Q&A: practice questions with answers

Concept: Compounded half-yearly: $A=P{(1+\frac{r}{200})}^{2t}$.

1. Time $=1.5$ years $=3$ half-years.
2. $\frac{15625}{8000}=\frac{125}{64}={\left(\frac{5}{4}\right)}^3$.
3. So $1+\frac{r}{200}=\frac{5}{4}\Rightarrow \frac{r}{200}=\frac{1}{4}\Rightarrow r=50\%$.

Trick: $8000={20}^3$ and $15625={25}^3$, so each half-year multiplies by $\frac{25}{20}$: $25\%$ per half-year $=50\%$ per year.

Answer: (B) $50\%$

Concept: Simplify nested roots from the innermost one outward.

1. $\sqrt{289}=17$.
2. $\sqrt{344+17}=\sqrt{361}=19$.
3. $\sqrt{125+19}=\sqrt{144}=12$.

Trick: Each step should give a perfect square: $361={19}^2$, $144={12}^2$.

Answer: (B) $12$

Concept: Required number = HCF of (number − remainder) for each number.

1. 152 − 2 = 150 and 181 − 6 = 175.
2. $150=2\times 3\times 5^2$, $175=5^2\times 7$.
3. HCF $=5^2=25$ (and $25>6$, so remainder 6 is valid).

Trick: Check options: 152 ÷ 25 leaves 2, 181 ÷ 25 leaves 6 ✓.

Answer: (C) 25

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

1. Arrange: 20, 47, 52, 53, 59, 69, 72, 81, 85, 86, 98.
2. $n=11$, so median $=6^{\text{th}}$ value.
3. 6th value = 69.

Trick: Odd count → no averaging needed; just pick the middle term.

Answer: (B) 69

Concept: Downstream speed $=u+v$, upstream speed $=u-v$.

1. Downstream $=7+5=12\text{km/h}$; upstream $=7-5=2\text{km/h}$.
2. Time down $=\frac{52.8}{12}=4.4$ h; time up $=\frac{52.8}{2}=26.4$ h.
3. Total $=4.4+26.4=30.8$ h.

Trick: Total $=d(\frac{1}{12}+\frac{1}{2})=52.8\times \frac{7}{12}=30.8$.

Answer: (D) $30.8$

Concept: Total = average $\times$ number of items; split total in the given ratio.

1. Total $=15135\times 3=45405$.
2. Ratio sum $=3+5+7=15$.
3. Cheapest $=\frac{3}{15}\times 45405=9081$.

Trick: Since $3+5+7=15$ and 5 is the middle term, average = middle share: $5$ parts = 15135, so $1$ part = 3027 and $3$ parts = 9081.

Answer: (C) ₹9,081

Concept: Overtaking a man: relative speed $=\frac{\text{train length}}{\text{time}}$; opposite directions: speeds add.

1. Relative speed $=\frac{100}{30}=\frac{10}{3}\text{m/s}=12\text{km/h}$, so train $=9+12=21\text{km/h}$.
2. Opposite direction: $21+51=72\text{km/h}=20\text{m/s}$.
3. Time $=\frac{100+220}{20}=16$ s.

Trick: $\frac{\text{m}}{\text{s}}\times \frac{18}{5}=\text{km/h}$: $\frac{10}{3}\times \frac{18}{5}=12$.

Answer: (D) 16

Concept: Take CP $=100$; SP $=$ MP $\times (1-\text{discount})$.

1. MP $=105$.
2. SP $=105\times 0.33=34.65$.
3. Loss $=100-34.65=65.35\%$.

Trick: Net effect $=5-67-\frac{5\times 67}{100}=-65.35\%$.

Answer: (C) $65.35\%$ loss

Concept: Solve innermost bracket first, multiplication before subtraction.

1. $22-22\times 4=22-88=-66$.
2. $\{22-(-66)\}=88$.
3. $[22-88]=-66$; $22-(-66)=88$.

Trick: Watch the signs: minus of a negative becomes plus.

Answer: (C) 88

Concept: Area of parallelogram $=\text{base}\times \text{height}$.

1. Base $=4(3{)}^2-7(3)(-2)+(-2{)}^2=36+42+4=82$ m.
2. Height $=3+(-2)=1$ m.
3. Area $=82\times 1=82{\text{m}}^2$.

Trick: Careful: $-7xy$ with $y=-2$ gives $+42$.

Answer: (B) $82{\text{m}}^2$

Concept: Mean proportional of $a$ and $b$ $=\sqrt{ab}$.

1. $ab=(18+10\sqrt{7})(18-4\sqrt{7})$.
2. $=324-72\sqrt{7}+180\sqrt{7}-40\times 7=324-280+108\sqrt{7}$.
3. $=44+108\sqrt{7}$, so mean proportional $=\sqrt{44+108\sqrt{7}}$.

Trick: Only the $\sqrt{7}$ coefficient needs care: $18\times 10-18\times 4=108$.

Answer: (A) $\sqrt{44+108\sqrt{7}}$

Concept: A quadrilateral is a polygon with exactly 4 sides.

1. Trapezoid, kite and rhombus all have 4 sides → quadrilaterals.
2. An octagon has 8 sides → not a quadrilateral.

Trick: 'Quadri' = 4, 'Octa' = 8.

Answer: (A) Octagon

Concept: Make the common term (red) equal in both ratios.

1. R : G = 7 : 4 and P : R = 20 : 11.
2. Make R $=77$: R : G $=77:44$, P : R $=140:77$.
3. G : P $=44:140=11:35$.

Trick: Directly: $\frac{G}{P}=\frac{G}{R}\times \frac{R}{P}=\frac{4}{7}\times \frac{11}{20}=\frac{11}{35}$.

Answer: (B) 11 : 35

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

1. $x=\frac{16\times 7.3}{4}=29.2$.
2. $y=\frac{32\times 7}{4}=56$.
3. $x:y=29.2:56=292:560=73:140$.

Trick: $x=4\times 7.3$, $y=8\times 7$ ⇒ $\frac{x}{y}=\frac{7.3}{14}=\frac{73}{140}$.

Answer: (A) 73 : 140

Concept: $SI=\frac{P\times R\times T}{100}$ and Amount $=P+SI$.

1. $SI=\frac{50000\times 12\times 4}{100}=24000$
2. Amount $=50000+24000=74000$

Trick: Total rate $=12\%\times 4=48\%$; amount $=50000\times 1.48=74000$.

Answer: (B) ₹74,000

Concept: $a\%$ of $x=\frac{a}{100}\times x$.

1. $\frac{4}{100}\times x=24$
2. $x=\frac{24\times 100}{4}=600$

Trick: $4\%=24\Rightarrow 1\%=6\Rightarrow 100\%=600$.

Answer: (C) 600

Concept: Successive discounts are applied one after another on the reduced price: $SP=MP\times \frac{100-d_1}{100}\times \frac{100-d_2}{100}\times \dots$

1. After $4\%$: $750\times 0.96=720$
2. After $25\%$: $720\times 0.75=540$
3. Cash discount $25\%$: $540\times 0.75=405$

Trick: $750\times \frac{24}{25}\times \frac{3}{4}\times \frac{3}{4}=405$ — cancel fractions directly.

Answer: (C) ₹405

Concept: A number is divisible by 3 if the sum of its digits is divisible by 3.

1. Digit sum $=5+1+8+8+0+6+9+p=37+p$
2. $37+p$ must be a multiple of 3 ⇒ $p=2,5,8$
3. Maximum value: $p=8$ (sum $=45$)

Trick: $37$ leaves remainder 1, so $p$ must leave remainder 2 when divided by 3 — largest such digit is 8.

Answer: (B) 8

Concept: Factorise by splitting the middle term: find two numbers with product $7\times 3=21$ and sum $-10$.

1. $-10x=-7x-3x$
2. $7x^2-7x-3x+3=7x(x-1)-3(x-1)=(7x-3)(x-1)$
3. $x=1$ or $x=\frac{3}{7}$

Trick: Sum of coefficients $7-10+3=0$ ⇒ one root is 1, other root $=\frac{c}{a}=\frac{3}{7}$.

Answer: (A) $1$ and $\frac{3}{7}$

Concept: Empirical formula: $\text{Mode}=3\text{Median}-2\text{Mean}$.

1. Rewrite: $\text{Mode}-\text{Median}=2(\text{Median}-\text{Mean})$, and $\text{Mode}-\text{Mean}=3(\text{Median}-\text{Mean})$
2. $\text{Median}-\text{Mean}=\frac{20.1}{3}=6.7$
3. $\text{Mode}-\text{Median}=2\times 6.7=13.4$

Trick: Mode–Mean gap splits $2:1$ — Mode–Median $=\frac{2}{3}$ of $20.1=13.4$.

Answer: (C) $13.4$