18 February 2025 · Shift 2

Train crossing a tunnel To cross a tunnel the train must cover the length of the tunnel PLUS its own length, because the tunnel is clear only when the last coach leaves the far end. Total distance =398+802=1200 m Speed =45 km/h $=45\times \frac{5}{18}=12.5$ m/s Time $=\frac{1200}{12.5}=96$ So the required time is 96 seconds and the correct option is (d).

Mark-up and discount The mark-up is measured on the cost price and the discount on the marked price, so take the cost price as ₹100. Marked price =100+18.8=118.8 Discount $=12.5\%$ of $118.8=\frac{118.8}{8}=14.85$ Selling price =118.8-14.85=103.95 Profit =103.95-100=3.95 on a cost of 100 Profit percentage $=3.95\%$ So the correct option is (c).

Compound interest, equal amounts Let Varun get V and Ashish get A. Both reach the SAME amount, Varun in 13 years and Ashish in 14 years at $10\%$ compounded annually. $V{\left(\frac{11}{10}\right)}^{13}=A{\left(\frac{11}{10}\right)}^{14}$ $\frac{V}{A}={\left(\frac{11}{10}\right)}^{14-13}=\frac{11}{10}$ Ashish has one extra year to grow, so his share must be the smaller one: Varun : Ashish =11:10. 11+10=21 units =1218, so 1 unit $=\frac{1218}{21}=58$ Ashish's share $=10\times 58=580$ So Raj gave Ashish ₹580 and the correct option is (d).

LCM by prime factorisation Break every number into prime factors first. $36=2^2\times 3^2$ and $30=2\times 3\times 5$ $114=2\times 3\times 19$ and $380=2^2\times 5\times 19$ The LCM keeps the HIGHEST power of each prime that appears: $2^2$, $3^2$, 5 and 19. LCM $=4\times 9\times 5\times 19=3420$ So the correct option is (b).

Third proportional The first three prime numbers are 2, 3 and 5 (1 is not a prime), so Y=2+3+5=10. If X is the third proportional of 20 and Y, the three numbers are in continued proportion: 20:Y=Y:X. Product of the means = product of the extremes: $20\times X=Y^2$ $X=\frac{{10}^2}{20}=\frac{100}{20}=5$ So the correct option is (c).

Work done in rotation Take the total work as the LCM of 22, 24 and 33, that is 264 units. One day's work: A $=\frac{264}{22}=12$, B $=\frac{264}{24}=11$, C $=\frac{264}{33}=8$ units. The turn order is A, C, B, so one cycle of 3 days does 12+8+11=31 units. 8 cycles =24 days do $8\times 31=248$ units, leaving 264-248=16 units. Day 25 belongs to A, who does 12 units; work left =16-12=4 units. Day 26 belongs to C, who needs $\frac{4}{8}=\frac{1}{2}$ day for the last 4 units. Total time $=24+1+\frac{1}{2}=25\frac{1}{2}$ days So the correct option is (c).

Fourth proportional If d is the fourth proportional to a, b and c, then a:b=c:d. Here 4:a=16a:81 Product of the extremes = product of the means: $4\times 81=a\times 16a$ $324=16a^2$ $a^2=\frac{324}{16}=20.25$ So the correct option is (a).

Weighted average A weighted average multiplies every score by its own weight and divides by the total weight - a plain average of 80, 70 and 60 would be wrong here. Total weight =20+35+45=100 Weighted total $=80\times 20+70\times 35+60\times 45$ =1600+2450+2700=6750 Weighted average $=\frac{6750}{100}=67.5$ So the correct option is (c).

SI and CI difference For 2 years the two interests differ only because compound interest also pays interest on the first year's interest. Simple interest of one year $=\frac{40}{2}=20$ Difference =50-40=10, and this 10 is exactly one year's interest on that 20. $\frac{20\times R}{100}=10$, so $R=50\%$ Principal $=\frac{100\times SI}{R\times T}=\frac{100\times 40}{50\times 2}=40$ So the principal is ₹40 and the correct option is (a).

BODMAS simplification By BODMAS, division and multiplication are done before addition and subtraction. $12÷3=4$ and $2\times 3=6$ $16+12÷3-2\times 3=16+4-6$ =20-6=14 So the correct option is (d).

LCM by prime factorisation Write each number as a product of primes. $51=3\times 17$ and $18=2\times 3^2$ $432=2^4\times 3^3$ and $144=2^4\times 3^2$ The LCM takes the highest power of every prime: $2^4\times 3^3\times 17$ LCM $=16\times 27\times 17=432\times 17=7344$ So the correct option is (a).

Equal SP, equal gain and loss When two articles are sold at the SAME price, one at a gain of $x\%$ and the other at a loss of $x\%$, the deal is always a LOSS. Loss percentage $=\frac{x^2}{100}$ Here x=5, so loss $=\frac{5^2}{100}=\frac{25}{100}=0.25\%$ Check with the actual figures: CP $=\frac{7980\times 100}{105}=7600$ and CP $=\frac{7980\times 100}{95}=8400$, total 16000 against a total SP of 15960. Loss $=\frac{40}{16000}\times 100=0.25\%$, which confirms the shortcut. The selling price itself never enters the formula. So the correct option is (b).

Area of a right triangle Bring all three sides to one unit: 0.25 m =25 cm, so the sides are 7 cm, 24 cm and 25 cm. Test the sides: $7^2+{24}^2=49+576=625={25}^2$ The Pythagoras relation holds, so the triangle is right-angled and 25 cm is the hypotenuse - no need for Heron's formula. The perpendicular sides are 7 cm and 24 cm, so area $=\frac{1}{2}\times 7\times 24$ Area =84 square cm So the correct option is (a).

Average of remaining group Total age of the class $=40\times 12=480$ years First group $=5\times 10=50$ years and second group $=5\times 14=70$ years Students left =40-5-5=30 Their total age =480-50-70=360 years Average of the rest $=\frac{360}{30}=12$ years One group is 2 years below the class average and the other 2 years above it, so the two shifts cancel and the rest still average 12 years. So the correct option is (b).

Price rise and consumption cut Expenditure = price × consumption, so take both the price and the expenditure as 100 to start with. New price =100+90=190 and the new expenditure =100+33=133 New consumption $=\frac{133}{190}\times 100=70$ Fall in consumption =100-70=30 Reduction $=\frac{30}{100}\times 100=30\%$ So the correct option is (a).

Simple interest rate Both sums run at the same rate for the same 3 years, so the extra interest comes only from the extra principal. Extra principal =10200-5400=4800 $\frac{4800\times R\times 3}{100}=720$ 144R=720 $R=\frac{720}{144}=5$ So the rate is $5\%$ per annum and the correct option is (c).

BODMAS with signed numbers By BODMAS, the division and the multiplication are settled before the addition and subtraction. $\left(-60\right)÷\left(-15\right)=4$, because a negative divided by a negative is positive. $\left(-2\right)\times 8=-16$ $\left(-9\right)-\left(-60\right)÷\left(-15\right)+\left(-2\right)\times 8=\left(-9\right)-4+\left(-16\right)$ =-13-16=-29 The tempting slip is to make the division negative, which gives -1 for the first two terms; keep the sign rule in mind. So the correct option is (d).

Profit stated in selling price Let the selling price of one chair be SP and its cost price be CP. Profit = total SP - total CP. $153\times SP-153\times CP=85\times SP$ $68\times SP=153\times CP$ $\frac{SP}{CP}=\frac{153}{68}=\frac{9}{4}$ Take SP=9 and CP=4, so profit =9-4=5 per chair. Profit percentage $=\frac{5}{4}\times 100=125\%$ Note the gain is compared with the COST price, not with the selling price - that is what rules out the neighbouring options. So the correct option is (b).

Percentage of a number $2\%$ of x means $\frac{2}{100}\times x$. $\frac{2x}{100}=312$ $x=\frac{312\times 100}{2}$ x=15600 A quick check: $2\%$ is $\frac{1}{50}$, so x must be 50 times 312, which is 15600. So the correct option is (a).

Trains in the same direction Moving in the SAME direction, the faster train gains on the slower one at the DIFFERENCE of their speeds. Relative speed =80-60=20 km/h $=20\times \frac{5}{18}=\frac{50}{9}$ m/s To pass completely, the faster train must cover both lengths: 200+250=450 m Time $=450÷\frac{50}{9}=450\times \frac{9}{50}$ Time =81 seconds So the correct option is (a).