Q&A: practice questions with answers

Concept: Open the brackets carefully and combine like terms.

1. $3(\frac{4}{3}{x}^2-24x+18)=4x^2-72x+54$
2. $-4(x^2+7x-18)=-4x^2-28x+72$
3. Add: $(4x^2-4x^2)+(-72x-28x)+(54+72)=-100x+126$

Trick: The $x^2$ terms cancel, so check only the $x$ and constant terms: $-72-28=-100$, $54+72=126$.

Answer: (B) $-100x+126$

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

1. SP of 50 = CP of 16, so SP is less than CP → loss.
2. Loss on 50 books $=$ CP of $(50-16)=34$ books
3. $\text{Loss}\%=\frac{34}{50}\times 100=68\%$

Trick: Smaller number on the CP side → loss; $\frac{\text{difference}}{\text{SP-side count}}\times 100$.

Answer: (C) 68% loss

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

1. In right $△BDC$: $\angle C={180}^{\circ }-{90}^{\circ }-{29}^{\circ }={61}^{\circ }$
2. In $△AEC$, $\angle AEB$ is the exterior angle at E.
3. $\angle AEB=\angle CAE+\angle ACE={25}^{\circ }+{61}^{\circ }={86}^{\circ }$

Trick: Exterior angle = sum of the two far interior angles – no need to find $\angle AEC$.

Answer: (B) ${86}^{\circ }$

Concept: Metal volume = outer volume − inner (hollow) volume. Open box: thickness is reduced from 2 sides in length and width, but only from the bottom in height.

1. Outer volume $=60\times 40\times 23=55200{\text{cm}}^3$
2. Inner: $54\times 34\times 20=36720{\text{cm}}^3$
3. Metal $=55200-36720=18480{\text{cm}}^3$
4. Weight $=18480\times 0.5=9240\text{g}=9.24\text{kg}$

Trick: Open box → height reduces by thickness only once ($23-3=20$).

Answer: (B) 9.24

Concept: Profit share ∝ capital × time.

1. $M+N=4064$, $M-N=508\Rightarrow M=2286,N=1778$; ratio $=9:7$
2. Let N's capital $=x$: $\frac{4(x+39000)}{7x}=\frac{9}{7}$
3. $4x+156000=9x\Rightarrow x=31200$
4. M's capital $=31200+39000=70200$

Trick: $2286:1778$ – divide both by 254 to get $9:7$ quickly.

Answer: (D) ₹70,200

Concept: Successive discounts are applied one after another on the reduced price.

1. After 10%: $5000\times \frac{90}{100}=4500$
2. After 5%: $4500\times \frac{95}{100}=4275$

Trick: Net discount $=10+5-\frac{10\times 5}{100}=14.5\%$, so SP $=5000\times 0.855=4275$.

Answer: (D) ₹4,275

Concept: Total literates = male literates + female literates.

1. Females $=374000-165000=209000$
2. Total literates $=21\%\text{ of }374000=78540$
3. Male literates $=40\%\text{ of }165000=66000$
4. Female literates $=78540-66000=12540$; $\frac{12540}{209000}\times 100=6\%$

Trick: Always find the female base first; percentage is of females, not of total population.

Answer: (B) $6\%$

Concept: Work per minute: filling pipes add, the leak (outlet) subtracts.

1. $\frac{1}{16}+\frac{1}{32}-\frac{1}{x}=\frac{1}{12}$
2. $\frac{3}{32}-\frac{1}{12}=\frac{9-8}{96}=\frac{1}{96}$
3. So $\frac{1}{x}=\frac{1}{96}\Rightarrow x=96$ minutes

Trick: LCM method: capacity $=96$ units; A $=6$, B $=3$, together $=8$ units/min ⇒ outlet $=9-8=1$ unit/min ⇒ $96$ min.

Answer: (D) 96 minutes

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

1. $\sqrt{490\times 10}=\sqrt{4900}$
2. $=70$

Trick: Check: $490:70=70:10$ (both equal 7).

Answer: (A) $70$

Concept: Solve innermost bracket first, then × , then +, then ÷ outside.

1. $(5-7)=-2$
2. $9\times (-2)=-18$
3. $26-18=8$
4. $616÷8=77$

Trick: Watch the negative sign: $26+9\times (-2)$ is $8$, not $70$.

Answer: (C) $77$

Concept: Average = sum of observations ÷ number of observations.

1. Sum $=40+49+56+74+39=258$
2. Average $=\frac{258}{5}=51.6\text{kg}$

Trick: Assume mean $50$: deviations $-10,-1,+6,+24,-11$ sum to $+8$; $50+\frac{8}{5}=51.6$.

Answer: (D) $51.6$

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

1. Ratio $=1340:1100:2440=67:55:122$
2. B's 55 parts $=814\Rightarrow 1$ part $=14.8$
3. C − A $=(122-67)=55$ parts $=814$

Trick: C − A investment $=2440-1340=1100=$ B's investment, so the profit difference equals B's share directly.

Answer: (D) ₹814

Concept: Check options: sum must be 84 and LCM must be 120.

1. All options sum to 84, so test LCM.
2. $60=2^2\times 3\times 5,24=2^3\times 3$
3. LCM $=2^3\times 3\times 5=120$ ✓

Trick: Both numbers must divide 120 – only 60 and 24 are factors of 120.

Answer: (B) 60, 24

Concept: Make decimal places equal, find LCM of the numbers without decimals, then put the decimal back.

1. $5.1=5.100$ and $0.102$ → $5100$ and $102$ (thousandths)
2. $5100=102\times 50$, so LCM $=5100$
3. Restore 3 decimal places: $5.100=5.1$

Trick: If one number is a multiple of the other, the larger is the LCM: $5.1=0.102\times 50$.

Answer: (B) $5.1$

Concept: $\text{SI}=\frac{P\times R\times T}{100}$; Amount = P + SI.

1. $8400=\frac{P\times 2.5\times 4.2}{100}\Rightarrow P=\frac{840000}{10.5}=80000$
2. New SI $=\frac{80000\times 8.1\times 10.25}{100}=66420$
3. Amount $=80000+66420=146420$

Trick: $4\frac{1}{5}=4.2$ and $10\frac{1}{4}=10.25$ – convert mixed fractions to decimals first.

Answer: (C) ₹1,46,420

Concept: Total = average × count; split total in the given ratio.

1. Total $=16860\times 3=50580$
2. Ratio sum $=3+5+7=15$
3. Costliest $=\frac{7}{15}\times 50580=23604$

Trick: Average of 15 parts over 3 items ⇒ 1 part $=\frac{16860}{5}=3372$; $7\times 3372=23604$.

Answer: (A) 23604

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

1. CP of one eraser $=\frac{150}{12}=12.5$
2. SP $=42$, profit $=29.5$
3. $\frac{29.5}{12.5}\times 100=236\%$

Trick: Per dozen: CP $=150$, SP $=42\times 12=504$; profit $=354$; $\frac{354}{150}\times 100=236\%$.

Answer: (A) 236

Concept: Area $=\sqrt{s(s-a)(s-b)(s-c)}$, where $s=\frac{a+b+c}{2}$.

1. $s=\frac{11+15+16}{2}=21$
2. $s-a=10,s-b=6,s-c=5$
3. Area $=\sqrt{21\times 10\times 6\times 5}=\sqrt{6300}\approx 79.4{\text{cm}}^2$

Trick: ${79}^2=6241$ and ${80}^2=6400$; $6300$ is nearer $6241$, so the area rounds to 79.

Answer: (A) $79{\text{cm}}^2$

Concept: Time $=\frac{\text{Distance}}{\text{Speed}}$. Let Pranay's speed $=p$, Aman's $=a$.

1. $\frac{60}{p}-\frac{60}{a}=6$ … (1)
2. $\frac{60}{a}-\frac{60}{2p}=9$ … (2)
3. Add: $\frac{60}{p}-\frac{30}{p}=15\Rightarrow \frac{30}{p}=15\Rightarrow p=2\text{km/h}$

Trick: Option check: $p=2$ ⇒ Pranay takes 30 h, Aman 24 h, doubled speed 15 h $=24-9$ ✓.

Answer: (B) $2\text{km/hr}$

Concept: Find both values separately, then subtract.

1. $65\%\text{ of }55=\frac{65\times 55}{100}=35.75$
2. $\frac{2}{5}\text{ of }15=6$
3. Difference $=35.75-6=29.75$

Trick: Quick: $0.65\times 55=33+2.75=35.75$ (60% plus 5%).

Answer: (C) $29.75$