Q&A: practice questions with answers

Concept: When a transversal cuts two parallel lines, interior angles on the same side (co-interior angles) are supplementary: $\angle 1+\angle 2={180}^{\circ }$.

1. $\angle 1={125}^{\circ }$ and $\angle 2$ are co-interior angles.
2. $\angle 2={180}^{\circ }-{125}^{\circ }={55}^{\circ }$.

Yaad rakhein: Co-interior angles always add to ${180}^{\circ }$; alternate and corresponding angles are equal.

Answer: (B) ${55}^{\circ }$

Concept: $\text{SP}=\text{CP}+\text{Profit}$. Doubling SP while CP stays the same gives a new profit $2\text{SP}-\text{CP}$.

1. Let CP $=C$, profit $=P$, so $\text{SP}=C+P$.
2. New profit: $2(C+P)-C=3P\Rightarrow C+2P=3P\Rightarrow C=P$.
3. $\text{Profit}\%=\frac{P}{C}\times 100=100\%$.

Trick: If doubling SP makes profit $n$ times, then $\text{CP}=(n-2)\times$ profit; here $n=3$, so CP = profit $\Rightarrow 100\%$.

Answer: (D) 100%

Concept: Rate of a pipe $=\frac{1}{\text{time}}$. Pipes 1 and 2 together equal pipe 3: $\frac{1}{x}+\frac{1}{x-5}=\frac{1}{x-9}$.

1. Let pipe 1 take $x$ h; pipe 2 takes $x-5$ h; pipe 3 takes $(x-5)-4=x-9$ h.
2. Test $x=15$: pipe 2 $=10$ h, pipe 3 $=6$ h.
3. $\frac{1}{15}+\frac{1}{10}=\frac{2+3}{30}=\frac{1}{6}$ ✓.

Trick: Option testing is fastest: $x-9$ must be positive, so $x>9$; only $x=15$ satisfies the equation.

Answer: (C) 15 hours

Concept: Since $\tan \theta \cot \theta =1$, $(\tan \theta +\cot \theta {)}^2={\tan }^2\theta +{\cot }^2\theta +2$.

1. ${\tan }^2\theta +{\cot }^2\theta ={21}^2-2=441-2=439$.
2. $6({\tan }^2\theta +{\cot }^2\theta )=6\times 439=2634$.

Trick: Like $a+\frac{1}{a}=k\Rightarrow a^2+\frac{1}{a^2}=k^2-2$, with $a=\tan \theta$.

Answer: (C) 2634

Concept: If $60\%$ males are 25 or older, the remaining $40\%$ are below 25.

1. Males $=45\%$ of $4800=\frac{45}{100}\times 4800=2160$.
2. Below 25 $=40\%$ of $2160=\frac{40}{100}\times 2160=864$.

Trick: Direct: $4800\times 0.45\times 0.40=864$.

Answer: (A) 864

Concept: If $x+y=S$ and $x-y=D$, then $x=\frac{S+D}{2}$ and $y=\frac{S-D}{2}$.

1. Larger $=\frac{45+19}{2}=\frac{64}{2}=32$.
2. Smaller $=\frac{45-19}{2}=\frac{26}{2}=13$.
3. Check: $32+13=45$, $32-13=19$ ✓.

Trick: Check options: only $32,13$ give both sum $45$ and difference $19$.

Answer: (B) 32, 13

Concept: Read the two required values from the table and subtract.

1. Dealer A in 2021 $=45$ cars.
2. Dealer B in 2022 $=75$ cars.
3. Difference $=75-45=30$.

Yaad rakhein: Pick the correct row and column first — mixing up years is the common error.

Answer: (C) 30

Concept: Difference in ratio parts equals difference in amounts; find value of one part.

1. C − D $=9-3=6$ parts $=$ ₹$720$.
2. 1 part $=\frac{720}{6}=$ ₹$120$.
3. A's share $=4\times 120=$ ₹$480$.

Trick: Always find 1 part from the given difference, then multiply.

Answer: (B) ₹480

Concept: $\text{SI}=\frac{P\times R\times T}{100}$, so $R=\frac{\text{SI}\times 100}{P\times T}$.

1. $R=\frac{4000\times 100}{20000\times 2}$.
2. $R=\frac{400000}{40000}=10\%$.

Trick: Interest for 2 years is $20\%$ of principal ($\frac{4000}{20000}$), so per year $=10\%$.

Answer: (A) 10%

Concept: Share of B $=\frac{\text{B's part}}{\text{total parts}}\times \text{total}$.

1. Total parts $=4+5=9$.
2. B gets $\frac{5}{9}\times 63=35$.

Trick: 1 part $=\frac{63}{9}=7$; B $=5\times 7=35$.

Answer: (C) 35

Concept: Form an equation using present ages and add the same years to both after time passes.

1. Daughter $=x$, woman $=4x$.
2. After 6 years: $4x+6=3(x+6)\Rightarrow 4x+6=3x+18\Rightarrow x=12$.
3. Woman $=48$; sum $=48+12=60$ years.

Trick: Sum must be a multiple of $5$ ($x+4x=5x$); check $60$: $x=12$ works in the condition.

Answer: (A) 60 years

Concept: BODMAS: Brackets, then Division/Multiplication, then Addition/Subtraction (left to right).

1. Bracket: $(6+4)=10$.
2. $10÷2=5$ and $6\times 4=24$.
3. $5-2+24=27$.

Yaad rakhein: Solve all $÷$ and $\times$ first, then add/subtract from left to right.

Answer: (A) 27

Concept: If every number is multiplied by $k$, the average is also multiplied by $k$.

1. Old average $=7$.
2. New average $=7\times 12=84$.

Trick: Same rule for add/subtract/divide: apply the operation directly to the average.

Answer: (B) 84

Concept: Net filling per hour $=\frac{1}{2}+\frac{1}{3}-\frac{1}{6}$ (outlet subtracts).

1. Net $=\frac{3+2-1}{6}=\frac{4}{6}=\frac{2}{3}$ tank per hour.
2. Time $=\frac{3}{2}$ h $=1$ h $30$ min.
3. 9:00 a.m. $+$ 1 h 30 min $=$ 10:30 a.m.

Trick: Take LCM $6$ as tank capacity: S $=3$, T $=2$, U $=-1$ units/h; net $4$ units/h, so $\frac{6}{4}=1.5$ h.

Answer: (A) 10:30 a.m.

Concept: Two successive discounts $a\%$ and $b\%$ equal a single discount of $(a+b-\frac{ab}{100})\%$.

1. $7+2-\frac{7\times 2}{100}$.
2. $=9-0.14=8.86\%$.

Trick: Equivalent discount is always a little less than the plain sum ($9\%$).

Answer: (D) 8.86%

Concept: Half-yearly compounding: rate $=\frac{R}{2}$, periods $=2T$; $A=P{(1+\frac{r}{100})}^n$.

1. Rate $=\frac{8}{2}=4\%$ per half-year, $n=2$.
2. $A=15000\times (1.04{)}^2=15000\times 1.0816$.
3. $A=$ ₹$16224$.

Trick: Two periods at $4\%$ = effective $8.16\%$; $15000\times 8.16\%=1224$, so $A=16224$.

Answer: (A) ₹16,224

Concept: Cross-multiply: $\frac{a}{b}=\frac{c}{d}\Rightarrow ad=bc$.

1. $\frac{3}{7}=\frac{x}{-35}\Rightarrow 7x=3\times (-35)$.
2. $x=\frac{-105}{7}=-15$.

Trick: $-35=7\times (-5)$, so multiply numerator by $-5$: $3\times (-5)=-15$.

Answer: (D) $-15$

Concept: ${(a+\frac{1}{a})}^2=a^2+\frac{1}{a^2}+2$.

1. $a^2+\frac{1}{a^2}=4^2-2$.
2. $=16-2=14$.

Yaad rakhein: $a+\frac{1}{a}=k\Rightarrow a^2+\frac{1}{a^2}=k^2-2$.

Answer: (D) 14

Concept: First 10 primes: $2,3,5,7,11,13,17,19,23,29$.

1. $2+3+5+7=17$.
2. $11+13+17+19=60$.
3. $23+29=52$; total $=17+60+52=129$.

Yaad rakhein: Remember: sum of first 10 primes $=129$ (and of primes below 100 $=1060$).

Answer: (B) 129

Concept: Euclid's method: $\text{HCF}(a,b)=\text{HCF}(b,a-b)$ / remainder, until remainder is $0$.

1. $2456-1248=1208$; $1248-1208=40$.
2. $1208=30\times 40+8$; remainder $8$.
3. $40=5\times 8$, remainder $0$ → HCF $=8$.

Trick: HCF divides the difference too; $1248$ and $2456$ are both divisible by $8$ but $\frac{1248}{16}=78$, $\frac{2456}{16}=153.5$ ✗, so not $16$.

Answer: (A) 8