Aesthetix Mid Term Assessment Test 1

Explanation
Christmas in 2019 was on a Friday. In a non-leap year, there are 365 days, which is equivalent to 52 weeks and 1 day. Since 2020 was a leap year, it had 366 days, which is equivalent to 52 weeks and 2 days. Therefore, 2021 will start on a Friday. Counting the days from January 1st to February 14th, which is Valentine's Day, there will be 45 days. Since 45 divided by 7 leaves a remainder of 3, it means that Valentine's Day in 2021 will be 3 days after Friday, which is Monday.

Explanation
If Arun and Bansan can finish the work together in 12 days, it means that their combined efficiency is 1/12 of the work per day. Let's assume Arun's usual efficiency is x and Bansan's usual efficiency is y.

According to the given information, if Arun works half as efficient (x/2) and Bansan works thrice as efficient (3y), the work would be completed in 9 days.

So, their combined efficiency in this scenario is (x/2 + 3y) and it is equal to 1/9 of the work per day.

Now, we can set up an equation to find the values of x and y.

(x/2 + 3y) = 1/9

Simplifying the equation, we get x + 6y = 2/9

Since we know that their combined efficiency is 1/12, we can also set up another equation:

x + y = 1/12

Solving these two equations simultaneously will give us the values of x and y.

Once we know Arun's usual efficiency (x), we can find out how many days he would take to finish the work alone at his usual efficiency by setting up the equation:

x = 1/t (where t is the number of days)

Simplifying this equation will give us the value of t, which is the answer to the question.

Explanation
The new average can be found by subtracting the sum of the mistakenly calculated numbers (65+70+79) from the sum of the original 80 numbers and then dividing by 80. This gives us (60*80 - (65+70+79))/80 = 78.

Explanation
To find the amount of glycerine needed, we can set up a proportion using the cost of the glycerine and the cost of the isopropyl alcohol. Let x be the amount of glycerine in litres. The cost of the glycerine is Rs. 60 per litre, so the cost of x litres of glycerine would be 60x. The cost of the isopropyl alcohol is Rs. 120 per litre, so the cost of 20 litres of isopropyl alcohol would be 120*20 = 2400. The total cost of the mixture is the sum of the cost of glycerine and the cost of isopropyl alcohol, which is 60x + 2400. The selling price is Rs. 110 per litre, and the profit is Rs. 10 per litre, so the selling price of the mixture is 110*(x+20). Setting up the proportion, we get (60x + 2400)/(x+20) = 110. Solving for x, we find x = 10. Therefore, 10 litres of glycerine should be mixed with 20 litres of isopropyl alcohol.

Explanation
To find the highest power of 21 in 100!, we need to determine the number of times 21 can be divided into 100!. Since 21 = 3 * 7, we need to count the number of times both 3 and 7 appear as factors in the prime factorization of numbers from 1 to 100. We know that the power of 3 in 100! is greater than the power of 7, so we only need to consider the power of 7. We can calculate the power of 7 in 100! by dividing 100 by 7, which gives us 14. Therefore, the highest power of 21 in 100! is 14.

Explanation
The train is crossing the platform, so the total distance covered by the train and the platform is equal to the length of the train. The train is 60m long and it takes 5 seconds to cross the platform. Therefore, the speed of the train is 60m/5s = 12m/s. We can convert the speed from m/s to km/hr by multiplying it by 3.6. So, the speed of the train is 12m/s * 3.6 = 43.2 km/hr. Since the train is running at a speed of 54 km/hr, the length of the platform can be calculated using the formula: length of platform = (speed of train * time taken to cross the platform) - length of train. Plugging in the values, we get: length of platform = (54 km/hr * 5s) - 60m = 270m - 60m = 210m. Therefore, the length of the platform is 210m, which is not one of the given options. Hence, the given answer of 90m is incorrect.

Explanation
The car travelled 140 - 60 = 80km at two-thirds of its original speed.
Let the normal speed of the car be x km/h.
The time taken to cover the first 60km is 60 / x hours.
The time taken to cover the remaining 80km is 80 / (2/3)x = 120 / x hours.
The total time taken for the journey is 60 / x + 120 / x = 180 / x hours.
It arrived 2 hours late, so the total time taken is the normal time + 2 hours.
Therefore, 180 / x = (140 / x) + 2.
Simplifying the equation, we get 180 = 140 + 2x.
Solving for x, we find x = 20.
Hence, the normal speed of the car is 20 km/h.

Explanation
The speed of the boat in still water is given as 18 km/hr, and the rate of the current is 7 km/hr. To find the distance travelled downstream in 36 minutes, we need to convert the time to hours by dividing it by 60 (36/60 = 0.6 hours). The speed of the boat relative to the current is the sum of the speed in still water and the rate of the current (18 + 7 = 25 km/hr). Finally, we can calculate the distance travelled by multiplying the speed relative to the current by the time (25 * 0.6 = 15 km). Therefore, the correct answer is 15 km.

Explanation
The filling rate of pipe alpha is 1/5 of the tank per hour, and the filling rate of pipe beta is 1/10 of the tank per hour. The draining rate of the hole is 1/20 of the tank per hour. When both pipes are opened simultaneously, the combined filling rate is (1/5 + 1/10) = 3/10 of the tank per hour. The draining rate is 1/20 of the tank per hour. Therefore, the net filling rate is (3/10 - 1/20) = 5/20 = 1/4 of the tank per hour. It will take 4 hours to fill the tank at this rate.

Explanation
Let's assume the number is "x". According to the given information, the difference between "x" and its three-seventh is 36. So, we can write the equation as x - (3/7)x = 36. Simplifying this equation, we get (4/7)x = 36. Multiplying both sides by 7/4, we find x = 63. Reversing the digits of 63 gives us the number 36.

Explanation
If Saran drove for at least 6 hours and both Saran and John reached their common destination at the same point of time, it means that John had less time to cover the same distance. Therefore, in order for John's speed to exceed Saran's by the highest percentage, John would need to cover the distance in the shortest amount of time possible. This would result in John driving for only 1 hour. Therefore, the highest possible value of the percentage by which John's speed could exceed Saran's is 20%.