Ratio, Proportion and Partnership

The cost price per liter of milk is Rs.3.50, and after adding water, the cost per liter reduces to Rs.2. Using the dilution formula: (Cost price × Quantity of milk) = (New cost × Total quantity), we get 3.5 × 40 = 2 × (40 + x). Solving for x gives x = 28 liters of water.

Let the total capacity be x litres. Initially, milk and water are in ratio 4:5, so milk = 4k and water = 5k with 9k = x. Adding 8 litres of milk fills the can, so new milk = 4k + 8, water = 5k, total = x. The new ratio is 6:5, so (4k + 8)/5k = 6/5. Solving gives k = 8, so x = 9k = 72 litres. However, 72 is not an option, so re-checking: The ratio after addition is 6:5, so (4k + 8)/(5k) = 6/5 => 5(4k + 8) = 6*5k => 20k + 40 = 30k => 40 = 10k => k = 4. So capacity x = 9k = 36 litres. But 36 is not an option either. Re-examining the problem: The can is filled after adding 8 litres milk, so total capacity = initial volume + 8 litres milk. Initial volume = 9k, so total capacity = 9k + 8. But the problem states the can is full after adding 8 litres milk, so total capacity = 9k + 8. The new ratio is (4k + 8) : 5k = 6 : 5. Cross-multiplied: 5(4k + 8) = 6*5k => 20k + 40 = 30k => 40 = 10k => k = 4. So initial volume = 9*4 = 36 litres, total capacity = 36 + 8 = 44 litres. Therefore, the capacity of the can is 44 litres, which corresponds to option B. Hence, the correct answer is B.

Let the amounts with P and Q be x. According to the problem, R has two-thirds of the total amount with P and Q combined, so R = (2/3)x. The total amount among P, Q, and R is Rs. 6000, so we have x + R = 6000. Substituting R, we get x + (2/3)x = 6000, which simplifies to (5/3)x = 6000. Solving for x gives x = (6000 * 3)/5 = 3600. Therefore, R's amount is (2/3)*3600 = 2400. Hence, the amount with R is Rs. 2400, which corresponds to option C.

First, find the total man-hours for 6 men working 4 hours: 6 × 4 = 24 man-hours to complete 50 tasks. So, 1 task requires 24/50 = 0.48 man-hours. For 10 men working 6 hours, total man-hours = 10 × 6 = 60. Number of tasks = 60 / 0.48 = 125 tasks.

Given that 12 men can complete the work in 20 days, the total work can be expressed as 12 men × 20 days = 240 man-days. Similarly, 20 women can complete the same work in 16 days, so total work = 20 women × 16 days = 320 woman-days. Since the total work is the same, 240 man-days = 320 woman-days. The capacity of one man per day is 1/240 of the work, and the capacity of one woman per day is 1/320 of the work. Therefore, the ratio of the capacity of a man to a woman is (1/240) : (1/320) = 320 : 240 = 4 : 3. Hence, the correct ratio is 4:3, which corresponds to option C.

The total work is the same in both cases. Work done by 20 women in 16 days equals work done by 16 men in 15 days. Calculating daily work rates and comparing, the ratio of a man's capacity to a woman's capacity is 2:1.

Each machine produces 270 ÷ 6 = 45 bottles per minute. Therefore, 10 machines produce 10 × 45 = 450 bottles per minute. In 4 minutes, they produce 450 × 4 = 1800 bottles.

Atif and Bilal start 110 km apart and cycle towards each other. Their speeds are in the ratio 6:5, so the total parts are 6 + 5 = 11. The distance each covers before meeting is proportional to their speeds. Therefore, Atif's share of the distance is (6/11) × 110 = 60 km. This matches option D. The original correct answer A (66 km) is incorrect based on the ratio calculation.

The total capacity of the tank is 12 buckets × 13.5 litres = 162 litres. To find how many 9-litre buckets are needed, divide 162 litres by 9 litres per bucket, which equals 18 buckets.

The shares of A, B, C, and D are in the ratio 5:2:4:3. Let the common multiplier be x. Then C's share is 4x and D's share is 3x. Given C gets Rs. 1000 more than D, we have 4x - 3x = 1000, so x = 1000. Therefore, B's share is 2x = 2 * 1000 = Rs. 2000. Among the options, Rs. 2000 corresponds to option C, so the correct answer is C.

The fourth proportion to 3, 7, and 9 is found by solving the proportion 3 : 7 = 9 : x. Cross-multiplying gives 3x = 7 × 9, so x = 63.

Let the number of 25p, 10p, and 5p coins be x, 2x, and 3x respectively. The total value is 25x + 10(2x) + 5(3x) = 25x + 20x + 15x = 60x pence. Since Rs. 30 = 3000 pence, 60x = 3000, so x = 50. Therefore, the number of 5p coins is 3x = 150.

Given the equation XY = kZ, and the initial ratio X:Y:Z = 2:3:4 with X = 4, we find the values of Y and Z. Since X corresponds to 2 parts and equals 4, each part is 2. Therefore, Y = 3 parts = 6 and Z = 4 parts = 8. Using XY = kZ, we get 4 * 6 = k * 8, so k = 3. When Y changes to 12 and Z remains 8, the equation becomes X * 12 = 3 * 8 = 24. Solving for X gives X = 24 / 12 = 2. Thus, the new value of X is 2, which corresponds to option A.

Given a:b = 3:5, let a=3x and b=5x. After adding 2, (3x+2)/(5x+2) = 2/3. Cross-multiplying gives 9x + 6 = 10x + 4, so x = 2. Thus, b = 5x = 10.

To find M:P, first express all ratios with a common term. Given M:N = 3:4, N:O = 5:7, and O:P = 7:9, we align N and O terms: N is 4 in first ratio and 5 in second, so multiply first ratio by 5 and second by 4 to get M:N = 15:20 and N:O = 20:28. Similarly, O:P = 7:9, multiply by 4 to get O:P = 28:36. Now, M:P = 15:36, which simplifies to 5:12. Hence, M:P = 5/12.

Let the amount received by a singer be x. Then, 3 dancers get as much as 2 singers, so 3 dancers = 2 singers = 2x, thus 1 dancer = 2x/3. Similarly, 3 comperes get as much as 2 dancers, so 3 comperes = 2 dancers = 2 * (2x/3) = 4x/3, thus 1 compere = (4x/3)/3 = 4x/9. Total amount = 4 singers * x + 5 dancers * (2x/3) + 3 comperes * (4x/9) = 4x + (10x/3) + (12x/9) = 4x + 3.333x + 1.333x = 8.666x = Rs.130000. So, x = 130000 / 8.666 = Rs.15000.

If a 3/4 full tank allows the car to travel 360 km, then a full tank would allow it to travel (360 km) ÷ (3/4) = 480 km. Therefore, with a 2/3 full tank, the car can travel 480 km × (2/3) = 320 km. However, since 320 km is option B, but the question asks how many km with 2/3 full tank, the correct answer is 320 km (option B). The original correct_answer was B, which matches this calculation, so the original answer is correct. The explanation was missing and has been added. Difficulty is set to easy based on the straightforward proportional calculation.

Let the two numbers be 5x and 9x. After decreasing each by 5, the ratio becomes (5x - 5) : (9x - 5) = 5 : 11. Solving the equation (5x - 5)/(9x - 5) = 5/11 gives x = 7. Therefore, the numbers are 5*7 = 35 and 9*7 = 63. However, none of the options match these numbers, so let's check the options by substituting back into the ratio condition. Option B (21, 37) gives (21 - 5) : (37 - 5) = 16 : 32 = 1 : 2, not 5 : 11. Option D (15, 27) gives (15 - 5) : (27 - 5) = 10 : 22 = 5 : 11, which matches the second ratio. But the original ratio 15 : 27 simplifies to 5 : 9, which matches the first ratio. Therefore, the correct answer is D (15, 27).

Using the allegation method, the difference between the cost of the second type and the mixture is 19 - 14.2 = 4.8, and the difference between the mixture and the first type is 14.2 - 13 = 1.2. The ratio of the quantities mixed is therefore 4.8 : 1.2, which simplifies to 4 : 1. This means the two types of rice are mixed in the ratio 4:1 (first type : second type) to get the mixture costing Rs. 14.2 per kg. This matches option B.

The sides of the triangle are in the ratio 4:3:2. Let the sides be 4x, 3x, and 2x. The perimeter is the sum of the sides: 4x + 3x + 2x = 9x. Given the perimeter is 81 cm, we have 9x = 81, so x = 9 cm. The largest side is 4x = 4 * 9 = 36 cm. Therefore, the correct answer is 36 cm, which corresponds to option C.

Given X:Y = 2:3 and Y:Z = 4:5, we first equalize the Y terms by finding the least common multiple of 3 and 4, which is 12. Scaling X:Y to 8:12 and Y:Z to 12:15, we combine to get X:Y:Z = 8:12:15.

Since the total costs are the same, the area ratio is inversely proportional to the cost per sq. ft. Therefore, area ratio = cost per sq. ft of commercial space : cost per sq. ft of flat = 9500 : 4500 = 19 : 9. So, the ratio of their areas is 9:19 (flat : commercial space).

Let the number of 1 rupee, 2 rupee, and 5 rupee coins be 3x, 5x, and 4x respectively. The total amount is calculated as 1*3x + 2*5x + 5*4x = 3x + 10x + 20x = 33x. Given the total amount is Rs. 264, we have 33x = 264, which gives x = 8. Therefore, the number of 1 rupee coins is 3x = 3*8 = 24. Hence, the correct answer is option C (24).

The mean proportion between two numbers a and b is given by the square root of their product, √(a×b). Here, √(5×45) = √225 = 15, which corresponds to option B.

Let the distance travelled by car be x km and by train be y km. Given y = x + 150 and x + y = 750. Substituting y, we get x + (x + 150) = 750, so 2x = 600, x = 300. Then y = 450. The ratio x:y = 300:450 = 2:3. However, the question asks for x:y, which is car to train, so the ratio is 2:3. The correct answer is A, not B.