13 February 2025 · Shift 3

Volume of a cylinder The capacity of a cylindrical tank is simply the volume of the cylinder. Formula: $V=\pi r^{2}h$ Substituting r=7, h=9 and $\pi =\frac{22}{7}$: $V=\frac{22}{7}\times 7^2\times 9$ $V=\frac{22}{7}\times 49\times 9=22\times 7\times 9$ V=1386 So the capacity of the tank is 1386 m³ and option (b) is correct.

Successive percentage change Take the original price as 100 units and turn every change into a multiplying factor. After the three falls: $100\times \frac{90}{100}\times \frac{50}{100}\times \frac{50}{100}=22.5$ The fourth month raises this by 45%: $22.5\times \frac{145}{100}=32.625$ Change =32.625-100=-67.375, so the price has fallen. Decrease $=\frac{67.375}{100}\times 100=67.375\%\approx 67.38\%$ The price decreases by 67.38%, so option (c) is correct.

Proportion Let the number added be x. Four numbers are in proportion when the first ratio equals the second. $\frac{9+x}{16+x}=\frac{13+x}{23+x}$ Cross-multiplying: (9+x)(23+x)=(13+x)(16+x) $207+32x+x^2=208+29x+x^2$ The $x^2$ terms cancel: 32x-29x=208-207 3x=1, so $x=\frac{1}{3}$ So option (c) is correct.

BODMAS simplification Solve the brackets from the inside out, following BODMAS. Innermost bracket: 18-12=6, so $\frac{18-12}{2}=\frac{6}{2}=3$ Curly bracket: $6÷2\times 3=3\times 3=9$ Square bracket: $23\times 12=276$ $276\times 9=2484$ So option (c) is correct.

CI and SI difference For three years the gap between compound and simple interest has a fixed formula. Formula: $CI-SI=P\times \frac{R^2\left(300+R\right)}{{10}^6}$ Substituting CI-SI=155 and R=10: $155=P\times \frac{{10}^2\times \left(300+10\right)}{{10}^6}$ $155=P\times \frac{100\times 310}{1000000}=\frac{31P}{1000}$ $P=\frac{155\times 1000}{31}=5000$ The principal is ₹5,000, so option (c) is correct.

Mixture cost price Cost per kg of a mixture = total cost ÷ total quantity. Cost of the first lot $=6\times 2730=16380$ Cost of the second lot $=12\times 1002=12024$ Total cost =16380+12024=28404 for 18 kg Cost per kg $=\frac{28404}{18}=1578$ So option (b) is correct.

Percentage basics "2% of x" means $\frac{2}{100}\times x$. $\frac{2}{100}\times x=252$ $x=252\times \frac{100}{2}$ $x=252\times 50=12600$ So option (a) is correct.

Profit percentage Profit = selling price − cost price =6200-5000=1200 Profit percentage is always taken on the cost price. Profit $\%=\frac{1200}{5000}\times 100$ $=\frac{120000}{5000}=24$ The profit is 24%, so option (b) is correct.

Trains crossing Both trains move the same way, so the faster one gains on the slower one at the difference of their speeds. Relative speed =100-80=20 km/h In m/s: $20\times \frac{5}{18}=\frac{50}{9}$ To cross completely the faster train must cover both lengths: 410+390=800 m Time $=800÷\frac{50}{9}=800\times \frac{9}{50}=144$ seconds In minutes: $\frac{144}{60}=2.4$ So option (c) is correct.

Average speed The two halves of the journey cover the same distance, so the average speed is not the ordinary average. Formula: average speed $=\frac{2xy}{x+y}$ $=\frac{2\times 24\times 40}{24+40}$ $=\frac{1920}{64}=30$ The average speed is 30 km/hr, so option (d) is correct.

Averages and spending Total pocket money $=2500\times 3=7500$ Total unspent money $=2100\times 3=6300$ Total money spent =7500-6300=1200 Let X spend x; then Y spends 2x and Z spends 3x. x+2x+3x=1200 6x=1200, so x=200 X spends ₹200, so option (c) is correct.

BODMAS rule By BODMAS, division and multiplication are done before addition and subtraction. $9÷3=3$ and $2\times 4=8$ The expression becomes 16+3-8 =19-8=11 So option (a) is correct.

Successive discount First discount: price after 20% off $=19200\times \frac{80}{100}=15360$ The second discount is worked out on ₹15,360, not on the marked price - that is the trap here. Second discount amount =15360-13824=1536 Second discount $=\frac{1536}{15360}\times 100=10$ The second discount is 10%, so option (b) is correct.

Time and work Measure the work in man-days: total work $=54\times 48=2592$ Work finished in the first 8 days $=54\times 8=432$ Remaining work =2592-432=2160 Men working now =54+36=90 Days needed $=\frac{2160}{90}=24$ So option (a) is correct.

CI and SI comparison Compare the two interests as rates on the same principal. Effective CI rate for 2 years at 4% $=4+4+\frac{4\times 4}{100}=8.16\%$ SI rate for 2 years at 4% $=4\times 2=8\%$ The principal is the same, so the interests are in the ratio of these rates. $\frac{SI}{255}=\frac{8}{8.16}$ $SI=255\times \frac{800}{816}$ $SI=255\times \frac{50}{51}=250$ The simple interest is ₹250, so option (b) is correct.

LCM with remainder Each division leaves 5, so x-5 is a common multiple of 12, 15, 18, 20 and 25. $12=2^2\times 3$, $15=3\times 5$, $18=2\times 3^2$, $20=2^2\times 5$, $25=5^2$ LCM $=2^2\times 3^2\times 5^2=900$ So x=900k+5 for some positive integer k. x must also be a multiple of 7, and the smallest such k is 4. $x=900\times 4+5=3605$, and $3605=7\times 515$ So option (b) is correct.

LCM by factorisation Break each number into prime factors. The LCM keeps the highest power of every prime that appears. LCM $=2^4\times 3\times 7\times 13$ $=16\times 3\times 7\times 13=16\times 273=4368$ So option (b) is correct.

Work on alternate days Take the total work as the LCM of 5, 6, 8 and 10, i.e. 120 units. Day 1 is worked by A, B and C: 24+20+15=59 units Day 2 is worked by A, B and D: 24+20+12=56 units So a 2-day cycle finishes 59+56=115 units. After 2 days the work left is 120-115=5 units. Day 3 begins a fresh cycle, so C helps again and the rate is 59 units per day. Extra time $=\frac{5}{59}$ day Total time $=2+\frac{5}{59}=2\frac{5}{59}$ days, so option (c) is correct.

Profit and selling price Cost price of 400 mangoes $=\frac{1220}{100}\times 400=4880$ Selling price of 400 mangoes =4880+820=5700 Selling price of one mango $=\frac{5700}{400}=14.25$ Selling price per dozen $=14.25\times 12=171$ So option (a) is correct.

Simple interest Formula: $SI=\frac{P\times R\times T}{100}$ Substituting SI=888, R=2 and T=8: $888=\frac{P\times 2\times 8}{100}=\frac{16P}{100}$ $P=\frac{888\times 100}{16}$ P=5550 The principal is ₹5,550, so option (d) is correct.