Q&A: practice questions with answers

Concept: Fourth proportional $y$ of $a,b,c$: $a:b=c:y\Rightarrow y=\frac{bc}{a}$.

1. $y=\frac{(x+1)(2x+3)}{x}=3x+3=3(x+1)$
2. Cancel $(x+1)$: $\frac{2x+3}{x}=3$
3. $2x+3=3x\Rightarrow x=3$

Trick: Option check: $x=3$ gives $3:4=9:12$ and $y=12=3\times 3+3$. ✔

Answer: (B) 3

Concept: Successive discounts multiply: $SP=MP\times (1-d_1)(1-d_2)$.

1. $SP=5000\times \frac{80}{100}\times \frac{70}{100}$
2. $=5000\times 0.56=2800$

Trick: Net discount of 20% and 30% $=20+30-\frac{20\times 30}{100}=44\%$; $56\%$ of ₹5,000 = ₹2,800.

Answer: (B) ₹2,800

Concept: $a^3-b^3=(a-b)(a^2+ab+b^2)$, so $\frac{a^3-b^3}{a^2+ab+b^2}=a-b$.

1. Here $a=1.4,b=1.2$
2. Value $=a-b=1.4-1.2=0.2$

Trick: Numerator cubes with minus, denominator $a^2+ab+b^2$ → answer is simply $a-b$.

Answer: (B) 0.2

Concept: Mean proportional of $a$ and $b$ $=\sqrt{ab}$.

1. $ab=0.15\times 0.000135=0.00002025$
2. $\sqrt{0.00002025}=0.0045$ (since ${45}^2=2025$)

Trick: Count decimals: 8 decimal places under the root → 4 decimal places in the answer.

Answer: (C) 0.0045

Concept: Form equations from present ages; each person gains the same years.

1. Let Priyal $=x$, brother $=x+5$
2. After 4 years: $(x+4)+(x+9)=39$
3. $2x+13=39\Rightarrow x=13$

Trick: Present sum $=39-8=31$; with difference 5, smaller age $=\frac{31-5}{2}=13$.

Answer: (B) 13 years

Concept: Solve brackets innermost first: ( ) → { } → [ ], then division.

1. $15-9=6$, $7\times 6=42$
2. $\{42-28\}=14$
3. $[12+14]=26$
4. $156÷26=6$

Trick: Inside-out order of brackets avoids sign mistakes.

Answer: (B) 6

Concept: $\text{Time}=\frac{\text{Distance}}{\text{Speed}}$.

1. Time $=\frac{12}{6}=2$ hours

Trick: Every hour he covers 6 km, so 12 km needs 2 such hours.

Answer: (B) 2 hours

Concept: $SP=MP\times (1-d_1)(1-d_2)$.

1. After 25%: $2000\times 0.75=1500$
2. After 10%: $1500\times 0.90=1350$

Trick: Net discount $=25+10-\frac{25\times 10}{100}=32.5\%$; $67.5\%$ of 2000 = 1350.

Answer: (A) ₹1,350

Concept: Exterior angle = sum of the two interior opposite angles; in a triangle the larger angle faces the larger side.

1. $\angle PRT=P+Q$, $\angle RQS=P+R$
2. Sum: $2P+Q+R={180}^{\circ }+P={220}^{\circ }\Rightarrow P={40}^{\circ }$, $Q+R={140}^{\circ }$
3. $\angle PRT>\angle RQS\Rightarrow Q>R$, so $Q>{70}^{\circ }$: $Q$ is the largest angle → $PR$ is the largest side
4. Only option (D) has $PR$ largest; it holds with $P<R<Q$, giving $QR<PQ<PR$

Trick: Side opposite an angle carries the other two letters: angle $Q$ ↔ side $PR$.

Answer: (D) $QR<PQ<PR$

Concept: Median $=l+\frac{\frac{N}{2}-cf}{f}\times h$.

1. $N=100$, $\frac{N}{2}=50$; cumulative frequencies 20, 40, 60, … → median class 58–60
2. $l=58,cf=40,f=20,h=2$
3. Median $=58+\frac{50-40}{20}\times 2=58+1=59$

Trick: Equal frequencies in equal classes → median is simply the middle of the whole range: $\frac{54+64}{2}=59$.

Answer: (C) 59

Concept: Net work per hour = sum of filling rates − emptying rate.

1. Net rate $=\frac{1}{4}+\frac{1}{15}-\frac{1}{15}=\frac{1}{4}$ per hour
2. Time for $\frac{1}{4}$ cistern $=\frac{1}{4}÷\frac{1}{4}=1$ hour

Trick: Second and third taps have equal rates and cancel — only the 4-hour tap matters.

Answer: (A) 1

Concept: Subtract equations having the same pen term to eliminate it.

1. $4n+6p=228$ and $7n+6p=333$
2. Subtract: $3n=105\Rightarrow n=35$

Trick: Same number of pens in both → extra 3 notebooks cost $333-228=105$.

Answer: (A) ₹ 35

Concept: Total string length = circumference; it is shared by the rectangle and square perimeters.

1. Length $=2\pi r=2\times \frac{22}{7}\times 21=132\text{cm}$
2. Rectangle: $l=2b\Rightarrow$ perimeter $=2(2b+b)=6b$; square perimeter $=s$
3. $6b+s=132$ and $6b-s=24\Rightarrow 12b=156\Rightarrow b=13\text{cm}$

Trick: Sum and difference given → bigger part $=\frac{132+24}{2}=78=6b$, so $b=13$.

Answer: (B) 13 cm

Concept: Mean proportional of $a,b$ $=\sqrt{ab}$.

1. $\sqrt{0.16\times 0.49}=\sqrt{0.16}\times \sqrt{0.49}$
2. $=0.4\times 0.7=0.28$

Trick: Both are perfect squares — take roots separately and multiply.

Answer: (C) 0.28

Concept: Effective SP/CP ratio $=(1+\text{profit})\times \frac{\text{true weight}}{\text{false weight}}$.

1. He charges for 1000 g at $115\%$ but gives only 950 g
2. Ratio $=1.15\times \frac{1000}{950}=\frac{1150}{950}=1.2105$
3. Profit $\approx 21.05\%$

Trick: Use $\frac{1000\times (100+15)}{950}-100=21.05\%$ directly.

Answer: (C) $21.05\%$

Concept: Final $=$ Original $\times (1+r_1)(1-r_2)$.

1. Factor $=1.20\times 0.76=0.912$
2. Original $=\frac{1938}{0.912}=2125$

Trick: Check option: $2125\times 1.2=2550$, $2550\times 0.76=1938$. ✔

Answer: (C) ₹2,125

Concept: Product of two numbers = HCF $\times$ LCM.

1. Other number $=\frac{27\times 324}{81}$
2. $=\frac{8748}{81}=108$

Trick: Check: HCF(81, 108) $=27$ and LCM $=324$. ✔

Answer: (A) 108

Concept: For 2 years at $r\%$: CI $=$ SI $+$ interest on interest; at 10%, CI $=21\%$ of P, SI $=20\%$ of P.

1. $0.21P=199.5\Rightarrow P=950$
2. SI $=950\times \frac{10\times 2}{100}=190$

Trick: SI : CI for 2 years at 10% $=20:21$ → SI $=\frac{199.5\times 20}{21}=190$.

Answer: (B) 190

Concept: Mode is the value that occurs most often.

1. Counts: 2 → 2 times, 3 → 3 times, 4 → 1 time, 5 → 4 times
2. Highest frequency is for 5

Trick: Tally quickly; the most repeated value wins.

Answer: (C) 5

Concept: Put $t=\frac{x}{x+1}$; then $t+\frac{1}{t}=\frac{5}{2}\Rightarrow t=2$ or $t=\frac{1}{2}$.

1. $\frac{x}{x+1}=2\Rightarrow x=2x+2\Rightarrow x=-2$
2. $\frac{x}{x+1}=\frac{1}{2}\Rightarrow 2x=x+1\Rightarrow x=1$

Trick: $t+\frac{1}{t}=\frac{5}{2}=2+\frac{1}{2}$ — read the two values of $t$ directly.

Answer: (D) 1 or −2