5 February 2025 · Shift 1

Average and total weight Averages cannot be added, but totals can - so convert every average into a total first. $M+K+T=45\times 3=135$ $M+K=43\times 2=86$ $K+T=48\times 2=96$ (M+K)+(K+T)=86+96=182 This sum counts Ketan twice, so 182=(M+K+T)+K K=182-135=47 So the correct option is (b).

Time and work, alternate days Take the whole work as the LCM of 20, 30 and 50, that is 300 units. 1 man's 1 day work $=\frac{300}{20}=15$ units 1 woman's 1 day work $=\frac{300}{30}=10$ units 1 boy's 1 day work $=\frac{300}{50}=6$ units 6 women work on every day: $6\times 10=60$ units a day. On day 1, 3, 5 ... the men join: $2\times 15+60=90$ units. On day 2, 4, 6 ... the boys join: $5\times 6+60=90$ units. So 90 units are done daily; in 3 days $3\times 90=270$ units. Work left =300-270=30 units, needing $\frac{30}{90}=\frac{1}{3}$ day. Total time $=3+\frac{1}{3}=3\frac{1}{3}$ days So the correct option is (d).

Profit percentage, cost price Profit percentage is always measured on the cost price, so CP is the 100% base. $SP=CP\times \frac{100+32}{100}=CP\times \frac{132}{100}$ $7392=CP\times \frac{132}{100}$ $CP=\frac{7392\times 100}{132}$ $CP=\frac{739200}{132}=5600$ So the correct option is (c).

Percentage of a number 'Of' means multiplication, and $2\%$ means $\frac{2}{100}$. $\frac{2}{100}\times x=348$ $x=\frac{348\times 100}{2}$ $x=\frac{34800}{2}=17400$ So the correct option is (a).

LCM, HCF and product For any two numbers, LCM $\times$ HCF = product of the two numbers. $168\times 14=42\times x$ 2352=42x $x=\frac{2352}{42}=56$ Check: 42 and 56 do have HCF 14 and LCM 168. So the correct option is (d).

Third proportional, surds If a, b, c are in continued proportion then $b^2=ac$, so the third proportional is $c=\frac{b^2}{a}$. Here $a=3+\sqrt{2}$ and $b=2\sqrt{7}$. $b^2={\left(2\sqrt{7}\right)}^2=4\times 7=28$ $c=\frac{28}{3+\sqrt{2}}$ Rationalise by multiplying above and below by $\left(3-\sqrt{2}\right)$: $c=\frac{28\left(3-\sqrt{2}\right)}{{\left(3\right)}^2-{\left(\sqrt{2}\right)}^2}=\frac{28\left(3-\sqrt{2}\right)}{9-2}$ $c=\frac{28\left(3-\sqrt{2}\right)}{7}=4\left(3-\sqrt{2}\right)$ So the correct option is (c).

Simple interest, principal $SI=\frac{P\times R\times T}{100}$ $824=\frac{P\times 8\times 4}{100}$ $824=\frac{32P}{100}$ $P=\frac{824\times 100}{32}$ $P=\frac{82400}{32}=2575$ So the correct option is (d).

Successive discounts Successive discounts multiply; they are never added. (i) $S{P}_1=1500\times \frac{80}{100}\times \frac{80}{100}$ $=1500\times \frac{64}{100}=960$ (ii) $S{P}_2=1500\times \frac{70}{100}\times \frac{90}{100}$ $=1500\times \frac{63}{100}=945$ Difference =960-945=15 So the correct option is (b).

Compound interest, equal amounts Under compound interest a sum is multiplied by the same factor every year. Let Piyush get P and Manoj get M. Their amounts are equal, so $P{\left(\frac{108}{100}\right)}^{12}=M{\left(\frac{108}{100}\right)}^{13}$ $\frac{P}{M}=\frac{108}{100}=\frac{27}{25}$ Total parts =27+25=52, and 52 parts =₹1612 1 part $=\frac{1612}{52}=31$ Piyush's share $=27\times 31=837$ So the correct option is (c).

Smallest number, LCM The smallest number divisible by several numbers is their LCM. $26=2\times 13$ $8=2^3$ 11=11 13=13 Highest powers available are $2^3$, 11 and 13. LCM $=8\times 11\times 13=88\times 13=1144$ So the correct option is (d).

Speed, time and delay For the same distance, time varies inversely as speed. Speed ratio (normal : reduced) =5:1 So time ratio (normal : reduced) =1:5 Let the normal time be t minutes; the slower trip takes 5t minutes. Extra time =5t-t=4t 4t=36 $t=\frac{36}{4}=9$ So the correct option is (d).

BODMAS simplification Work from the innermost bracket outwards, following BODMAS. 8-7=1 $\frac{12}{1}\times 1=12$ $\frac{72}{3}+12=24+12=36$ $50÷5=10$ $10\times 36=360$ So the correct option is (d).

Integers with BODMAS By BODMAS, do the division and the multiplication first, keeping the signs. $\left(-60\right)÷\left(-15\right)=4$ $\left(-3\right)\times 7=-21$ The expression becomes (-9)-4+(-21) =-9-4-21=-34 So the correct option is (a).

Profit stated on selling price The gain is given in selling price, so fix the selling price of one chair as ₹100. SP of 152 chairs $=152\times 100=15200$ Gain = SP of 52 chairs $=52\times 100=5200$ CP of 152 chairs =15200-5200=10000 Profit percentage $=\frac{5200}{10000}\times 100$ $=52\%$ So the correct option is (c).

Weighted average of heights Two averages cannot be averaged directly when the groups differ in size - convert each into a total first. Females $=\frac{60}{100}\times 500=300$ Males =500-300=200 Total height of females $=300\times 162=48600$ cm Total height of males $=200\times 168=33600$ cm Average height $=\frac{48600+33600}{500}=\frac{82200}{500}$ =164.4 cm So the correct option is (d).

Mean proportional The mean proportional between a and b is $\sqrt{ab}$. $\sqrt{2.8\times 0.7}=\sqrt{1.96}$ =1.4 So the correct option is (a).

CI minus SI for 3 years For 3 years, CI - SI $=P\times \frac{R^2\left(300+R\right)}{{10}^6}$ $244=P\times \frac{5^2\left(300+5\right)}{1000000}$ $244=P\times \frac{25\times 305}{1000000}$ $244=P\times \frac{7625}{1000000}$ $P=\frac{244\times 1000000}{7625}$ P=32000 So the correct option is (c).

Successive percentage change Successive percentage changes multiply, so carry the price along as a chain of factors. Take the original price as 100. After a $55\%$ fall: $100\times \frac{45}{100}=45$ After a $10\%$ fall: $45\times \frac{90}{100}=40.5$ After a $20\%$ fall: $40.5\times \frac{80}{100}=32.4$ After a $65\%$ rise: $32.4\times \frac{165}{100}=53.46$ Net change =53.46-100=-46.54 So the price decreases by $46.54\%$. So the correct option is (c).

Volume of a cube Volume of a cube $=a^3$, where a is its edge. Original volume $=a^3$ New edge =3a New volume $={\left(3a\right)}^3=27a^3$ New volume $÷$ original volume $=\frac{27a^3}{a^3}=27$ So the correct option is (b).

Average speed both ways The two halves of the journey cover the same distance, so the average speed is the harmonic mean of the two speeds - never their simple average. Average speed $=\frac{2xy}{x+y}$ $=\frac{2\times 78\times 91}{78+91}$ $=\frac{14196}{169}$ =84 km/hr So the correct option is (c).