7 February 2025 · Shift 1

Average of grouped numbers Sum of all thirteen numbers $=13\times 57=741$ Sum of the first three $=3\times 48=144$ Sum of the next seven $=7\times 58=406$ Sum of the last three =741-144-406=191 Let the 12th number be x, then the 11th =2x and the 13th =x+3 2x+x+x+3=191 4x=188 x=47, so the 11th =94 and the 13th =50 Required average $=\frac{94+50}{2}=\frac{144}{2}=72$

Mean proportional Mean proportional between a and b $=\sqrt{ab}$ Here a=68 and $b=272=4\times 68$ $\sqrt{68\times 272}=\sqrt{4\times 68\times 68}$ $=2\times 68=136$ So the mean proportional is 136.

LCM by prime factors Break every number into its prime factors. $40=2^3\times 5$ $20=2^2\times 5$ $120=2^3\times 3\times 5$ $335=5\times 67$ LCM takes the highest power of each prime $=2^3\times 3\times 5\times 67$ $=120\times 67=8040$

Average with a new member Total age of 59 students $=59\times 22=1298$ years Total age of the 60 persons $=60\times 23=1380$ years Age of the teacher =1380-1298=82 years Shortcut: teacher's age = new average + students $\times$ rise $=23+59\times 1=82$ years

Train crossing a train Both move the same way, so relative speed =120-60=60 km/h 60 km/h $=60\times \frac{5}{18}=\frac{50}{3}$ m/s Distance to cover = sum of the lengths =150+350=500 m Time $=\frac{500}{\frac{50}{3}}=500\times \frac{3}{50}=30$ seconds 30 seconds $=\frac{30}{60}=0.5$ minute

Comparing ratios Turn every ratio into a decimal. $\frac{19}{21}=0.905$ $\frac{23}{26}=0.885$ $\frac{11}{13}=0.846$ $\frac{17}{42}=0.405$ $\frac{1}{7}=0.143$ Largest first: 0.905>0.885>0.846>0.405>0.143 So the descending order is 19 : 21, 23 : 26, 11 : 13, 17 : 42, 1 : 7.

LCM and a common remainder A number leaving the same remainder with both divisors = (a multiple of their LCM) + remainder. $64=2^6$ and $80=2^4\times 5$ LCM $=2^6\times 5=320$ Smallest such number =320+9=329 Check: $329=64\times 5+9$ and $329=80\times 4+9$

Meeting on a circular track Time taken by Ram for one round $=\frac{800}{25}=32$ seconds Time taken by Mohan for one round $=\frac{800}{40}=20$ seconds Both stand at the starting point again after the LCM of these two times. $32=2^5$ and $20=2^2\times 5$ LCM $=2^5\times 5=160$ So they meet first at the starting point after 160 seconds.

BODMAS simplification Open the innermost bracket first (BODMAS). 16-12=4, so $\frac{16-12}{2}=\frac{4}{2}=2$ $8÷4=2$ Curly bracket $=2\times 2=4$ $18\times 19=342$ Value $=342\times 4=1368$

Successive percentage change Take the original price as ₹100. After the 35% fall: $100\times \frac{65}{100}=65$ After the 10% fall: $65\times \frac{90}{100}=58.5$ After the 30% fall: $58.5\times \frac{70}{100}=40.95$ After the 65% rise: $40.95\times \frac{165}{100}=67.5675$ Net change =67.5675-100=-32.4325 So the price decreases by about 32.43%.

Percentage of a number 2% of x means $\frac{2}{100}\times x$ $\frac{2x}{100}=324$ $x=\frac{324\times 100}{2}$ $x=162\times 100=16200$

Volume of a hemisphere Radius $r=\frac{7}{2}=3.5$ cm Volume of a hemisphere $=\frac{2}{3}\pi r^3$ $=\frac{2}{3}\times \frac{22}{7}\times \frac{7}{2}\times \frac{7}{2}\times \frac{7}{2}$ $=\frac{2\times 22\times 343}{3\times 7\times 8}=\frac{15092}{168}$ =89.83 (to two decimal places) So the volume is 89.83 cm³.

Profit and selling price Cost of 100 chickoos = ₹1,260 Cost of 400 chickoos $=4\times 1260=5040$, i.e. ₹5,040 Selling price of 400 chickoos =5040+860=5900, i.e. ₹5,900 Selling price of one chickoo $=\frac{5900}{400}=14.75$ Selling price per dozen $=14.75\times 12=177$ So one dozen chickoos sell for ₹177.

Difference of CI and SI For three years, difference $=P\times \frac{r^2\left(300+r\right)}{{100}^3}$ Here the difference is ₹183 and r=5 $183=P\times \frac{25\times 305}{1000000}$ $183=P\times \frac{7625}{1000000}$ $P=\frac{183\times 1000000}{7625}$ P=24000 So the principal is ₹24,000.

Order of operations BODMAS: do $÷$ and $\times$ before + and -. $8÷4=2$ $2\times 3=6$ 16+2-6=12

Work and wages Let the whole work be LCM of 5 and 10 =10 units. Abhishek does $\frac{10}{5}=2$ units a day Deepanshi does $\frac{10}{10}=1$ unit a day All three finish in 2 days, so together they do $\frac{10}{2}=5$ units a day Chitransh does 5-(2+1)=2 units a day Wages are shared in the ratio of work done =2:1:2 Chitransh's share $=\frac{2}{5}\times 1000=400$, i.e. ₹400

CI and SI for two years For 2 years the compound rate $=2r+\frac{r^2}{100}$ $=2\times 4+\frac{16}{100}=8.16$, so the CI is 8.16% of the principal For 2 years the simple rate $=2\times 4=8$, so the SI is 8% of the principal The principal is the same, so SI : CI =8:8.16 SI $=178.5\times \frac{8}{8.16}=175$ So the simple interest is ₹175.

Cost price from gain percent Let the cost price be x. A gain of 20% makes the selling price $\frac{120}{100}$ of the cost price. $x\times \frac{120}{100}=216$ $x=\frac{216\times 100}{120}=180$ So the cost of the article is ₹180.

Successive discounts Marked price = ₹30,000 After the first discount of 20%: $30000\times \frac{80}{100}=24000$ After the second discount of 10%: $24000\times \frac{90}{100}=21600$ So the customer pays ₹21,600.

Rate of simple interest $SI=\frac{P\times R\times T}{100}$, with the same R and T=5 for both. Extra principal with Sachin =9400-5400=4000 This extra principal alone earns the extra interest of ₹960. $\frac{4000\times R\times 5}{100}=960$ 200R=960 R=4.8 So the rate of interest is 4.8% per annum.