Advanced Math Quiz: Mid Term Assessment Test

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
Let's solve the problem step by step to determine the maximum marks of the examination.
Step 1: Define the Variables
Let the maximum marks of the examination be MMM.
Step 2: Calculate Marks Secured Before and After Revaluation
Mark secured 40% of the total marks initially.
Initial Marks=0.40M\text{Initial Marks} = 0.40MInitial Marks=0.40M
After revaluation, his marks increased by 50%.
Marks after Revaluation=0.40M+0.50×0.40M=0.40M+0.20M=0.60M
Step 3: Condition After Revaluation
After revaluation, he failed by 35 marks, meaning the passing marks are 0.60M+35.
Passing Marks=0.60M+35
Step 4: Condition on Further Increase
His post-revaluation score has increased by 20%.
Increased Marks=0.60M+0.20×0.60M=0.60M+0.12M=0.72M
After this increase, he has 7 marks more than the passing marks:
0.72M=Passing Marks+7=(0.60M+35)+7
0.72M=0.60M+42
Step 5: Solve for M
Subtract 0.60M from both sides:
0.72M−0.60M=42
Now, solve for M:
M=42/0.12=350M
Conclusion
The maximum mark of the examination is 350.

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
Let's solve the problem step by step to find the new average.
Step 1: Calculate the Original Sum The average of 80 numbers is 60. Therefore, the original sum of the numbers is:
Original Sum = 80 × 60 = 4800
Step 2: Correct the Mistake The three numbers 12, 15, and 17 were mistakenly calculated as 65, 70, and 79. Let's calculate the incorrect total of these three numbers:
Incorrect Sum = 65 + 70 + 79 = 214
Now, let's calculate the correct sum of these three numbers:
Correct Sum = 12 + 15 + 17 = 44
Step 3: Calculate the Corrected Sum To find the corrected sum, subtract the incorrect sum and add the correct sum:
Corrected Sum = 4800 - 214 + 44 = 4800 - 170 = 4630
Step 4: Calculate the New Average Now, calculate the new average with the corrected sum:
New Average = 4630 / 80 = 57.875
Conclusion The new average is 57.875.

Explanation
You need to determine how much glycerine, costing $60 per litre, should be mixed with 20 litres of isopropyl alcohol, costing $120 per litre, so that the sanitizer can be sold at $110 per litre after taking a profit of $10 per litre.
Step 1: Calculate the Selling Price per Litre Before Profit The selling price per litre, after taking a profit of $10, would be:
Selling Price = $110 - $10 = $100 per litre
Step 2: Set Up the Equation for the Cost Let the amount of glycerine to be mixed be x litres.
The cost of isopropyl alcohol = $120 × 20 = $2400
The cost of glycerine = $60 × x
The total cost for the mixture = $2400 + $60x
Since the final mixture is sold at $100 per litre and the total amount of the mixture will be 20 + x litres:
(2400 + 60x) / (20 + x) = 100
Step 3: Solve the Equation Multiply both sides by 20 + x to eliminate the fraction:
2400 + 60x = 100(20 + x)
Expand and simplify:
2400 + 60x = 2000 + 100x
Subtract 60x from both sides:
2400 = 2000 + 40x
Subtract 2000 from both sides:
400 = 40x
Divide by 40:
x = 10
Conclusion You need to mix 10 litres of glycerine costing $60 per litre with 20 litres of isopropyl alcohol costing $120 per litre to produce a sanitizer that can be sold at $110 per litre after making a profit of $10 per litre.

Explanation
Understanding the Given Information:
ABCD is a rectangle.
ABE is an equilateral triangle with a side AB of length 12 cm.
E is the midpoint of CD, meaning DE = EC.
Five parallel lines divide the rectangle ABCD into six regions of equal area.
Calculating the Height of the Equilateral Triangle ABE:
Since ABE is an equilateral triangle with side length 12 cm, its height can be calculated using the formula for the height of an equilateral triangle:
Height of ABE = (√3 / 2) * 12 = 6 * √3 cm.
Determine the Height of Rectangle ABCD:
Let the height of the rectangle ABCD be "h". Since E is the midpoint of CD, DE = EC = 6 cm.
Area of the Rectangle ABCD:
The area of rectangle ABCD is:
Area of ABCD = Length * Height = 12 * h.
The rectangle is divided into six strips of equal area, so each strip has an area of:
Area of each strip = (12 * h) / 6 = 2h.
Calculate the Shaded Area:
The shaded strips alternate with the non-shaded strips. Since there are six strips and three are shaded:
Shaded Area = 3 * Area of each strip = 3 * 2h = 6h.
Determine the Height "h" of Rectangle ABCD:
The height of rectangle ABCD is equal to the sum of the height of the equilateral triangle and the additional portion extending below it. Given that E is at the midpoint of CD, we use the full height of the equilateral triangle for the calculation:
h = 6 * √3 cm.
Substitute the Value of "h":
Now, substitute h = 6 * √3 into the shaded area calculation:
Shaded Area = 6 * 6 * √3 = 36 * √3 square cm.
Conclusion
The area of the shaded region in the given figure is:
36 * √3 square cm.

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 correct pin code is 231361 because:
When you calculate the square root of 231361, you get 481.
Since 481 is a whole number (no decimals), 231361 is a perfect square.
This confirms that the pin code the postmaster remembers is 231361.

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%.

Explanation
Step 1: Express Salaries in Terms of B's Salary
Let B's salary be B.
A's salary is 20% more than B's: A = B + 20% of B = 1.20B
A's salary is also 20% less than C's: A = C - 20% of C = 0.80C Therefore, we can express C's salary in terms of B: C = A / 0.80 = 1.20B / 0.80 = 1.50B
Step 2: Adjust Salaries According to Changes
C's salary is decreased by 4%: New C's Salary = 1.50B - 4% of 1.50B = 1.50B * 0.96 = 1.44B
B's salary is increased by 10%: New B's Salary = B + 10% of B = 1.10B
Step 3: Calculate the Percentage by Which C's Salary Exceeds B's Salary
Now, find the difference in the salaries: Difference in Salaries = 1.44B - 1.10B = 0.34B
The percentage by which C's salary exceeds B's salary: Percentage = (0.34B / 1.10B) * 100 ≈ 30.91%
Conclusion
The percentage by which C's salary would exceed B's salary is nearest to 31%.