4 February 2025 · Shift 3

Discount and selling price Discount is always reckoned on the marked price, never on the selling price. Formula: Selling price $=MP-\frac{D\times MP}{100}$ Discount $=\frac{4.2\times 7895}{100}=331.59$ Selling price = 7895 - 331.59 = 7563.41 Hence the selling price is ₹7,563.41, so the correct option is (a).

LCM by prime factorisation Break every number into prime factors first. $56=2^3\times 7$, $30=2\times 3\times 5$, $108=2^2\times 3^3$, $120=2^3\times 3\times 5$ The LCM keeps the highest power of each prime that occurs: $2^3$, $3^3$, 5 and 7. $LCM=8\times 27\times 5\times 7=7560$ Hence the LCM is 7560, so the correct option is (b).

Ratio and proportion Let the two numbers be 19x and 24x. After reducing each by 36: $\frac{19x-36}{24x-36}=\frac{3}{4}$ Cross-multiplying: 4(19x-36) = 3(24x-36) 76x - 144 = 72x - 108 4x = 36, so x = 9. The numbers are $19\times 9=171$ and $24\times 9=216$. Sum = 171 + 216 = 387, so the correct option is (a).

Time and work Take the whole painting as 1 unit of work. One day's work of Ritu and Mona together $=\frac{1}{18}$. One day's work of Ritu alone $=\frac{1}{27}$. Mona's one day's work $=\frac{1}{18}-\frac{1}{27}=\frac{3-2}{54}=\frac{1}{54}$ Mona finishes the whole work in 54 days. Hence Mona alone needs 54 days, so the correct option is (d).

Profit per cent Let the selling price be 100, because the cost price is given as a part of it. Cost price $=\frac{75}{100}\times 100=75$ Profit = SP - CP = 100 - 75 = 25 Formula: profit per cent $=\frac{Profit}{CP}\times 100$ Profit per cent $=\frac{25}{75}\times 100=\frac{100}{3}=33\frac{1}{3}$ Hence the profit is $33\frac{1}{3}$ per cent, so the correct option is (a).

Average with a wrong entry Total recorded first $=7\times 55=385$ The entry 35 was read as 53, so the total is too large by 53 - 35 = 18. Correct total = 385 - 18 = 367 Correct average $=\frac{367}{7}=52.428...=52.43$ Hence the correct average score is 52.43, so the correct option is (c).

Average age Total age of the 54 students $=54\times 37=1998$ years. With the teacher the group has 55 people, so their total age $=55\times 38=2090$ years. Age of the teacher = 2090 - 1998 = 92 years. Hence the teacher is 92 years old, so the correct option is (a).

Speed of sound The answer is wanted in metres per second, so first change the distance to metres. 1.35 km $=1.35\times 1000=1350$ m Formula: $Speed=\frac{Distance}{Time}$ $Speed=\frac{1350}{3}=450$ Hence sound travels at 450 m/s, so the correct option is (b).

Compound interest division Let Ajay get A and Prakash get P, so A + P = 1224. The two amounts are equal: $A{\left(1+\frac{4}{100}\right)}^{18}=P{\left(1+\frac{4}{100}\right)}^{19}$ Dividing both sides by ${\left(\frac{26}{25}\right)}^{18}$ gives $A=P\times \frac{26}{25}$. So A : P = 26 : 25. Total parts = 26 + 25 = 51, so one part $=\frac{1224}{51}=24$. Prakash's share $=25\times 24=600$ Hence Vijay gave Prakash ₹600, so the correct option is (d).

Simple interest Formula: $SI=\frac{P\times R\times T}{100}$ Substituting: $318=\frac{P\times 6\times 2}{100}$ $318=\frac{12P}{100}$ $P=\frac{318\times 100}{12}=2650$ Hence the principal is ₹2,650, so the correct option is (b).

Profit and unit price 400 apples make $\frac{400}{100}=4$ hundreds. Cost price of all 400 $=4\times 1200=4800$ Selling price of all 400 = 4800 + 800 = 5600 Selling price of one apple $=\frac{5600}{400}=14$ Selling price of one dozen $=14\times 12=168$ Hence a dozen apples sell for ₹168, so the correct option is (b).

Percentage of a number '4% of x' means $\frac{4}{100}\times x$. So $\frac{4x}{100}=72$ $4x=72\times 100=7200$ $x=\frac{7200}{4}=1800$ Hence x is 1800, so the correct option is (c).

Direct proportion In direct proportion the earning for one hour stays the same, so find it first. Earning in 1 hour $=\frac{324}{6}=54$ Earning in 29 hours $=54\times 29=1566$ Hence ₹1,566 is earned in 29 hours, so the correct option is (b).

Boat and stream Let the speed of the boat in still water be x km/h. Downstream speed = x + 9 and upstream speed = x - 9. Total time: $\frac{56}{x+9}+\frac{28}{x-9}=7$ $56\left(x-9\right)+28\left(x+9\right)=7\left(x^2-81\right)$ $84x-252=7x^2-567$ Dividing by 7: $x^2-12x-45=0$ (x-15)(x+3) = 0, and a speed cannot be negative, so x = 15. Hence the boat's speed in still water is 15 km/h, so the correct option is (d).

Volume of a sphere Formula: volume of a sphere $=\frac{4}{3}\pi r^3$ Original volume $=\frac{4}{3}\pi r^3$ New radius is 3r, so new volume $=\frac{4}{3}\pi {\left(3r\right)}^3=27\times \frac{4}{3}\pi r^3$ Ratio $=\frac{\frac{4}{3}\pi r^3}{27\times \frac{4}{3}\pi r^3}=\frac{1}{27}$ Hence the required ratio is 1 : 27, so the correct option is (c).

Compound interest division Let Vijay get V and Ajay get A, so V + A = 1617. The two amounts are equal: $V{\left(1+\frac{10}{100}\right)}^{13}=A{\left(1+\frac{10}{100}\right)}^{14}$ Dividing both sides by ${\left(\frac{11}{10}\right)}^{13}$ gives $V=A\times \frac{11}{10}$. So V : A = 11 : 10. Total parts = 11 + 10 = 21, so one part $=\frac{1617}{21}=77$. Vijay's share $=11\times 77=847$ Hence Mahesh gave Vijay ₹847, so the correct option is (d).

HCF by prime factorisation Write each number as a product of primes. $153=3^2\times 17$, $117=3^2\times 13$, $405=3^4\times 5$ The only prime common to all three is 3, and its lowest power in the three is $3^2$. $HCF=3^2=9$ Hence the HCF is 9, so the correct option is (a).

LCM and HCF relation For any two numbers: product of the numbers $=LCM\times HCF$. Let the other number be x, so $33\times x=66\times 11$. 33x = 726 $x=\frac{726}{33}=22$ Hence the other number is 22, so the correct option is (c).

Successive percentage change Take the original price as ₹100 and apply one change at a time. After a fall of 40%: $100\times \frac{60}{100}=60$ After a fall of 10%: $60\times \frac{90}{100}=54$ After a fall of 50%: $54\times \frac{50}{100}=27$ After a rise of 50%: $27\times \frac{150}{100}=40.5$ Net change = 40.5 - 100 = -59.5 on an original of 100. Hence the price decreases by 59.5%, so the correct option is (b).

BODMAS simplification By BODMAS, clear the innermost bracket first, then division and multiplication from left to right. 18 - 14 = 4, so $\frac{18-14}{4}=\frac{4}{4}=1$ Inside the curly bracket: $6÷6\times 1=1\times 1=1$ In the round bracket: $23\times 5=115$ Whole expression $=115\times 1=115$ Hence the value is 115, so the correct option is (d).