Entrance Exam Maths 001

Explanation
Continued proportion mentioned in the question means that remainders will be in the same proportion as the original number are.

Explanation
The sum of the digits of a 2-digit number is 11, so the possible numbers are 29, 38, 47, 56, and 65. If we add 45 to each of these numbers, the new number obtained will be a number formed by interchanging the digits. The only number that satisfies this condition is 38.

Explanation
The first number is twice the second and thrice the third. Let's assume the second number is x. Therefore, the first number is 2x and the third number is (2x/3). The average of the three numbers is 517, so we can write the equation (2x + x + 2x/3) / 3 = 517. Solving this equation, we find x = 309. Therefore, the first number is 2x = 618 and the third number is (2x/3) = 206. The difference between the first and third number is 618 - 206 = 412, which is not among the given options. Therefore, the correct answer is 564.

Explanation
When arranging fractions in ascending order, we compare their values. In this case, the fractions are: 1/2, 2/3, 5/9, 6/13, and 7/9. To determine which fraction is the fourth when arranged in ascending order, we compare their values. The fraction 2/3 is greater than 1/2, 5/9, and 6/13, but smaller than 7/9. Therefore, the fourth fraction when arranged in ascending order is 2/3.

Explanation
The given question requires finding the sum of numbers from 5 to 105. This can be solved using the arithmetic series formula, which states that the sum of an arithmetic series is equal to the average of the first and last term, multiplied by the number of terms. In this case, the first term is 5, the last term is 105, and the number of terms is (105-5)/1 + 1 = 101. Therefore, the sum is (5+105)/2 * 101 = 5555.

Explanation
To find the least number that leaves a remainder of 6 when divided by 12, 27, and 35, we need to find the least common multiple (LCM) of these three numbers. The LCM of 12, 27, and 35 is 3780. Adding 6 to this LCM gives us the least number that satisfies the given conditions, which is 3786. Therefore, the correct answer is 3786.

Explanation
The number 1200 is the smallest 4-digit number that can be divided by 12, 15, 25, and 30 without leaving any remainder. This is because 1200 is divisible by each of these numbers individually.

Explanation
The number must be a multiple of 2, 3, 4, and 5 plus 3. The least common multiple of 2, 3, 4, and 5 is 60. Adding 3 to 60 gives us 63. When 63 is divided by 9, it leaves no remainder. Therefore, 63 is the least number that satisfies the given conditions.

Explanation
To find the maximum number of pieces that can be cut, we need to find the greatest common divisor (GCD) of the lengths of the rods. The GCD of 78, 104, 117, and 169 is 13. Therefore, each part can be cut into lengths of 13 cm. To find the maximum number of pieces, we divide the length of each rod by 13 and sum them up: 78/13 + 104/13 + 117/13 + 169/13 = 6 + 8 + 9 + 13 = 36. Hence, the maximum number of pieces that can be cut is 36.

Explanation
The greatest number that can divide both 564 and 467 and leave a remainder of 4 and 7 respectively is 20. This means that when 564 is divided by 20, the remainder is 4, and when 467 is divided by 20, the remainder is 7.

Explanation
The four devices beep at different intervals: 30 minutes, 1 hour, 1 hour 30 minutes, and 1 hour 45 minutes. To find when they will all beep together again, we need to find the least common multiple (LCM) of these intervals. The LCM of 30 minutes, 1 hour, 1 hour 30 minutes, and 1 hour 45 minutes is 6 hours, or 360 minutes. Counting from 12 noon, the devices will beep together again at 9:00 AM.

Explanation
Based on the information given, we know that 60% of the employees are women. Since the total number of employees who earn more than 20000 per month is 60% of the total employees, we can infer that 60% of the employees earn more than 20000 per month. Therefore, the remaining employees, which are the men, must earn less than 20000 per month. Since 60% of the employees earn more than 20000 per month, it means that 100% - 60% = 40% of the employees earn less than 20000 per month. Since the question asks for the fraction of men who earn less than 20000 per month, and we know that the men make up 40% of the employees, the fraction of men who earn less than 20000 per month is 40% = 5/8.

Explanation
The question provides information about the age distribution and the percentage of women who play basketball. We are given that 40% of women are above 30 years of age and 80% of women are less than or equal to 50 years of age. Additionally, 20% of all women play basketball. We are also given that 30% of women above the age of 50 play basketball. From this information, we can deduce that the remaining 70% of players must be less than or equal to 50 years of age. Therefore, the percentage of players who are less than or equal to 50 years is 70%.

Explanation
The vendor sells 50% of the apples he had, which leaves him with 50% of the original quantity. He then throws away 20% of the remaining apples, which is equivalent to 20% of 50%, or 10% of the original quantity. This means that after the first day, the vendor has 40% of the original quantity left. On the next day, he sells 60% of the remaining apples, which is equivalent to 60% of 40%, or 24% of the original quantity. The vendor then throws away the rest, which is 100% - 24% = 76% of the original quantity. Therefore, the vendor throws away a total of 10% + 76% = 86% of the original quantity. Since this is not one of the given options, the closest option is 27%, which is the answer.

Explanation
The initial ratio of milk to water is 1:4, which means there is 1 part milk and 4 parts water in the mixture. To have a resulting mixture with 15% milk, the milk should be 15% of the total quantity. However, it is not possible to achieve this because even if we add 100% milk to the mixture, the resulting mixture will still have less than 15% milk. Therefore, it is not possible to achieve a resulting mixture with 15% milk.

Explanation
Rahul saves the remaining amount after spending a certain percentage of his monthly income on household items, buying clothes, and medicines. Since the remaining amount is given as Rs. 11250, we can calculate the total percentage spent by adding the percentages mentioned in the question (50% + 20% + 5% = 75%). The remaining percentage, which represents the amount saved, is 100% - 75% = 25%. We can calculate Rahul's monthly income by dividing the amount saved (Rs. 11250) by the percentage saved (25%) and multiplying by 100, which gives us Rs. 45000. Therefore, the correct answer is 45000.

Explanation
The cost price of the mangoes purchased at the rate of 8 mangoes/Rs is 1/8 Rs per mango, and the cost price of the mangoes purchased at the rate of 6 mangoes/Rs is 1/6 Rs per mango. Since the investment is equal in both cases, let's assume that the investment is Rs 48 (LCM of 8 and 6). Therefore, in the first case, 48/8 = 6 mangoes are purchased, and in the second case, 48/6 = 8 mangoes are purchased. The total cost price in both cases is 48 Rs. Now, when the whole quantity is sold at the rate of 3.5 mangoes/Rs, the selling price is 48/3.5 = 13.71 Rs. The profit is the difference between the selling price and the cost price, which is 13.71 - 48 = 35.71 Rs. The profit percentage is (35.71/48) * 100 = 74.4%. Hence, the net percentage profit is approximately 74.4%.

Explanation
To find the least price for which he must sell the table to avoid any loss on the two together, we need to calculate the cost price of the chair and table.

Let's assume the cost price of the chair is x.

According to the given information, the selling price of the chair is Rs. 45, which is 25% gain.

So, we can write the equation:
x + 25% of x = 45

Solving this equation, we get x = 36.

Now, let's assume the cost price of the table is y.

According to the given information, the selling price of the table is Rs. 57.5, which is 15% gain.

So, we can write the equation:
y + 15% of y = 57.5

Solving this equation, we get y = 50.

To avoid any loss on the two together, the total selling price should be equal to the total cost price.

So, the least price for which he must sell the table is Rs. 43.

Explanation
Let the cost price of the pen be x. According to the given information, selling the pen at a profit of 60% gives a profit of 0.6x.

We are also given that this profit is Rs 38 more than one third of its cost price, which is (1/3)x.

So, we can write the equation: 0.6x = (1/3)x + 38

Simplifying the equation, we get: 0.3x = 38

Dividing both sides by 0.3, we find: x = 38/0.3 = 126.67

Since the cost price of the pen cannot be in decimal form, we round it to the nearest whole number, which is 127. Therefore, the cost price of the pen is 127.

Note: The given answer of 30 is incorrect.

Explanation
When the student weighing 45 kgs left the class, the total weight of the remaining 59 students increased by 200g. This means that the weight of the student who left the class was less than the average weight of the remaining students. In order for the average weight to increase, the weight of the student who left must be less than the average weight. Therefore, the average weight of the remaining 59 students must be greater than 45 kgs, and the only option that satisfies this condition is 57 kgs.

Explanation
If the average weight of the class of 24 students is 35 kg, then the total weight of all the students combined is 24 * 35 = 840 kg. When the weight of the teacher is included, the average increases by 400 g, which is equal to 0.4 kg. Therefore, the total weight of the class with the teacher is 840 + 0.4 = 840.4 kg. Since the weight of the teacher is the difference between the two totals, the weight of the teacher is 840.4 - 840 = 0.4 kg, which is equal to 400 g, or 45 kg.

Explanation
The correct answer is 2X=Y+Z. This can be deduced by considering the total sum of the six numbers. Since the average of the six numbers is X, the total sum is 6X. The average of three of these numbers is Y, so their total sum is 3Y. The average of the remaining three numbers is Z, so their total sum is 3Z. Therefore, the total sum of all six numbers can be expressed as 3Y + 3Z. Since the total sum is also 6X, we can equate the two expressions and simplify to get 2X = Y + Z.

Explanation
The batsman's average increased by 3 after scoring 87 runs in the 17th match. This means that his average before the 17th match was 3 less than his current average. Therefore, his average before the 17th match was 39 - 3 = 36.

Explanation
Train Y starts 2.5 hours after Train X. In that time, Train X would have traveled a distance of 2.5A km. The relative speed between the two trains is (B-A) km/h. To overtake Train X, Train Y needs to cover the distance of 2.5A km, which will take (2.5A) / (B-A) hours. Simplifying the expression gives us 5A / (2(B-A)) hours. Therefore, Train Y will take 5A / (2(B-A)) hours to overtake Train X.

Explanation
If A and B meet after 6 hours when they both start walking towards each other, it means that their combined speed is equal to the total distance divided by the time taken, which is 60 km / 6 hours = 10 km/h.

Now, if A travels with 2/3rd of his speed and B travels with double his speed, their combined speed would be (2/3) * A + 2B. We can set up the equation (2/3) * A + 2B = 10 km/h.

Since we know that A + B = 10 km/h, we can substitute B = 10 - A into the equation to get (2/3) * A + 2(10 - A) = 10 km/h. Simplifying this equation gives us A = 6 km/h.

Explanation
The truck covers a distance of 376 km in 8 hours, which means its speed is 376/8 = 47 kmph. The car's speed is 18 kmph more than the truck's speed, so the car's speed is 47 + 18 = 65 kmph. The distance covered by the car is 14 km more than the truck's distance, so the car's distance is 376 + 14 = 390 km. To find the time taken by the car, we divide the distance by the speed: 390/65 = 6 hours. Therefore, the car would take 6 hours to cover the given distance.

Explanation
The two trains are traveling towards each other, so their relative speed is the sum of their individual speeds. The relative speed is 100 km/h + 136 km/h = 236 km/h. The trains started at different times, so the second train has been traveling for 45 minutes longer than the first train. In 45 minutes, the first train travels (100 km/h * 45/60 h) = 75 km. So, when the second train starts, the first train is already 75 km ahead. The time it takes for the trains to meet is the distance between them divided by their relative speed: 75 km / 236 km/h = 0.318 h. In this time, the second train travels (136 km/h * 0.318 h) = 43.248 km. Adding this to the 75 km, the two trains will be together approximately 75 km + 43.248 km = 118.248 km from Delhi. However, this is the distance traveled by the second train from its starting point. To find the distance from Delhi, we subtract this from the total distance traveled by the second train: 136 km/h * 45/60 h = 102 km. So, the two trains will be together approximately 102 km + 118.248 km = 220.248 km from Delhi. However, this is not one of the given answer options. Therefore, the question is either incomplete or not readable, and it is not possible to determine the correct answer.

Explanation
The speed of the boat in still water is 24 km/hr and the speed of the stream is 8 km/hr. It takes a total of 20 hours to row upstream from point X to Point Y and downstream from Point Y to Point Z. Since the distance from X to Y is one third of the distance between Y and Z, we can assume the distance from X to Y as 1x and the distance from Y to Z as 3x.

To find the time taken to row upstream, we use the formula: Time = Distance / Speed.
So, the time taken to row upstream is (1x / (24-8)) = (1x / 16) hours.

To find the time taken to row downstream, we use the same formula: Time = Distance / Speed.
So, the time taken to row downstream is (3x / (24+8)) = (3x / 32) hours.

The total time taken to row upstream and downstream is 20 hours.
So, (1x / 16) + (3x / 32) = 20.

Simplifying the equation, we get: (2x + 3x) / 32 = 20.
Solving for x, we get x = 16.

Therefore, the total distance traveled by the boat is (1x + 3x) = 4x = 4 * 16 = 64 km.
Since the boat travels both upstream and downstream, the total distance traveled is 64 km * 2 = 128 km.

Hence, the correct answer is 512 km.

Explanation
Aditya will have a larger amount by 290. This can be explained by calculating the interest earned by each person. Aditya invested 10000 in scheme A, which offers simple interest. The interest earned can be calculated using the formula: Interest = Principal * Rate * Time. In this case, the interest earned by Aditya would be 10000 * 12% * 3 = 3600. Therefore, the total amount Aditya will have after 3 years is 10000 + 3600 = 13600. On the other hand, Bhushan invested 10000 in scheme B, which offers compound interest. The compound interest formula is: Amount = Principal * (1 + Rate/100)^Time. Using this formula, Bhushan will have 10000 * (1 + 10/100)^3 = 13310. Therefore, Aditya will have a larger amount by 13600 - 13310 = 290.

Explanation
The man divided his savings into three parts: one for his wife, and one for each of his sons. He divided it in such a way that each son would receive the same amount at 5% compound interest per annum when they turn 18. Since the amount will accumulate interest over the years, the share of each son would be greater than their initial share. The share of B, who is 12 years old, would be less than the share of A, who is 14 years old, because B has less time for the amount to accumulate interest. Therefore, the correct answer is Rs. 16000.

Explanation
Vijay can complete the work in 24 days, Rakesh can complete it in 30 days, and Vinod can complete it in 40 days. Vijay and Vinod worked together for 4 days, which means they completed 4/24 + 4/40 = 1/6 of the work. They then handed it to Rakesh who worked for some days and handed it back to Vijay and Vinod 6 days before completing the work. This means that Rakesh worked for 24 - 6 = 18 days. Since Vijay and Vinod worked for 4 days initially, Rakesh worked for a total of 18 - 4 = 14 days. However, since Vijay and Vinod took over the work again and completed it, Rakesh only worked for 14 - 4 = 10 days. Therefore, the correct answer is 10 days.

Explanation
Arnab is twice as fast as Bhanu, and Bhanu is one-third as fast as Chandu. This means that Arnab is twice as fast as Bhanu, and Bhanu is three times slower than Chandu. Let's assume that Chandu's speed is x. Therefore, Bhanu's speed is x/3 and Arnab's speed is 2(x/3) = 2x/3.

If they work together, their combined speed is x + x/3 + 2x/3 = 2x.

Since they can complete the work in 30 days, their combined speed is 1/30 of the work per day.

2x = 1/30

x = 1/60

Therefore, Chandu can complete the work alone in 60 days.

Bhanu's speed is x/3 = (1/60)/3 = 1/180 of the work per day.

Therefore, Bhanu can complete the work alone in 180 days.

Arnab's speed is 2x/3 = (2/60)/3 = 1/90 of the work per day.

Therefore, Arnab can complete the work alone in 90 days.

Hence, the correct answer is 90, 180, 60.

Explanation
Let's assume A's marks in Science to be x. Since A got twice as many marks in English as in Science, his marks in English would be 2x. The ratio of his marks in English and Mathematics is given as 2:3. Let's assume his marks in Mathematics to be y. Therefore, the equation can be written as 2x + y + x = 180. Simplifying this equation, we get 3x + y = 180. Since the ratio of English and Mathematics marks is 2:3, we can write the equation as 2/3y + y = 180. Solving this equation, we find y = 90. Substituting this value in the equation 3x + y = 180, we can find x = 30. Therefore, A's marks in Science is 30.

Explanation
Let's assume the number of Hindi books is x.

According to the given ratio, we can set up the equation:

7/2 = 1512/x

Cross-multiplying, we get:

7x = 3024

Simplifying, x = 432

After the addition of English books, the new ratio becomes 15:4.

So, the new number of English books can be calculated using the equation:

15/4 = (1512 + added books)/432

Cross-multiplying, we get:

15 * 432 = 4 * (1512 + added books)

6480 = 6048 + 4 * added books

432 = 4 * added books

added books = 108

Therefore, the number of English books added is 108.

Explanation
In the given scenario, the amount of Rs. 117 was divided among A, B, and C in the ratio of 2:3:4 instead of the intended ratio of 1/2, 1/3, 1/4. To determine who gains the most, we need to compare the actual amount received by each person with the intended amount.

Using the intended ratio, A should have received (1/2) * 117 = Rs. 58.5, B should have received (1/3) * 117 = Rs. 39, and C should have received (1/4) * 117 = Rs. 29.25.

However, in the mistaken division, A received Rs. 2, B received Rs. 3, and C received Rs. 4.

Comparing these amounts, we can see that C gained the most, as he received Rs. 4 instead of the intended Rs. 29.25. Therefore, C gains Rs. 4 - Rs. 29.25 = Rs. 25.

Hence, the correct answer is C, Rs. 25.

Explanation
The perimeter of a square is equal to 4 times the length of its side. If the perimeter of the square is twice the perimeter of a rectangle with length 10 cm and breadth 4 cm, then the perimeter of the square is 2 * (2 * 10 + 2 * 4) = 2 * (20 + 8) = 2 * 28 = 56 cm. The side of the square is therefore 56 / 4 = 14 cm. The diameter of the semi-circle is equal to the side of the square, so it is also 14 cm. The circumference of a semi-circle is equal to half the circumference of a full circle, so the circumference of the semi-circle is 1/2 * π * 14 = 22/7 * 14 / 2 = 22 cm. Therefore, the correct answer is 36 cm.

Explanation
The perimeter of a square is equal to four times the length of one side. If the perimeter of the square is twice the perimeter of the rectangle, then the length of one side of the square must be equal to twice the sum of the length and breadth of the rectangle. In this case, the length of the rectangle is 10 cm and the breadth is 4 cm, so the length of one side of the square is 2 * (10 + 4) = 28 cm. The diameter of the semi-circle is equal to the side of the square, so it is also 28 cm. The circumference of a semi-circle is equal to half the circumference of a full circle, so the circumference of the semi-circle is 0.5 * π * 28 = 14π cm. Since the question asks for the answer in cm^2, the answer is 14π cm^2, which is approximately equal to 189 cm^2.

Explanation
Let's assume the quantity of alcohol in the original mixture is 6x liters and the quantity of water is 5x liters. After replacing 22 liters of the mixture with water, the quantity of alcohol remains the same (6x liters) but the quantity of water becomes (5x + 22) liters.

Given that the new ratio is 9:13, we can set up the equation:
6x / (5x + 22) = 9/13

Solving this equation, we find that x = 26. Therefore, the quantity of water after replacement is 5x + 22 = 5(26) + 22 = 130 + 22 = 152 liters.

However, since we replaced 22 liters of the mixture with water, the actual quantity of water after replacement is 152 - 22 = 130 liters.

Therefore, the correct answer is 52 liters.

Explanation
The cost of Type 1 rice is Rs. 30 per kg and Type 2 rice is Rs. 40 per kg. Since the two types of rice are mixed in the ratio of 1:4, the total cost of the mixture can be calculated as follows: (1/5 * 30) + (4/5 * 40) = 6 + 32 = 38. Therefore, the price per kg of the mixed variety of rice is Rs. 38.