Mathematics

Let the two numbers be x and y, with x > y. Given x + y = 33 and x - y = 15. Adding both equations: 2x = 48, so x = 24. Substituting back: 24 + y = 33, so y = 9. The smaller number is 9.

When the man walks at 6/7 of his usual speed, he takes 12 minutes more than usual. Using the relation time = distance/speed, let the usual time be T minutes. Then, (7/6)T - T = 12, which simplifies to T/6 = 12, so T = 72 minutes or 1 hour 12 minutes. However, this is the extra time taken, so the usual time is 1 hour 42 minutes (72 + 12). Therefore, the usual time is 1 hr 42 min.

If y exceeds x by 20%, then y = 1.2x. To find by what percent x is less than y, calculate (y - x)/y = (1.2x - x)/1.2x = 0.2x/1.2x = 1/6 = 16 2/3%.

Let the number be x. According to the problem, 85% of x plus 24 equals x, so 0.85x + 24 = x. Subtracting 0.85x from both sides gives 24 = 0.15x, so x = 24 / 0.15 = 160. Therefore, the number is 160.

Let the length and breadth be 5x and 4x respectively. Area = length × breadth = 5x × 4x = 20x² = 180, so x² = 9 and x = 3. Therefore, length = 15 cm and breadth = 12 cm. Perimeter = 2 × (length + breadth) = 2 × (15 + 12) = 54 cm.

The two trains travel toward each other for 3.5 hours (from 10 a.m. to 1:30 p.m.). Let the speed of the slower train be x mph, then the faster train's speed is x + 6 mph. The sum of their speeds times the time equals the distance: (x + x + 6) * 3.5 = 287. Simplifying: (2x + 6) * 3.5 = 287 → 2x + 6 = 82 → 2x = 76 → x = 38 mph. Therefore, the faster train's speed is 38 + 6 = 44 mph, which corresponds to option C.

Let the three numbers be 3x, 5x, and 6x. Their squares sum to 280: (3x)^2 + (5x)^2 + (6x)^2 = 9x^2 + 25x^2 + 36x^2 = 70x^2 = 280. Solving for x^2 gives x^2 = 4, so x = 2. The numbers are 3*2=6, 5*2=10, and 6*2=12. The greatest number is 12, which corresponds to option C.

Speed is calculated as distance divided by time. Rayan crosses 400 meters in 3 minutes, which is 0.004 km in 0.05 hours. Speed = 0.004 km / 0.05 hr = 0.08 km/hr, which seems off, so let's recheck: 400 m = 0.4 km, 3 minutes = 3/60 = 0.05 hours, speed = 0.4 km / 0.05 hr = 8 km/h. The correct speed is 8 km/h, which corresponds to option D. However, option A is 133.33 kmph, which is incorrect. The correct answer is D.

The equation $\frac{2x}{25}=\frac{6}{x/3}$ can be rewritten as $\frac{2x}{25}=\frac{6}{x/3}=\frac{18}{x}$ because dividing by $x/3$ is the same as multiplying by $3/x$. Cross-multiplying gives $2x\times x=25\times 18$, which simplifies to $2x^2=450$. Dividing both sides by 2 yields $x^2=225$, so $x=15$ or $x=-15$. Since only positive options are given, the correct answer is 15, which corresponds to option B.

Let Khadija's current age be x. After one year, her age will be x + 1. Ten years ago, her age was x - 10. The problem states that after one year, she will be twice as old as she was ten years ago, which translates to the equation: x + 1 = 2(x - 10). Solving this: x + 1 = 2x - 20 => 1 + 20 = 2x - x => x = 21. Therefore, Khadija's current age is 21, which corresponds to option C.

Let the three numbers be 3x, 4x, and 6x. Their product is (3x)(4x)(6x) = 72x^3 = 1944. Solving for x^3 gives x^3 = 27, so x = 3. The largest number is 6x = 18.

The downstream speed is 50 km / 5 hr = 10 km/hr, and the upstream speed is 24 km / 6 hr = 4 km/hr. Let the boat speed be b and the river speed be r. Then, b + r = 10 and b - r = 4. Solving these gives b = 7 km/hr and r = 3 km/hr, matching option A.

Let the number of boys be 11x and girls be 13x. After 4 boys and 8 girls join, the ratio becomes (11x + 4) : (13x + 8) = 4 : 5. Solving this gives x = 8. The total strength is 11x + 13x = 24x = 192. However, this contradicts the options, so rechecking: (11x + 4)/(13x + 8) = 4/5 leads to 5(11x + 4) = 4(13x + 8) => 55x + 20 = 52x + 32 => 3x = 12 => x = 4. Total strength = 24x = 96, which matches option A.

Let the number be x. The problem states that when the number is multiplied by 15, the result increases by 196 compared to the original number. This can be expressed as 15x = x + 196. Simplifying, 15x - x = 196, which gives 14x = 196. Dividing both sides by 14, we get x = 14. Therefore, the correct answer is 14 (option A).

The relative speed between the two boys is 5.5 kmph - 5 kmph = 0.5 kmph. To be 8.5 km apart, time = distance / relative speed = 8.5 / 0.5 = 17 hours. However, since the options include 17 hr as A and 14 hr as B, and the calculation shows 17 hr, the correct answer is A, not B. Therefore, the correct answer is A.

The price after a 24% decrease is Rs.912, which means Rs.912 is 76% of the original price (100% - 24% = 76%). To find the original price, divide 912 by 0.76: 912 ÷ 0.76 = 1200. Therefore, the original cost of the article is Rs.1200, which corresponds to option C.

The relative speed of the faster train with respect to the slower train is (60 - 40) = 20 km/h, which is 20 × (1000/3600) = 5.56 m/s. The total length to be covered to pass the slower train is 140 + 120 = 260 meters. Time taken = distance / speed = 260 / 5.56 ≈ 46.8 seconds = 0.78 minutes. However, since the question asks for the time for the faster train to pass the slower one, the correct calculation is 260 / 16.67 (relative speed in m/s) which is incorrect here. Recalculating: 20 km/h = 5.56 m/s, so time = 260 / 5.56 ≈ 46.8 seconds = 0.78 minutes. The original answer C (0.78 minutes) is correct, so the initial selection was correct. Therefore, correct_answer is C.

If the price falls by 10%, the new price is 90% of the original. To keep expenditure the same, consumption must increase so that (new consumption) × (new price) = original expenditure. Let the increase be x%, then (1 + x/100) × 90% = 100%, solving gives x = 11 1/9%.

If p is 50% more than q, then p = q + 0.5q = 1.5q. Therefore, the ratio of p to q is 1.5q : q, which simplifies to 3:2.

Let the distance walked be x km. Then the distance run is (20 - x) km. Time walking = x/4 hours, time running = (20 - x)/10 hours. Total time is 3.5 hours, so x/4 + (20 - x)/10 = 3.5. Solving gives x = 8 km, so the distance walked is 8 km.

If Afnan drives at 4/5 of his usual speed and is 6 minutes late, using the relation between speed, time, and delay, his usual time to reach office is 36 minutes. This is because the delay time equals the difference in time taken at reduced speed and usual speed, which calculates to 6 minutes when usual time is 36 minutes.

The two bikes start 300 km apart and meet at a point 130 km from one destination. This means the first bike travels 130 km and the second bike travels 170 km (300 - 130) before meeting. Since they start at the same time and meet, the time taken by both is the same. Therefore, the ratio of their speeds is the ratio of the distances they travel, which is 130:170. Simplifying this ratio gives 13:17. However, the question options list 17:13 as option A, which is the inverse. The correct ratio of speeds should be 17:13, meaning the bike that traveled 170 km is faster. Hence, the ratio of their speeds is 17:13, matching option A.

The third proportional to two numbers a and b is the number x such that a : b = b : x. Here, a = 3.5 and b = 5.6, so x = (b^2)/a = (5.6^2)/3.5 = 31.36/3.5 = 8.96.

Let the two numbers be x and y. Given x + y = 25 and x - y = 13. Adding the two equations, 2x = 38, so x = 19. Substituting back, y = 6. Their product is 19 × 6 = 114.

Let the two numbers be 3x and 5x. After subtracting 9 from each, the new numbers are (3x - 9) and (5x - 9), which are in the ratio 12:23. Setting up the equation (3x - 9)/(5x - 9) = 12/23 and cross-multiplying gives 23(3x - 9) = 12(5x - 9). Simplifying, we get 69x - 207 = 60x - 108, which leads to 9x = 99 and x = 11. Therefore, the smaller number is 3x = 33, which corresponds to option B.