Q&A: practice questions with answers

Concept: If $a:b=c:d$, the fourth proportional is $d=\frac{b\times c}{a}$.

1. $a=0.48,b=0.84,c=32$
2. $d=\frac{0.84\times 32}{0.48}$
3. $\frac{0.84}{0.48}=\frac{7}{4}$, so $d=\frac{7}{4}\times 32=56$

Trick: Remove decimals first: $0.84:0.48=84:48=7:4$.

Answer: (C) 56

Concept: Percentage less $=\frac{\text{Difference}}{\text{Base value}}\times 100$; base is the value being compared with (village P).

1. Illiterates in P $=380$, in R $=150$
2. Difference $=380-150=230$
3. $\frac{230}{380}\times 100=\frac{1150}{19}=60\frac{10}{19}\%$

Trick: Only the last column is needed; the ratio and literacy columns are distractors here.

Answer: (C) $60\frac{10}{19}\%$

Concept: Arc length $l=\frac{\theta }{{360}^{\circ }}\times 2\pi r$.

1. $r=42\text{cm},\theta ={60}^{\circ }$
2. $l=\frac{60}{360}\times 2\times \frac{22}{7}\times 42$
3. $=\frac{1}{6}\times 264=44\text{cm}$

Trick: ${60}^{\circ }$ is $\frac{1}{6}$ of the circle, so arc $=\frac{\text{circumference}}{6}=\frac{264}{6}$.

Answer: (A) $44\text{cm}$

Concept: $\text{Loss}\%=\frac{\text{CP}-\text{SP}}{\text{CP}}\times 100$.

1. CP $=625$, SP $=550$
2. Loss $=625-550=75$
3. $\text{Loss}\%=\frac{75}{625}\times 100=12\%$

Trick: $\frac{75}{625}=\frac{3}{25}=\frac{12}{100}$.

Answer: (A) $12\%$

Concept: Successive decreases: New value $=\text{Initial}\times (1-\frac{a}{100})(1-\frac{b}{100})$.

1. After year 1: $80000\times 0.95=76000$
2. After year 2: $76000\times 0.92=69920$

Trick: Net change $=-5-8+\frac{5\times 8}{100}=-12.6\%$; $80000\times 0.874=69920$.

Answer: (B) 69920

Concept: Average of two values $=\frac{\text{sum}}{2}$; then subtract.

1. Institute X (2010, 2011): $\frac{320+240}{2}=280$
2. Institute Y (2011, 2012): $\frac{320+200}{2}=260$
3. Difference $=280-260=20$

Trick: Difference of averages $=\frac{560-520}{2}=20$ — subtract the sums first.

Answer: (A) 20

Concept: Solid hemisphere: $\text{TSA}=3\pi r^2$, $V=\frac{2}{3}\pi r^3$.

1. $3\times \frac{22}{7}\times r^2=4158\Rightarrow r^2=441\Rightarrow r=21$
2. $V=\frac{2}{3}\times \frac{22}{7}\times {21}^3=\frac{2}{3}\times \frac{22}{7}\times 9261$
3. $V=19404{\text{cm}}^3$

Trick: $\frac{4158}{66}=63=3\times 21$ gives $\frac{r^2}{7}=63$, so $r=21$ quickly.

Answer: (D) 19404

Concept: Combined average $=\frac{\text{total weight}}{\text{total students}}$.

1. Totals: $8\times 48=384$, $4\times 44=176$, $4\times 58=232$
2. Sum $=384+176+232=792$
3. Average $=\frac{792}{16}=49.5\text{kg}$

Trick: Deviation from 48: $4\times (-4)+4\times (+10)=24$; $48+\frac{24}{16}=49.5$.

Answer: (A) $49.5\text{kg}$

Concept: $(a-b{)}^2-(a+b{)}^2=-4ab$.

1. Here $a=1,b=2x$
2. $(1-2x{)}^2-(1+2x{)}^2=-4\times 1\times 2x$
3. $=-8x$

Trick: Check with $x=1$: $(-1{)}^2-3^2=-8$, which matches $-8x$.

Answer: (B) $-8x$

Concept: Share from a ratio: part $=\frac{\text{ratio term}}{\text{sum of terms}}\times \text{total}$.

1. Seller A, Brand1 $=\frac{3}{4}\times 4000=3000$
2. Seller B, Brand2 $=\frac{1}{3}\times 6000=2000$
3. Difference $=3000-2000=1000$

Trick: Read carefully: Brand1 of A versus Brand2 of B.

Answer: (C) 1000

Concept: Equal distances at speeds $x$ and $y$: average speed $=\frac{2xy}{x+y}$.

1. $x=30,y=70$
2. Average $=\frac{2\times 30\times 70}{30+70}=\frac{4200}{100}$
3. $=42\text{km/h}$

Trick: Average speed for equal distances is always less than the simple mean (50 here).

Answer: (A) $42\text{km/h}$

Concept: If A is $r\%$ more than B, then B is less than A by $\frac{r}{100+r}\times 100\%$.

1. Let Madhu $=100$, then Rupu $=120$
2. Madhu is less by $\frac{20}{120}\times 100$
3. $=\frac{50}{3}=16\frac{2}{3}\%$

Trick: $20\%=\frac{1}{5}$; answer $=\frac{1}{5+1}=\frac{1}{6}=16\frac{2}{3}\%$.

Answer: (D) $16\frac{2}{3}\%$

Concept: Under SI, each instalment $x$ earns interest till the due date; $\text{Debt}=\sum x(1+\frac{rt}{100})$.

1. 1st instalment earns 2 years, 2nd earns 1 year, 3rd earns 0
2. $x(1+0.245)+x(1+0.1225)+x=3.3675x$
3. $3.3675x=9429\Rightarrow x=2800$

Trick: Formula: $x=\frac{100A}{100n+\frac{n(n-1)}{2}r}=\frac{942900}{300+36.75}=2800$.

Answer: (B) 2800

Concept: Compare the column totals of all schools.

1. Totals: V $=917$, VI $=924$, VII $=904$, VIII $=919$, IX $=903$, X $=910$
2. Smallest total $=903$ (class IX)

Trick: VII (904) and IX (903) are very close — compare the exact totals.

Answer: (B) IX

Concept: Least number divisible by all = LCM (highest power of every prime).

1. $20=2^2\times 5,28=2^2\times 7,34=2\times 17,60=2^2\times 3\times 5,75=3\times 5^2$
2. LCM $=2^2\times 3\times 5^2\times 7\times 17$
3. $=300\times 119=35700$

Trick: The LCM must be divisible by 17 and 7, i.e. by 119: only $35700=119\times 300$ works.

Answer: (D) 35700

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

1. $ab=0.03\times 0.0003=0.000009$
2. $\sqrt{0.000009}=0.003$

Trick: Decimal places in the square root = half of 6, i.e. 3: $\sqrt{9}=3\to 0.003$.

Answer: (C) 0.003

Concept: $a^3+b^3+c^3-3abc=(a+b+c)[(a+b+c{)}^2-3(ab+bc+ca)]$.

1. $a+b+c=5,ab+bc+ca=7$
2. $(a+b+c{)}^2-3(ab+bc+ca)=25-21=4$
3. Value $=5\times 4=20$

Trick: Remember the short form $S(S^2-3P)$ with $S=a+b+c$, $P=ab+bc+ca$.

Answer: (B) 20

Concept: Central angle $=\frac{\text{percentage}}{100}\times {360}^{\circ }$; the total surveyed is not needed.

1. Football $=30\%$
2. Angle $=\frac{30}{100}\times {360}^{\circ }={108}^{\circ }$

Trick: $1\%={3.6}^{\circ }$, so $30\times {3.6}^{\circ }={108}^{\circ }$.

Answer: (A) ${108}^{\circ }$

Concept: Net effect of $+a\%$ then $-b\%$: $a-b-\frac{ab}{100}$.

1. Let CP $=100$; MP $=150$
2. SP $=150\times 0.6=90$
3. Loss $=100-90=10\Rightarrow 10\%$

Trick: $+50-40-\frac{50\times 40}{100}=-10\%$.

Answer: (C) $10\%$

Concept: $\text{SP}=\text{MP}\times (1-\frac{d}{100})$.

1. $z\times 0.88=6622$
2. $z=\frac{6622}{0.88}=7525$

Trick: Check: 12% of 7525 $=903$; $7525-903=6622$.

Answer: (C) ₹7,525