x}{x}\times 100$.

1. SP of 50 = CP of 39, so $x=50,y=39$.
2. $\frac{39-50}{50}\times 100=-\frac{11}{50}\times 100=-22\%$
3. Negative sign → loss of $22\%$.

Trick: SP of more items = CP of fewer items → always a loss; here $\frac{11}{50}=22\%$.

Answer: (C) 22% loss

Concept: Consecutive odd numbers: $n$ and $n+2$.

1. $n^2+(n+2{)}^2=34\Rightarrow 2n^2+4n+4=34\Rightarrow n^2+2n-15=0$
2. $(n+5)(n-3)=0\Rightarrow n=3$ (natural number)
3. Numbers 3 and 5; sum $=8$.

Trick: Try small odd squares: $9+25=34$ → 3 and 5.

Answer: (A) 8

Concept: Same time, equal SI ⇒ Principal × Rate is equal for both parts.

1. $x\times 14\times 6=(22800-x)\times 7\times 6$
2. $2x=22800-x\Rightarrow 3x=22800\Rightarrow x=7600$
3. Sum at $14\%$ = ₹7,600 (at $7\%$ = ₹15,200).

Trick: Principals are in inverse ratio of rates: $\frac{1}{14}:\frac{1}{7}=1:2$ → $\frac{1}{3}\times 22800=7600$.

Answer: (A) 7,600

Concept: $\text{MP}\times (1-d)=\text{SP}=\text{CP}\times (1+g)$.

1. CP $=1600\times 0.65\times 0.95=988$
2. Required SP for $25\%$ gain $=988\times 1.25=1235$
3. New list price $=\frac{1235}{0.95}=1300$

Trick: Direct: $\text{MP}=1600\times 0.65\times 0.95\times \frac{1.25}{0.95}=1600\times 0.65\times 1.25=1300$ (the 0.95 cancels).

Answer: (D) ₹1,300

Concept: Margin = (winner% − loser%) of valid votes.

1. Let registered voters $=R$. Polled $=0.87R$; valid $=0.95\times 0.87R=0.8265R$.
2. Margin $=(52-48)\%=4\%$ of valid $=0.04\times 0.8265R=0.03306R$.
3. $0.03306R=1653\Rightarrow R=\frac{1653}{0.03306}=50000$

Trick: Test option (C): $50000\times 0.87\times 0.95\times 0.04=1653$ ✓.

Answer: (C) 50000

Concept: Curved (total) surface area of a sphere $=4\pi r^2$.

1. $r=2.8\text{cm}$, $r^2=7.84$
2. $4\times \frac{22}{7}\times 7.84=4\times 22\times 1.12$
3. $=88\times 1.12=98.56{\text{cm}}^2$

Trick: $r=2.8=\frac{14}{5}$ makes $\frac{22}{7}$ cancel neatly: $4\times \frac{22}{7}\times \frac{196}{25}=\frac{2464}{25}=98.56$.

Answer: (A) $98.56{\text{cm}}^2$

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

1. Sanjay’s investment $=\frac{1900}{15600}\times 62400=1900\times 4=7600$
2. Ritik’s investment $=62400-7600=54800$
3. Difference $=54800-7600=47200$

Trick: Investment ÷ profit $=\frac{62400}{15600}=4$ → multiply each profit share by 4.

Answer: (B) ₹47,200

Concept: Increasing a number by $100\%$ doubles it.

1. $x\times (1+\frac{100}{100})=2740$
2. $2x=2740\Rightarrow x=1370$

Trick: $+100\%$ = ×2, so just halve: $\frac{2740}{2}=1370$.

Answer: (B) 1370

Concept: $\text{Profit}\%=\frac{\text{SP}-\text{CP}}{\text{CP}}\times 100$, both for one article.

1. CP of one sharpener $=\frac{150}{12}=12.5$ (₹).
2. SP of one sharpener = ₹19, so profit $=19-12.5=6.5$.
3. $\text{Profit}\%=\frac{6.5}{12.5}\times 100=52\%$.

Trick: Compare per dozen: SP $=19\times 12=228$ vs CP 150, gain 78; $\frac{78}{150}=52\%$.

Answer: (D) 52

Concept: Angle sum of a triangle is ${180}^{\circ }$; an exterior angle equals the sum of the two interior opposite angles.

1. In right $△BDC$: $\angle C={180}^{\circ }-{90}^{\circ }-{65}^{\circ }={25}^{\circ }$.
2. $\angle AEB$ is the exterior angle of $△AEC$ at E.
3. $\angle AEB=\angle EAC+\angle ECA={48}^{\circ }+{25}^{\circ }={73}^{\circ }$.

Trick: In a right triangle the two acute angles add to ${90}^{\circ }$: $\angle C={90}^{\circ }-{65}^{\circ }$ instantly.

Answer: (B) ${73}^{\circ }$

Concept: Interest of the $n^{th}$ year = rate% of the amount at the end of year $(n-1)$.

1. Amount after 1 year $=1.12P$.
2. Interest in 2nd year $=12\%$ of $1.12P=0.1344P$.
3. $0.1344P=9828\Rightarrow P=\frac{9828}{0.1344}=73125$.

Trick: Check: $73125\times 1.12=81900$; 12% of 81900 = 9828.

Answer: (C) ₹73,125

Concept: Mode is the observation that occurs most often.

1. Count repeats: 75 appears 3 times.
2. 65, 82, 84, 89 appear only twice each; all others once.
3. So the mode is 75.

Trick: Scan for each number in options only: 81, 70, 67 occur once, 75 thrice.

Answer: (B) 75

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

1. Sum $=158\times 159$.
2. Average $=\frac{158\times 159}{158}=159$.

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

Answer: (D) 159

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

1. ${47}^{-11}÷{47}^{16}={47}^{-11-16}={47}^{-27}$.
2. ${47}^{-27}\times {47}^{-10}={47}^{-37}$.

Trick: Add exponents with sign: $-11-16-10=-37$.

Answer: (B) ${47}^{-37}$

Concept: Expand and simplify; the $x^2$ terms cancel, leaving a linear equation.

1. $2(4x^2-6)=8x^2-12$; $4(2x^2+9x-3)=8x^2+36x-12$.
2. Difference: $8x^2-12-8x^2-36x+12=-36x$.
3. $-36x=18\Rightarrow x=-\frac{1}{2}$.

Trick: Notice $8x^2$ on both sides cancels at once – only $-36x$ remains.

Answer: (C) $-\frac{1}{2}$

Concept: $(a^m{)}^n=a^{mn}$; equal powers of the same base mean equal exponents.

1. $x^z=({54}^{0.2}{)}^z={54}^{0.2z}$.
2. $y^3=({54}^{0.47}{)}^3={54}^{1.41}$.
3. $0.2z=1.41\Rightarrow z=7.05$.

Trick: $z=\frac{3\times 0.47}{0.2}=\frac{1.41}{0.2}=7.05$.

Answer: (A) 7.05

Concept: Take distance MN = LCM of travel times; then use relative speed for the part left after both start.

1. A takes 9 h (6:45 AM–3:45 PM); B takes 6 h. Let MN = 18 units: A = 2 units/h, B = 3 units/h.
2. By 8:45 AM, A covers $2\times 2=4$ units; remaining 14 units.
3. Time to meet $=\frac{14}{2+3}=2.8\text{h}=2\text{h}48\text{min}$.
4. 8:45 AM + 2 h 48 min = 11:33 AM.

Trick: Distance = LCM(9, 6) = 18 keeps all speeds whole numbers.

Answer: (C) 11:33 AM

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

1. $32:40:68=8:10:17$, total 35 parts.
2. One part $=\frac{3210}{35}$.
3. Z − Y $=7$ parts $=\frac{7\times 3210}{35}=\frac{3210}{5}=642$.

Trick: Difference is $\frac{7}{35}=\frac{1}{5}$ of total profit: $\frac{3210}{5}=642$.

Answer: (C) ₹642