Q&A: practice questions with answers

Concept: Required number $=\text{HCF}(a-r_1,b-r_2)$.

1. $48-9=39$ and $46-7=39$
2. $\text{HCF}(39,39)=39$
3. Check: $48=39+9$, $46=39+7$ ✔

Trick: Subtract the remainders; when both results are equal, that number itself is the answer.

Answer: (D) $39$

Concept: Successive discounts multiply: price left $=(1-d_1)(1-d_2)(1-d_3)$.

1. $0.74\times 0.71=0.5254$
2. $0.5254\times 0.98=0.514892$
3. Single discount $=(1-0.514892)\times 100=48.5108\%\approx 48.51\%$

Trick: Combine two at a time: $26+29-\frac{26\times 29}{100}=47.46$, then $47.46+2-\frac{47.46\times 2}{100}=48.51$.

Answer: (C) $48.51\%$

Concept: Area of a quadrilateral $=\frac{1}{2}\times d\times (h_1+h_2)$, where $h_1,h_2$ are perpendiculars on diagonal $d$.

1. $\frac{1}{2}\times AC\times (11+12)=299$
2. $AC\times 23=598$
3. $AC=26\text{cm}$

Trick: Diagonal splits it into two triangles on the same base AC – just add their heights.

Answer: (A) $26$

Concept: For 2 years: $\text{CI}-\text{SI}=P{\left(\frac{R}{100}\right)}^2$.

1. $44=P\times {\left(\frac{4}{100}\right)}^2=\frac{16P}{10000}$
2. $P=\frac{44\times 10000}{16}=27500$

Trick: $R=4\%$ → $R^2=16$; $P=\frac{\text{diff}\times 10000}{16}$.

Answer: (C) ₹$27500$

Concept: A ratio needs both quantities in the same unit.

1. $17\text{ days}=17\times 24=408\text{ hours}$
2. Ratio $=\frac{39}{408}$
3. Divide by 3: $\frac{13}{136}$

Trick: Convert the bigger unit to the smaller one first, then cancel the HCF.

Answer: (D) $13:136$

Concept: $SP=MP\times (1-\frac{\text{discount}}{100})$

1. Let $CP=100$, so $MP=145$.
2. Discount $60\%$: $SP=145\times 0.40=58$
3. Loss $=100-58=42$, i.e. $42\%$ loss.

Trick: Net effect $=45-60-\frac{45\times 60}{100}=-42\%$ (minus = loss).

Answer: (B) $42\%$ loss

Concept: $22=2\times 11$, so the number must be even and divisible by 11 (alternate-digit difference is 0 or a multiple of 11).

1. 7019 and 8371 are odd – rejected.
2. 7986: $(7+8)-(9+6)=0$ ✓ and even ✓.
3. 7664: $(7+6)-(6+4)=3$ ✗.

Trick: For a composite divisor, test its co-prime factors separately.

Answer: (A) 7986

Concept: $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$

1. $\frac{0.038025}{0.004225}=\frac{38025}{4225}=9$
2. $\sqrt{9}=3$
3. Check: $\sqrt{0.038025}=0.195$, $\sqrt{0.004225}=0.065$, and $\frac{0.195}{0.065}=3$.

Trick: Same number of decimal places → just divide the whole numbers inside the root.

Answer: (D) 3

Concept: $(a\pm b{)}^2=a^2+b^2\pm 2ab$

1. $(a-b{)}^2=169-20=149$
2. $(a+b{)}^2=169+20=189$
3. Since $a>b$, $\frac{a-b}{a+b}=\sqrt{\frac{149}{189}}$

Trick: Square the required ratio: ${\left(\frac{a-b}{a+b}\right)}^2=\frac{a^2+b^2-2ab}{a^2+b^2+2ab}$.

Answer: (C) $\sqrt{\frac{149}{189}}$

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

1. Net rate $=\frac{1}{3}+\frac{1}{57}-\frac{1}{3}=\frac{1}{57}$ per hour.
2. Full cistern takes 57 hours.
3. One-third takes $\frac{57}{3}=19$ hours.

Trick: The 3-hour filler and 3-hour emptier cancel each other – only the 57-hour tap works.

Answer: (A) 19

Concept: Equal roots when discriminant $D=b^2-4ac=0$.

1. Here $a=7$, $b=-4$, $c=k$.
2. $(-4{)}^2-4\times 7\times k=0$
3. $16=28k\Rightarrow k=\frac{16}{28}=\frac{4}{7}$

Trick: For equal roots, $k=\frac{b^2}{4a}=\frac{16}{28}$.

Answer: (A) $\frac{4}{7}$

Concept: Income = Saving + Expenditure.

1. Saving $=30\%$, so expenditure $=70\%$ of income.
2. $70\%=560\Rightarrow 1\%=8$
3. Income $=100\times 8=800$

Trick: $560\times \frac{10}{7}=800$.

Answer: (B) 800

Concept: Use the 3-4-5 right triangle.

1. $\sin A=\frac{4}{5}\Rightarrow \cos A=\frac{3}{5}$, $\tan A=\frac{4}{3}$
2. $2+\tan A=\frac{10}{3}$, $4+\cos A=\frac{23}{5}$
3. Product $=\frac{10}{3}\times \frac{23}{5}=\frac{46}{3}$

Trick: Value is clearly greater than $2\times 4=8$, so only $\frac{46}{3}$ or $\frac{208}{15}$ fit – compute once.

Answer: (C) $\frac{46}{3}$

Concept: Form two linear equations in present ages P and Q.

1. $4P-Q=35$ …(i)
2. After 8 years: $2(Q+8)=6(P+8)-40\Rightarrow 2Q=6P-8\Rightarrow Q=3P-4$ …(ii)
3. Put (ii) in (i): $4P-3P+4=35\Rightarrow P=31$
4. $Q=3\times 31-4=89$

Trick: Check options: $Q=89\Rightarrow P=\frac{89+35}{4}=31$; then $2\times 97=194=6\times 39-40$ ✓.

Answer: (C) 89

Concept: Correct total = wrong total − wrong value + correct value.

1. Wrong total $=10\times 55=550$
2. Correct total $=550-48+84=586$
3. Correct average $=\frac{586}{10}=58.6$

Trick: Change in average $=\frac{84-48}{10}=3.6$, so $55+3.6=58.6$.

Answer: (D) 58.6

Concept: Take total work = LCM of days; work backwards from the end.

1. Total work $=\text{LCM}(16,10)=80$ units; Ramesh $=5$, Suresh $=8$ units/day.
2. Last 4 days Suresh alone: $4\times 8=32$ units.
3. Remaining $80-32=48$ units done together at 13 units/day: $\frac{48}{13}$ days.
4. Total $=\frac{48}{13}+4=\frac{100}{13}=7\frac{9}{13}$ days.

Trick: "Leaves x days before completion" → subtract the other person's x-day work first.

Answer: (B) $7\frac{9}{13}$ days

Concept: Sum = average × count.

1. First two numbers: $2\times 77=154$
2. Remaining two $=655-154-101=400$
3. Average $=\frac{400}{2}=200$

Trick: Subtract known parts from the total, then divide by how many are left.

Answer: (C) 200

Concept: Exterior angle bisectors at B and C meet at the excentre P, and $\angle P={90}^{\circ }-\frac{\angle A}{2}$.

1. $\angle CBD={180}^{\circ }-B$, $\angle BCE={180}^{\circ }-C$; halves are ${90}^{\circ }-\frac{B}{2}$ and ${90}^{\circ }-\frac{C}{2}$.
2. In $△BPC$: $\angle P={180}^{\circ }-({90}^{\circ }-\frac{B}{2})-({90}^{\circ }-\frac{C}{2})=\frac{B+C}{2}={90}^{\circ }-\frac{A}{2}$
3. $\angle P={90}^{\circ }-{42}^{\circ }={48}^{\circ }$

Trick: Internal bisectors: ${90}^{\circ }+\frac{A}{2}$; external bisectors: ${90}^{\circ }-\frac{A}{2}$.

Answer: (C) ${48}^{\circ }$

Concept: For $n$ odd, median = ${\left(\frac{n+1}{2}\right)}^{\text{th}}$ value after arranging in order.

1. Ascending order: 24, 26, 38, 39, 43, 49, 60, 61, 62, 72, 95
2. $n=11$, so median is the $6^{\text{th}}$ value.
3. Median $=49$

Trick: Always sort first – the 6th number in the given (unsorted) list is not the median.

Answer: (D) 49

Concept: In $a:b::c:d$, product of extremes = product of means.

1. $7:14::8.7:x\Rightarrow x=\frac{14\times 8.7}{7}=17.4$
2. $5:35::7:y\Rightarrow y=\frac{35\times 7}{5}=49$
3. $x:y=17.4:49=174:490=87:245$

Trick: $x=2\times 8.7$ and $y=7\times 7$ – the ratios are just "double" and "seven times".

Answer: (C) $87:245$