10 February 2025 · Shift 1

Compound interest, equal amounts Let Varun get V and Ashish get A, so V+A=1612. Both sums grow at $8\%$ compounded annually and the two amounts are equal: $V{\left(1+\frac{8}{100}\right)}^{18}=A{\left(1+\frac{8}{100}\right)}^{19}$ Cancelling 18 common factors gives $\frac{V}{A}=1+\frac{8}{100}=\frac{27}{25}$ Varun's share $=1612\times \frac{27}{27+25}=1612\times \frac{27}{52}=837$ So Raj gave Varun ₹837. Hence the correct option is (c).

Fourth proportional If a:b::c:d, then d is the fourth proportional and $a\times d=b\times c$. Here the proportion is 5:35::x:84. $5\times 84=35\times x$ $x=\frac{420}{35}=12$ Hence the correct option is (b).

Direct proportion The two subjects keep a fixed ratio, Maths : English =60:75=4:5. Let this year's Mathematics marks be x, with English marks 60. $\frac{x}{60}=\frac{4}{5}$ $x=\frac{4\times 60}{5}=48$ Hence the correct option is (a).

Time and work, replacement Take the total work as LCM(25,30)=150 units. Then A does $\frac{150}{25}=6$ units a day and B does $\frac{150}{30}=5$ units a day. Work done together in the first 6 days $=\left(6+5\right)\times 6=66$ units. Remaining work =150-66=84 units, done by A and P in 8 days. A contributes $6\times 8=48$ units, so P does 84-48=36 units in 8 days. P's rate $=\frac{36}{8}=\frac{9}{2}$ units a day. P alone needs $150÷\frac{9}{2}=\frac{100}{3}=33\frac{1}{3}$ days. Hence the correct option is (c).

Circular track, opposite directions P takes 1 minute 30 seconds =90 s for one round; Q takes 45 s. Moving in opposite directions, the fractions of a round they cover per second add up. Rounds closed together in one second $=\frac{1}{90}+\frac{1}{45}=\frac{1+2}{90}=\frac{1}{30}$ They meet when this total reaches one whole round, i.e. after 30 seconds. Hence the correct option is (c).

BODMAS with integers By BODMAS, do division and multiplication before addition and subtraction. $\left(-60\right)÷\left(-12\right)=5$ and $\left(-3\right)\times 7=-21$ The expression becomes (-9)-5+(-21) =-9-5-21=-35 Hence the correct option is (c).

LCM with a common remainder A number that leaves remainder 4 with 12, 16 and 24 must be (a common multiple) $+4$. LCM(12,16,24)=48, so the number has the form 48k+4. Largest such number below 5000: $5000÷48=104$ with remainder 8, so $48\times 104=4992$. Required number =4992+4=4996, which lies between 4000 and 5000. Hence the correct option is (d).

Weighted average Amount spent by a group = its members × its average spend. Total amount $=81\times 467+29\times 117+70\times 378$ =37827+3393+26460=67680 Total members =81+29+70=180 Overall average $=\frac{67680}{180}=376$ So ₹376 is spent per member. Hence the correct option is (c).

Curved surface area of a cylinder Formula: curved surface area of a right circular cylinder $=2\pi rh$. $2\pi r\times 12=96\pi$ 24r=96 $r=\frac{96}{24}=4$ So the radius is 4 cm. Hence the correct option is (b).

LCM by prime factorisation Factorise each number into primes. $36=2^2\times 3^2$, $45=3^2\times 5$, $465=3\times 5\times 31$, $310=2\times 5\times 31$ The LCM takes the highest power of every prime that appears. LCM $=2^2\times 3^2\times 5\times 31$ $=4\times 9\times 5\times 31=5580$ Hence the correct option is (a).

Percentage of a number "$4\%$ of x" means $\frac{4}{100}\times x$. $\frac{4}{100}\times x=24$ $x=\frac{24\times 100}{4}=600$ Hence the correct option is (b).

Compound and simple interest For 2 years the effective compound rate $=3+3+\frac{3\times 3}{100}=6.09\%$ of the principal. For the same 2 years the simple interest rate $=3\times 2=6\%$ of the principal. Both interests sit on the same principal, so SI : CI =6:6.09. SI $=203\times \frac{6}{6.09}=203\times \frac{600}{609}=200$ So the corresponding simple interest is ₹200. Hence the correct option is (a).

Average expenditure Let the average expenditure of all six persons be a (in ₹). Five persons spent $5\times 30=150$ and the sixth spent a+50. The total is also 6a, so 150+(a+50)=6a 200=5a a=40 Total money spent $=6\times 40=240$, i.e. ₹240. Hence the correct option is (b).

False weight, profit percent The weight is $24\%$ light, so when he charges for 100 g he actually hands over 76 g. He sells at cost price, so his whole gain is the 24 g he keeps back. Profit % = (goods kept back ÷ goods actually given) × 100 $=\frac{24}{76}\times 100=\frac{2400}{76}=\frac{600}{19}$ $\frac{600}{19}=31\frac{11}{19}$, so the profit is $31\frac{11}{19}\%$. Hence the correct option is (b).

Price, consumption, expenditure Expenditure = price × consumption. Take price =100 and consumption =100, so expenditure =10000. New price $=100\times \frac{150}{100}=150$ New expenditure is allowed to be $10000\times \frac{123}{100}=12300$ New consumption $=\frac{12300}{150}=82$ Reduction =100-82=18 on a base of 100, i.e. $18\%$. Hence the correct option is (a).

Smallest common multiple The smallest number divisible by all of them is their LCM. $33=3\times 11$, $72=2^3\times 3^2$, 11=11, $18=2\times 3^2$ LCM $=2^3\times 3^2\times 11$ $=8\times 9\times 11=792$ Hence the correct option is (c).

Cost price and required gain Cost price = selling price - gain. CP =1500-150=1350, i.e. ₹1,350. For a $24\%$ gain, SP $=1350\times \left(1+\frac{24}{100}\right)$ $=1350\times \frac{124}{100}=1674$ So he must sell it for ₹1,674. Hence the correct option is (d).

Speed and time, inverse ratio For a fixed distance, time is inversely proportional to speed. Speed becomes $\frac{3}{5}$ of normal, so time becomes $\frac{5}{3}$ of the normal time. Let the normal time be 3x minutes; then the slower trip takes 5x minutes. Extra time =5x-3x=2x=10 x=5 Normal time =3x=15 minutes. Hence the correct option is (d).

Equivalent discount on an offer "Buy three, get two free" means the customer takes 5 shirts but pays for only 3. Let each shirt be marked at 100, so 5 shirts are worth 500 and he pays 300. Discount =500-300=200 Discount % $=\frac{200}{500}\times 100=40$ So the offer equals a $40\%$ discount. Hence the correct option is (d).

Simple interest, difference method Formula: SI $=\frac{P\times R\times T}{100}$. The rate and the time are the same for both, so the extra interest comes only from the extra principal. Extra principal =9400-5400=4000 $\frac{4000\times R\times 4}{100}=360$ 160R=360 $R=\frac{360}{160}=2.25$ So the rate is $2.25\%$ per annum. Hence the correct option is (b).