Q&A: practice questions with answers

Concept: One-day work = $\frac{1}{\text{days}}$; work done = rate × time.

1. A's 1-day work $=\frac{1}{12}$, B's $=\frac{1}{6}$. Together = $\frac{1}{12}+\frac{2}{12}=\frac{3}{12}=\frac{1}{4}$ per day.
2. In 2 days: $2\times \frac{1}{4}=\frac{1}{2}$ work done. Remaining $=\frac{1}{2}$.
3. A alone: $\frac{1}{2}÷\frac{1}{12}=6$ days.

Trick: LCM method: total work = LCM(12, 6) = 12 units; A = 1, B = 2 units/day. 2 days → 6 units; left 6 units ÷ 1 = 6 days.

Answer: (D) 6

Concept: Gap between two scores in % = gap in marks (pass mark lies between them).

1. 91% score is 187 above pass mark; 36% score is 60.5 below pass mark.
2. So $(91\%-36\%)=55\%$ of max $=187+60.5=247.5$.
3. Max marks = $247.5\times \frac{100}{55}=450$.

Trick: Passed by + failed by = difference of % → just add the two marks and divide by the % gap.

Answer: (D) 450

Concept: Sum 5 years ago + $2\times 5$ = present sum; then use $\frac{\text{sum}-\text{difference}}{2}$ for the younger.

1. Present sum = $60+5+5=70$ years.
2. $\text{Ayushi}-\text{Nidhi}=14$.
3. Nidhi = $\frac{70-14}{2}=28$ years.

Trick: Younger = $\frac{\text{Sum}-\text{Difference}}{2}$; elder = $\frac{\text{Sum}+\text{Difference}}{2}$.

Answer: (A) 28

Concept: Mode = the value that occurs most often.

1. Count: 52 → 4 times; 50 → 3 times; 44, 45, 47, 53 → 2 times each; others once.
2. Highest frequency = 4 for 52.

Trick: Only count the option values: 50 (3), 52 (4), 47 (2), 44 (2) – done in seconds.

Answer: (B) 52

Concept: Same direction: relative speed = difference; distance = sum of lengths.

1. Relative speed = $72-36=36$ km/h $=36\times \frac{5}{18}=10$ m/s.
2. Distance in 27 s = $10\times 27=270$ m $=L+180$.
3. $L=270-180=90$ m.

Trick: km/h → m/s: multiply by $\frac{5}{18}$ (36 km/h = 10 m/s, 72 km/h = 20 m/s).

Answer: (B) 90 m

Concept: Mode = 3 Median − 2 Mean.

1. Mode = $3\times 51-2\times 19$
2. $=153-38=115$

Trick: Remember "3 Me − 2 M": median gets 3, mean gets 2.

Answer: (A) 115

Concept: Overall profit % = weighted sum of each part's profit % (equal cost per item).

1. Take 100 items of CP ₹1: 40 at +17% → +6.8.
2. Remaining 60; 25% of 60 = 15 at +22% → +3.3.
3. Rest 45 at −10% → −4.5.
4. Net = 6.8 + 3.3 − 4.5 = 5.6 on 100 → 5.6%.

Trick: Take total = 100 so each part's share × its % gives the answer directly.

Answer: (D) 5.6%

Concept: Profit ratio = capital × time.

1. Kirti: $35,000\times 12$; Vaishnavi: $V\times 9$.
2. $35,000\times 12:9V=7:6$
3. $9V=4,20,000\times \frac{6}{7}=3,60,000\to V=$ ₹40,000.

Trick: Check options: $40,000\times 9=3,60,000$ and 4,20,000 : 3,60,000 = 7 : 6 ✓.

Answer: (B) ₹40,000

Concept: ${\sec }^2\theta -1={\tan }^2\theta$.

1. ${\sec }^2{60}^{\circ }-1={\tan }^2{60}^{\circ }$
2. $=(\sqrt{3}{)}^2=3$

Trick: Directly: $\sec {60}^{\circ }=2\to 4-1=3$.

Answer: (D) 3

Concept: Same base ⇒ equate powers; $\sqrt{27}=3^{3/2}$.

1. $2a-b=\frac{3}{2}$ and $a+b=\frac{3}{2}$.
2. Add: $3a=3\to a=1$.
3. $b=\frac{3}{2}-1=\frac{1}{2}$.

Trick: Subtracting the two equations gives $a-2b=0\to a=2b$; then $3b=\frac{3}{2}\to b=\frac{1}{2}$.

Answer: (B) $\frac{1}{2}$

Concept: $a:b=c:d\Rightarrow a\times d=b\times c$.

1. $8\times x=12\times 20$
2. $x=\frac{240}{8}=30$

Trick: 8 → 20 is ×2.5, so $12\times 2.5=30$.

Answer: (C) 30

Concept: A number is divisible by 4 if its last two digits are divisible by 4.

1. Last two digits = A4.
2. A = 0 → 04 = 4, divisible by 4 ✓ (also 24, 44, 64, 84).
3. Least value = 0.

Trick: Check the smallest option first – 04 works, stop.

Answer: (A) 0

Concept: Add/subtract whole parts and fraction parts separately.

1. Whole parts: $2+1-1=2$.
2. Fractions: $\frac{1}{2}+\frac{3}{4}-\frac{1}{4}=\frac{1}{2}+\frac{1}{2}=1$.
3. Total = $2+1=3$.

Trick: In decimals: $2.5+1.75-1.25=3$.

Answer: (C) 3

Concept: x as % of y = $(\frac{x}{y})\times 100$; maximum marks are irrelevant.

1. $\frac{B}{A}=\frac{50}{80}$
2. $=\frac{5}{8}\times 100=62.5\%$

Trick: $\frac{5}{8}=0.625\to 62.5\%$ (remember $\frac{1}{8}=12.5\%$).

Answer: (A) 62.5%

Concept: Quarterly: rate = $\frac{R}{4}$ per quarter, n = number of quarters.

1. Rate = $\frac{12}{4}=3\%$ per quarter; 6 months = 2 quarters.
2. $A=24,000\times (1.03{)}^2=24,000\times 1.0609$
3. = ₹25,461.60

Trick: Effective rate for 2 periods at 3% = $3+3+\frac{3\times 3}{100}=6.09\%$ → $24,000\times 6.09\%=1,461.60$.

Answer: (A) ₹25,461.60

Concept: Successive discounts multiply: price $\times (1-d_1)(1-d_2)(1-d_3)$.

1. After 10%: $50,000\times 0.90=45,000$.
2. After 12%: $45,000\times 0.88=39,600$.
3. After 5% cashback: $39,600\times 0.95=$ ₹37,620.

Trick: Answer must be a multiple of $0.9\times 0.88\times 0.95\times 50,000$ – only 37,620 fits; $39,600-1,980=37,620$.

Answer: (B) ₹37,620

Concept: Area of regular hexagon = $\frac{3\sqrt{3}}{2}{a}^2$.

1. Square side = $\sqrt{225}=15$ cm; perimeter = 60 cm.
2. Hexagon side = $\frac{60}{6}=10$ cm.
3. Area = $\frac{3\sqrt{3}}{2}\times 100=150\sqrt{3}$ cm².

Trick: Hexagon = 6 equilateral triangles: $6\times \frac{\sqrt{3}}{4}\times {10}^2=150\sqrt{3}$.

Answer: (D) $150\sqrt{3}$

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

1. Net rate (A+B−C) = $\frac{1}{10}+\frac{1}{12}-\frac{1}{15}=\frac{6+5-4}{60}=\frac{7}{60}$.
2. In 4 h: $\frac{28}{60}=\frac{7}{15}$ filled; left $=\frac{8}{15}$.
3. $A+B$ rate $=\frac{11}{60}$ → time = $\frac{8}{15}\times \frac{60}{11}=\frac{32}{11}=2\frac{10}{11}$ h.
4. Total = $4+2\frac{10}{11}=6\frac{10}{11}$ hours.

Trick: Tank = LCM 60 units: A 6, B 5, C −4. 4 h → 28 units; 32 left $÷11=2\frac{10}{11}$ h.

Answer: (D) $6\frac{10}{11}$ hours

Concept: Product of two numbers = HCF × LCM.

1. $21\times x=7\times 84=588$
2. $x=\frac{588}{21}=28$

Trick: Check: HCF(21, 28) = 7 and LCM = 84 ✓. 42 fails (HCF would be 21).

Answer: (C) 28

Concept: Solve innermost brackets first; 'of' means multiply.

1. $(18÷9\times 3)=2\times 3=6$.
2. $18÷(6\times 2)=1.5$; $1.5÷3=0.5$.
3. $16÷(5+3-6)=16÷2=8$; so $\{\}=0.5+8=8.5$.
4. 6 of 8.5 = 51.

Trick: ÷ and × go left to right: $18÷9\times 3=6$, not $18÷27$.

Answer: (C) 51