The Lilavati and Its Mathematical Problems

To solve the equation, we can use the quadratic formula: x = (-b ± √(b^2 - 4ac)) / 2a. Plugging in the values from the equation, we get x = (-2 ± √(2^2 - 4(1)(-8))) / 2(1) = (-2 ± √36) / 2 = (-2 ± 6) / 2. Therefore, x = 2 or x = -4.

The farmer plants wheat on 120 * 2/5 = 48 acres of land and rice on 120 * 1/3 = 40 acres of land. Therefore, the total area used for farming is 48 + 40 = 88 acres. The remaining 120 - 88 = 32 acres are left fallow.

The first train travels for 1 hour before the second train leaves. Therefore, the second train has to cover a distance of 800 - 60 = 740 miles in order to meet the first train. The relative speed of the two trains is 70 + 60 = 130 miles per hour. Therefore, the two trains will meet in 740 / 130 = 6 hours. Since the second train leaves at 11:00 AM, the two trains will meet at 11:00 AM + 6 hours = 5:00 PM.

The area of a rectangle is given by the formula A = l * w. Plugging in the values from the question, we get A = 20 * 15 = 300 square meters.

Let the cost price of the shirt be x. The profit is 20% of the cost price, which is 0.2 * x. Therefore, the selling price of the shirt is x + 0.2 * x = 1.2 * x. Since the selling price is ₹100, we have 1.2 * x = 100. Solving for x, we get x = 100 / 1.2 = ₹80.

Let the sum of money be P and the rate of interest be r. According to the question, P * (1 + r * 10) = 2P. Dividing both sides by P, we get 1 + r * 10 = 2. Subtracting 1 from both sides, we get r * 10 = 1. Therefore, r = 1 / 10 = 0.1 = 10%.

The speed of the train can be calculated using the formula speed = distance / time. The distance traveled by the train in 10 seconds is the length of the telegraph pole, which is negligible. Therefore, the distance traveled by the train in 20 seconds is 200 meters. The speed of the train is therefore 200 / 20 = 10 meters per second. Converting this to kilometers per hour, we get 10 * 3600 / 1000 = 72 km/h.

Let the number of ₹1 coins be x, the number of ₹2 coins be y, and the number of ₹5 coins be z. We know that x + y + z = 100 and x + 2y + 5z = 170. We also know that x = 2y. Substituting x = 2y into the second equation, we get 2y + 2y + 5z = 170. Simplifying this, we get 4y + 5z = 170. Since x = 2y, we can substitute x = 2y into the first equation to get 2y + y + z = 100. Simplifying this, we get 3y + z = 100. Subtracting the second equation from the first equation, we get z = 30. Therefore, the man has 30 ₹5 coins.

The farmer plants wheat on 100 * 40% = 40 acres of land and rice on 100 * 30% = 30 acres of land. Therefore, the total area used for farming is 40 + 30 = 70 acres. The remaining 100 - 70 = 30 acres are left fallow.

The first train travels for 1 hour before the second train leaves. Therefore, the second train has to cover a distance of 800 - 60 = 740 miles in order to meet the first train. The relative speed of the two trains is 70 + 60 = 130 miles per hour. Therefore, the two trains will meet in 740 / 130 = 6 hours. Since the second train leaves at 11:00 AM, the two trains will meet at 11:00 AM + 6 hours = 5:00 PM.

The area of a rectangle is given by the formula A = l * w. Plugging in the values from the question, we get A = 20 * 15 = 300 square meters.

Let the cost price of the shirt be x. The profit is 20% of the cost price, which is 0.2 * x. Therefore, the selling price of the shirt is x + 0.2 * x = 1.2 * x. Since the selling price is ₹100, we have 1.2 * x = 100. Solving for x, we get x = 100 / 1.2 = ₹80.

Let the sum of money be P and the rate of interest be r. According to the question, P * (1 + r * 10) = 2P. Dividing both sides by P, we get 1 + r * 10 = 2. Subtracting 1 from both sides, we get r * 10 = 1. Therefore, r = 1 / 10 = 0.1 = 10%.

The speed of the train can be calculated using the formula speed = distance / time. The distance traveled by the train in 10 seconds is the length of the telegraph pole, which is negligible. Therefore, the distance traveled by the train in 20 seconds is 200 meters. The speed of the train is therefore 200 / 20 = 10 meters per second. Converting this to kilometers per hour, we get 10 * 3600 / 1000 = 72 km/h.

Let the number of ₹1 coins be x, the number of ₹2 coins be y, and the number of ₹5 coins be z. We know that x + y + z = 100 and x + 2y + 5z = 170. We also know that x = 2y. Substituting x = 2y into the second equation, we get 2y + 2y + 5z = 170. Simplifying this, we get 4y + 5z = 170. Since x = 2y, we can substitute x = 2y into the first equation to get 2y + y + z = 100. Simplifying this, we get 3y + z = 100. Subtracting the second equation from the first equation, we get z = 30. Therefore, the man has 30 ₹5 coins.