Q&A: practice questions with answers

Concept: Find each quantity separately, then subtract.

1. $65\%\text{ of }75=\frac{65\times 75}{100}=48.75$
2. $\frac{3}{5}\text{ of }25=15$
3. Difference $=48.75-15=33.75$

Trick: $65\%$ of $75$ $=75\%$ of $65$ $=\frac{3}{4}\times 65=48.75$.

Answer: (D) $33.75$

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

1. $4.7=\frac{470}{100}$, $0.04=\frac{4}{100}$
2. $\text{LCM}(470,4)=940$ (since $470=2\times 5\times 47$, $4=2^2$).
3. LCM $=\frac{940}{100}=9.4$

Trick: LCM must be a multiple of the larger number $4.7$: $9.4=2\times 4.7$ and $9.4=235\times 0.04$ ✓.

Answer: (A) $9.4$

Concept: $\text{Average speed}=\frac{\text{Total distance}}{\text{Total time}}$ (not the average of speeds).

1. Times: $\frac{488}{61}=8$ h, $\frac{468}{78}=6$ h, $\frac{210}{35}=6$ h ⇒ total $20$ h.
2. Total distance $=488+468+210=1166\text{km}$.
3. Average speed $=\frac{1166}{20}=58.3\text{km/h}$

Trick: Each part divides exactly – compute times first, the rest is one division.

Answer: (D) $58.3$

Concept: If $a:b=b:x$, then $x$ is the third proportional of $a$ and $b$: $x=\frac{b^2}{a}$.

1. $x=\frac{{30}^2}{50}=\frac{900}{50}=18$
2. Check: $50:30=5:3$ and $30:18=5:3$ ✓

Yaad rakhein: Third proportional $=\frac{(\text{second}{)}^2}{\text{first}}$.

Answer: (D) $18$

Concept: Write each statement as an equation and solve them together.

1. $3M=5D+22$ … (i); $3D=M-2$ ⇒ $M=3D+2$ … (ii)
2. Put (ii) in (i): $3(3D+2)=5D+22$ ⇒ $9D+6=5D+22$ ⇒ $4D=16$ ⇒ $D=4$.
3. $M=3\times 4+2=14$; difference $=14-4=10$ years.

Trick: Check: $3\times 14=42=5\times 4+22$ ✓ and $3\times 4=12=14-2$ ✓.

Answer: (C) $10$

Concept: Surface area of a sphere $=4\pi r^2$, so it varies as the square of the radius.

1. New radius $=12r$.
2. New area $=4\pi (12r{)}^2=144\times 4\pi r^2$.
3. So the surface area becomes 144 times.

Trick: Area ratio $=(\text{radius ratio}{)}^2={12}^2=144$; volume would be ${12}^3$.

Answer: (B) 144

Concept: Let one-way walking time $=w$ and one-way driving time $=d$.

1. Walk + drive: $w+d=4.5\text{h}$.
2. Driving both ways saves 2 h: $2d=4.5-2=2.5$, so $d=1.25\text{h}$.
3. $w=4.5-1.25=3.25\text{h}$.
4. Walking both ways $=2w=6.5\text{h}$ = 6 hours 30 minutes.

Trick: Walk both ways $=2(\text{total})-(\text{drive both})=9-2.5=6.5\text{h}$.

Answer: (A) 6 hours 30 minutes

Concept: Solve brackets from innermost outward: ( ) → { } → [ ], then ÷, ×, −.

1. $(6-5)=1$, so $8\times 1\times 2=16$.
2. $\{16-16\}=0$; $3\times 0÷32=0$.
3. $[4+0]=4$; $9\times 4=36$.
4. $67-36=31$.

Trick: Once a bracket becomes 0, everything multiplied by it vanishes – check that first.

Answer: (C) 31

Concept: Form two equations in pen price $p$ and pencil price $c$.

1. $10p+3c=129$.
2. $8(p-6)+2(c+9)=72\Rightarrow 8p+2c=102\Rightarrow 4p+c=51$.
3. $c=51-4p$; $10p+3(51-4p)=129\Rightarrow p=12,c=3$.
4. $9p+16c=108+48=156$.

Trick: Check: $10\times 12+3\times 3=129$ ✓.

Answer: (A) ₹156

Concept: $\text{SI}=\frac{P\times R\times T}{100}$, with R and T in the same time unit (here months).

1. $P=1200$, $R=3\%$ per month, $T=4$ months.
2. $\text{SI}=\frac{1200\times 3\times 4}{100}=144$.

Trick: 3% per month for 4 months = 12% of ₹1200 = ₹144.

Answer: (D) 144

Concept: Discount = discount% of marked price.

1. $\text{Discount}=10\%\text{ of }500=\frac{10}{100}\times 500=50$.

Trick: 10% of any amount = move decimal one place left: 500 → 50.

Answer: (B) ₹50

Concept: Evaluate each power and root first, then add/subtract left to right.

1. $2^2=4$, $\sqrt{5^2}=5$, $3\sqrt{25}=3\times 5=15$.
2. $4+5-15-10=-16$.

Trick: $\sqrt{a^2}=a$ for positive $a$.

Answer: (B) $-16$

Concept: Check for a Pythagorean triplet before using Heron's formula.

1. $54:72:90=3:4:5$ (each $\times 18$), so the triangle is right-angled.
2. ${54}^2+{72}^2=2916+5184=8100={90}^2$ ✓.
3. Area $=\frac{1}{2}\times 54\times 72=1944{\text{cm}}^2$.

Trick: Heron check: $s=108$, $\sqrt{108\times 54\times 36\times 18}=1944$.

Answer: (B) 1944

Concept: $\text{SI}=\frac{PRT}{100}$; find the principal first, then the new amount $=P+\text{SI}$.

1. $6726=\frac{P\times 5.9\times 9.5}{100}\Rightarrow P=\frac{672600}{56.05}=12000$.
2. New SI $=\frac{12000\times 5.6\times \frac{22}{7}}{100}=2112$.
3. Amount $=12000+2112=14112$.

Trick: $5.6\times \frac{22}{7}=17.6\%$, and $17.6\%$ of 12000 $=2112$.

Answer: (D) ₹14,112

Concept: Product of two numbers = HCF × LCM.

1. $55\times x=11\times 440$.
2. $x=\frac{4840}{55}=88$.

Trick: Check: HCF(55, 88) = 11 and LCM = 440 ✓.

Answer: (D) 88

Concept: Overhead (decoration) is added to cost price. $\text{Loss}\%=\frac{\text{Loss}}{\text{CP}}\times 100$.

1. CP $=849+6=855$; SP $=441$ → loss.
2. Loss $=855-441=414$.
3. $\text{Loss}\%=\frac{414}{855}\times 100=\frac{920}{19}\%\approx 48.42\%$.

Trick: SP < CP means loss – options with 'profit' are out at once.

Answer: (A) $\frac{920}{19}\%$ loss

Concept: Square roots of decimals: take the root of the digits and halve the decimal places.

1. $\sqrt{169}=13$.
2. $\sqrt{0.0196}=0.14$ (since ${14}^2=196$).
3. $\sqrt{4.41}=2.1$.
4. $13+0.14-2.1=11.04$.

Trick: 4 decimal places under the root → 2 decimal places in the answer.

Answer: (C) 11.04

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

1. Mohsin's share $=\frac{2600}{14400}=\frac{13}{72}$; Abdul's $=\frac{59}{72}$.
2. Mohsin's investment $=\frac{13}{72}\times 79200=14300$; Abdul's $=64900$.
3. Difference $=64900-14300=50600$.

Trick: Difference $=\frac{59-13}{72}\times 79200=46\times 1100=50600$.

Answer: (B) ₹50,600

Concept: Savings = Income − Expenditure; apply each % change separately.

1. Savings $=25\%$ of 91700 $=22925$; expenditure $=68775$.
2. New income $=1.27\times 91700=116459$.
3. New expenditure $=1.40\times 68775=96285$.
4. New savings $=116459-96285=20174$; change $=20174-22925=-2751$ → decrease by ₹2,751.

Trick: Change in savings $=0.27\times 91700-0.40\times 68775=24759-27510=-2751$.

Answer: (A) decrease by ₹2,751

Concept: Take one day's work as a fraction of the whole work and subtract group rates.

1. Shifa $=\frac{1}{30}$, Shifa + Afiya $=\frac{1}{18}$, all three $=\frac{1}{9}$.
2. Mariyam $=\frac{1}{9}-\frac{1}{18}=\frac{1}{18}$.
3. Shifa + Mariyam $=\frac{1}{30}+\frac{1}{18}=\frac{3+5}{90}=\frac{4}{45}$.
4. Days $=\frac{45}{4}=11.25$.

Trick: LCM method: total work 90 units → S = 3, S+A = 5, S+A+M = 10 → M = 5; S+M = 8 → $\frac{90}{8}=\frac{45}{4}$.

Answer: (D) $\frac{45}{4}$