18 February 2025 · Shift 1

Percentage and ratio Given: 4% of m + 6% of n = half of (7% of m + 14% of n). In symbols: $\frac{4m}{100}+\frac{6n}{100}=\frac{1}{2}\left(\frac{7m}{100}+\frac{14n}{100}\right)$ Multiply throughout by 100: $4m+6n=\frac{1}{2}\left(7m+14n\right)$ Multiply throughout by 2: 8m+12n=7m+14n $8m-7m=14n-12n\Rightarrow m=2n$ Hence $\frac{m}{n}=\frac{2}{1}$, so m:n=2:1 and the correct option is (c).

Weighted average The overall average is the total amount divided by the total number of members, not the average of the three averages. Total members =13+10+45=68 Total amount $=13\times 367+10\times 112+45\times 401$ =4771+1120+18045=23936 Overall average $=\frac{23936}{68}=352$ Hence the average spent per member is ₹352 and the correct option is (c).

Time and work Take the total work as the LCM of 16 and 24, i.e. 48 units. A's one day work $=\frac{48}{16}=3$ units and B's one day work $=\frac{48}{24}=2$ units. Working alternately, one pair of days (A then B) finishes 3+2=5 units. In 9 such pairs, i.e. 18 days, work done $=9\times 5=45$ units. Work left =48-45=3 units, and the 19th day belongs to A, who does exactly 3 units in a day. So A completes the job on the 19th day itself, and the correct option is (d).

Trains crossing each other The trains run in the SAME direction, so the relative speed is the difference of the speeds. Relative speed =110-80=30 km/h 30 km/h $=30\times \frac{5}{18}=\frac{25}{3}$ m/s To cross completely, the faster train must cover the sum of the two lengths =240+410=650 m. Time = distance ÷ relative speed $=\frac{650}{\frac{25}{3}}=650\times \frac{3}{25}=78$ s 78 s $=\frac{78}{60}=1.3$ minutes, so the correct option is (d).

Simple interest For interest the day of lending is not counted, so count from 4 February to 17 April. Time =25+31+17=73 days. 2023 is an ordinary year, so $T=\frac{73}{365}=\frac{1}{5}$ year. Formula: $SI=\frac{P\times R\times T}{100}$ $SI=\frac{2000\times 6.75}{100}\times \frac{1}{5}=135\times \frac{1}{5}=27$ Hence the simple interest is ₹27 and the correct option is (c).

LCM by prime factors Break each number into prime factors and keep the highest power of every prime. LCM $=2^4\times 3^3\times 5$ $=16\times 27\times 5=2160$ Hence the LCM is 2160 and the correct option is (c).

Loss percentage Selling price is less than cost price, so there is a loss. Loss =2160-1998=162 Loss percentage is always taken on the COST price, not on the selling price. Loss percentage $=\frac{162}{2160}\times 100=\frac{16200}{2160}=7.5$ Hence the loss is 7.5% and the correct option is (b).

Volume of a cuboid First bring the capacity to cubic metres: 1000 litres = 1 m³. 62000 litres $=\frac{62000}{1000}=62$ m³ Formula: volume of a cuboid $=l\times b\times h$ $62=2.5\times b\times 10$ $62=25b\Rightarrow b=\frac{62}{25}=2.48$ Hence the breadth of the tank is 2.48 m and the correct option is (d).

Percentage change in a fraction Let the original fraction be $\frac{x}{y}$. A 10% fall makes the numerator $\frac{90x}{100}$ and a 15% rise makes the denominator $\frac{115y}{100}$. $\frac{90x}{115y}=\frac{2}{5}$ $\frac{x}{y}=\frac{2}{5}\times \frac{115}{90}=\frac{230}{450}$ $\frac{230}{450}=\frac{23}{45}$ (dividing both parts by 10) Hence the original fraction is $\frac{23}{45}$ and the correct option is (b).

Discount on marked price A discount of 20% means the customer pays 80% of the marked price, and the discount is always figured on the marked price. Price paid $=25960\times \frac{100-20}{100}=25960\times \frac{80}{100}$ $=25960\times 0.8=20768$ Hence the customer pays ₹20,768 and the correct option is (c).

Price, consumption and expenditure Formula: Expenditure = Price × Consumption. Take price =100 and consumption =100, so expenditure $=100\times 100=10000$. Price rises by 70%: new price =170. Expenditure may rise by only 19%: new expenditure $=10000\times \frac{119}{100}=11900$. New consumption $=\frac{11900}{170}=70$ Fall in consumption =100-70=30, i.e. $\frac{30}{100}\times 100=30$ per cent. Hence the consumption must be reduced by 30% and the correct option is (b).

LCM of lap times A runner is at the starting point only at whole multiples of his own lap time, so all four are there together at the LCM of the four times. $280=2^3\times 5\times 7$, $350=2\times 5^2\times 7$, $420=2^2\times 3\times 5\times 7$, $560=2^4\times 5\times 7$ LCM $=2^4\times 3\times 5^2\times 7$ $=16\times 3\times 25\times 7=8400$ Hence they meet at the starting point after 8400 seconds and the correct option is (d).

Average of readings Formula: average = sum of the readings ÷ number of readings. Sum =1.35+1.29+1.32+1.36=5.32 Number of readings =4 Average $=\frac{5.32}{4}=1.33$ Hence the average refractive index is 1.33 and the correct option is (b).

Compound interest and ratio Let Anand's share be x and Anil's share be y, so x+y=1230. Anand keeps his money for 12 years and Anil for 13 years, and the two amounts are equal: $x{\left(1+\frac{5}{100}\right)}^{12}=y{\left(1+\frac{5}{100}\right)}^{13}$ Cancel ${\left(\frac{21}{20}\right)}^{12}$ from both sides: $x=y\times \frac{21}{20}$ So x:y=21:20; total parts =21+20=41. One part $=\frac{1230}{41}=30$ Anil's share $=20\times 30=600$ Hence Manoj gave ₹600 to Anil and the correct option is (a).

LCM and HCF relation Formula: LCM × HCF = product of the two numbers. $54\times 9=18\times x$ 486=18x $x=\frac{486}{18}=27$ Check: HCF of 18 and 27 is 9 and their LCM is 54, so the pair fits. Hence the other number is 27 and the correct option is (c).

Simple and compound interest For 2 years, the extra amount in CI is just the interest earned on the first year's interest. Difference =45-42=3 First year's interest $=\frac{42}{2}=21$ So $21\times \frac{R}{100}=3\Rightarrow R=\frac{300}{21}=\frac{100}{7}$ per cent. Formula: $SI=\frac{P\times R\times T}{100}$ $42=\frac{P\times \frac{100}{7}\times 2}{100}=\frac{2P}{7}$ $P=\frac{42\times 7}{2}=147$ Hence the principal is ₹147 and the correct option is (a).

Combining two ratios A:B=9:4 and A:C=3:2. The common term is A, so make it the same in both ratios. Multiply A:C=3:2 by 3 to get A:C=9:6. Now A:B:C=9:4:6. Multiplying every term by 3 gives 27:12:18, which is the same ratio. Hence A:B:C=27:12:18 and the correct option is (b).

Profit and selling price 400 apples are 4 hundreds, so cost price $=1240\times 4=4960$ Selling price of all 400 = cost price + profit =4960+840=5800 Selling price of one apple $=\frac{5800}{400}=14.5$ Selling price of one dozen $=14.5\times 12=174$ Hence the selling price per dozen is ₹174 and the correct option is (c).

BODMAS with integers By BODMAS, division and multiplication are done before addition and subtraction. $\left(-60\right)÷\left(-12\right)=5$ (a negative divided by a negative is positive) $\left(-4\right)\times 9=-36$ The expression becomes (-9)-5+(-36) =-9-5-36=-50 Hence the value is -50 and the correct option is (c).

BODMAS simplification Work from the innermost bracket outwards. Innermost: 19-12=7, so $\frac{19-12}{7}=\frac{7}{7}=1$ Curly bracket: $8÷8\times 1=1\times 1=1$ Square bracket: $\left(17\times 10\right)\times 1=170\times 1=170$ Hence the value is 170 and the correct option is (d).