Q&A: practice questions with answers

Concept: Shares in a ratio are equal 'parts'; find the value of one part from the given difference.

1. Ratio $12:3:14$; smallest $=3$ parts, largest $=14$ parts.
2. Difference $=14-3=11$ parts $=61699$ ⇒ 1 part $=\frac{61699}{11}=5609$.
3. Total $=(12+3+14)\times 5609=29\times 5609=162661$.

Trick: Total must be a multiple of 29: only $162661=29\times 5609$ works.

Answer: (A) ₹1,62,661

Concept: If $2A=4B=7C=k$, then $A:B:C=\frac{1}{2}:\frac{1}{4}:\frac{1}{7}$.

1. Multiply by LCM $28$: $A:B:C=14:7:4$.
2. Total parts $=14+7+4=25$; 1 part $=\frac{21300}{25}=852$.
3. $A=14\times 852=11928$.

Trick: For $aA=bB=cC$, $A:B:C=bc:ca:ab=28:14:8=14:7:4$.

Answer: (D) ₹11,928

Concept: Increasing a number by $60\%$ means multiplying it by $\frac{160}{100}$.

1. $x\times \frac{160}{100}=3640$.
2. $x=\frac{3640\times 100}{160}=\frac{3640\times 5}{8}=455\times 5=2275$.

Trick: $60\%=\frac{3}{5}$ ⇒ number : new $=5:8$; $\frac{3640}{8}\times 5=2275$.

Answer: (D) $2275$

Concept: In 'buy $x$ get $y$ free', the customer gets $x+y$ items for the price of $x$; effective discount $=\frac{y}{x+y}\times 100\%$.

1. (i) $\frac{3}{8}=37.5\%$; (ii) $\frac{3}{9}=33.3\%$.
2. (iii) $\frac{5}{25}=20\%$; (iv) $\frac{6}{14}=42.86\%$.
3. Highest discount is (iv) – most beneficial.

Trick: Compare $\frac{\text{free}}{\text{total}}$, not $\frac{\text{free}}{\text{bought}}$ – but here both rankings give (iv).

Answer: (C) (iv)

Concept: Consecutive even numbers: $n$ and $n+2$.

1. $n^2+(n+2{)}^2=452$ ⇒ $2n^2+4n+4=452$ ⇒ $n^2+2n-224=0$.
2. $(n+16)(n-14)=0$ ⇒ $n=14$ (natural).
3. Numbers $14,16$: $196+256=452$ ✓; sum $=30$.

Trick: $\frac{452}{2}=226\approx {15}^2$, so numbers lie around $15$: try $14,16$.

Answer: (B) $30$

Concept: Do division and multiplication before addition and subtraction (BODMAS), keeping track of signs.

1. $(-60)÷(-12)=5$ and $(-2)\times 9=-18$.
2. Expression $=-9-5+(-18)$.
3. $=-14-18=-32$.

Trick: Negative ÷ negative = positive; then subtracting a positive moves left on the number line.

Answer: (B) $-32$

Concept: Make the decimal places equal, find the LCM of the whole numbers, then put back the decimal places.

1. $8.6=8.60=\frac{860}{100}$, $0.12=\frac{12}{100}$.
2. $860=2^2\times 5\times 43$, $12=2^2\times 3$ ⇒ $\text{LCM}=2^2\times 3\times 5\times 43=2580$.
3. $\text{LCM}=\frac{2580}{100}=25.80$.

Trick: LCM must be a multiple of $8.6$: $8.6\times 3=25.8$, and $\frac{25.8}{0.12}=215$ ✓.

Answer: (C) $25.8$

Concept: Same time, equal interest: $P_1\times R_1=P_2\times R_2$, so principal is inversely proportional to rate.

1. $P_1\times 12\times 7=P_2\times 13\times 7$ ⇒ $P_1:P_2=13:12$.
2. Total parts $=25$; 1 part $=\frac{20000}{25}=800$.
3. Sum at $12\%$ $=13\times 800=10400$.

Trick: Rates swap into the ratio: rates $12:13$ ⇒ sums $13:12$ (time does not matter).

Answer: (C) ₹10,400

Concept: Volume of a sphere $=\frac{4}{3}\pi r^3$, where $r=\frac{d}{2}$.

1. $r=\frac{12}{2}=6\text{cm}$.
2. $V=\frac{4}{3}\pi \times 6^3=\frac{4}{3}\pi \times 216$.
3. $V=4\times 72\pi =288\pi {\text{cm}}^3$.

Yaad rakhein: Sphere: $V=\frac{4}{3}\pi r^3$, $\text{SA}=4\pi r^2$; hemisphere volume is half.

Answer: (B) $288\pi {\text{cm}}^3$

Concept: Remove brackets by multiplying each term, then combine like terms.

1. $2(\frac{7}{2}{x}^2-28x+17)=7x^2-56x+34$.
2. $-7(x^2+6x-11)=-7x^2-42x+77$.
3. Add: $(7x^2-7x^2)+(-56x-42x)+(34+77)=-98x+111$.

Trick: $x^2$ terms cancel; constant $=34+77=111$ (positive) → only option (C) fits.

Answer: (C) $-98x+111$

Concept: With a false weight, profit is found by comparing what the buyer pays with what the trader really gives: $\text{CP}$ of goods given vs $\text{SP}$ received.

1. Let CP of 1000 g = ₹100. He charges for 1000 g at 44% profit: SP = ₹144.
2. But he gives only 900 g (10% less), whose CP = ₹90.
3. Profit $=144-90=54$ on cost 90 → $\frac{54}{90}\times 100=60\%$.

Trick: Overall factor $=\frac{1.44}{0.9}=1.6$ → 60% profit.

Answer: (D) 60%

Concept: No overall profit/loss means the loss on one article equals the gain on the other.

1. Let CP of first $=x$, second $=1200-x$.
2. Loss = gain: $0.09x=0.27(1200-x)$.
3. $0.36x=324$ → $x=900$.

Trick: Losses balance: CP ratio $=27:9=3:1$ → first $=\frac{3}{4}\times 1200=900$.

Answer: (D) 900

Concept: After successive discounts $d_1,d_2$: $\text{SP}=\text{MP}\times (1-\frac{d_1}{100})(1-\frac{d_2}{100})$.

1. $\text{SP}=\text{MP}\times 0.88\times 0.91=0.8008\text{MP}$.
2. $0.8008\text{MP}=4600$ → $\text{MP}=\frac{4600}{0.8008}$.
3. $\text{MP}\approx 5744.26$.

Trick: Net discount of 12% and 9% $=12+9-\frac{12\times 9}{100}=19.92\%$ → MP is just under $4600\times 1.25=5750$; only ₹5,744.26 fits.

Answer: (D) ₹5,744.26

Concept: Sum of interior angles of an $n$-sided polygon $=(2n-4)$ right angles $=(n-2)\times {180}^{\circ }$.

1. $2n-4=28$.
2. $2n=32$ → $n=16$.

Trick: Right angles $÷2$ + 2 = sides: $\frac{28}{2}+2=16$.

Answer: (D) 16

Concept: Under compound interest, if money becomes $k$ times in $t$ years, it becomes $k^n$ times in $nt$ years.

1. 3 times in 16 years.
2. $9=3^2$ → time $=2\times 16=32$ years.

Trick: CI multiplies: $3\to 9\to 27$ takes $16\to 32\to 48$ years.

Answer: (D) 32 years

Concept: Same direction: relative speed = difference; opposite direction: relative speed = sum. Distance = sum of the lengths.

1. Overtaking the man: $\frac{150}{36}=\frac{25}{6}\text{m/s}=15\text{km/h}$ relative → train $=15+9=24\text{km/h}$.
2. Opposite train: relative speed $=24+43=67\text{km/h}=\frac{67\times 5}{18}\text{m/s}$.
3. Time $=\frac{335\times 18}{67\times 5}=\frac{6030}{335}=18\text{s}$.

Trick: $335=67\times 5$, so time $=\frac{335\times 18}{67\times 5}=18$ straight away.

Answer: (B) 18

Concept: Mode is the value that occurs most often in the data.

1. Counts: 11 → 3 times, 12 → 2 times, all others once.
2. Highest frequency is for 11.

Trick: Just tally the repeats – the most repeated value is the mode.

Answer: (D) 11

Concept: Convert each statement into an equation and solve simultaneously (Mary $=M$, Diya $=D$).

1. $3M=9D+27$ → $M=3D+9$.
2. $4D=M-7$ → $4D=3D+2$ → $D=2$.
3. $M=15$; difference $=15-2=13$.

Trick: Check: $3\times 15=45=9\times 2+27$ ✓ and $4\times 2=8=15-7$ ✓.

Answer: (A) 13

Concept: Use $\sec \theta -\tan \theta =\frac{1-\sin \theta }{\cos \theta }$ and $1-{\sin }^2\theta ={\cos }^2\theta$.

1. $(1+\sin \theta )(\sec \theta -\tan \theta )=\frac{(1+\sin \theta )(1-\sin \theta )}{\cos \theta }=\frac{{\cos }^2\theta }{\cos \theta }=\cos \theta$.
2. Expression $=\sec \theta \times \cos \theta =1$.

Trick: Put $\theta =0^{\circ }$: $1\times 1\times (1-0)=1$.

Answer: (C) $1$

Concept: For equal distances at speeds $a$ and $b$: average speed $=\frac{2ab}{a+b}$.

1. $\frac{2\times 18\times 36}{18+36}=\frac{1296}{54}$.
2. $=24\text{km/h}$.

Trick: Speeds in ratio $1:2$ → average $=\frac{4}{3}\times 18=24$.

Answer: (B) 24