Question

This problem is from the Treviso Arithmetic, the first printed arithmetic text, dated 1478: The Holy Father sent a courier from Rome to Venice, commanding him that he should reach Venice in 7 days. And the most illustrious Signoria of Venice also sent another courier to Rome, who should reach Rome in 9 days. And from Rome to Venice is 250 miles. It happened that by order of these lords the couriers started their journeys at the same time. It is required to find in how many days they will meet, and how many miles each will have traveled \({ }^{37}\)

Short answer

Answer: The two couriers will meet after approximately 3.94 days. By that time, the courier sent by the Holy Father would have traveled approximately 140.79 miles, and the courier sent by the Signoria of Venice would have traveled approximately 109.21 miles.

Step-by-step solution

Determine the speed of each courier

First, we need to calculate how fast each courier is traveling. We know that the courier sent by the Holy Father should travel 250 miles in 7 days, so we can find his speed with the formula: Speed = Distance / Time Speed = 250 / 7 He travels approximately 35.71 miles per day. Similarly, the courier sent by the Signoria of Venice should travel 250 miles in 9 days. We can find his speed by substituting the appropriate values into the same formula: Speed = 250 / 9 He travels approximately 27.78 miles per day.

Find the time they meet

Since both couriers start at the same time, we can use relative speed to determine when they meet. This is the combined speed of both couriers. The relative speed is the sum of their individual speeds: Relative Speed = 35.71 + 27.78 = 63.49 miles per day Now, we should find how many days it will take for them to travel the total distance of 250 miles with this combined speed. We can use the same formula from Step 1: Time = Distance / Speed Time = 250 / 63.49 The couriers will meet after approximately 3.94 days.

Calculate the distance each courier travels by the time they meet

To find the distance each courier has traveled in the 3.94 days, we can use the formula: Distance = Speed * Time For the courier from Rome: Distance_Rome = 35.71 * 3.94 ≈ 140.79 miles For the courier from Venice: Distance_Venice = 27.78 * 3.94 ≈ 109.21 miles So by the time they meet, the courier sent by the Holy Father would have traveled approximately 140.79 miles, and the courier sent by the Signoria of Venice would have traveled approximately 109.21 miles.

Key concepts

Relative Speed

When two objects are moving towards each other, their relative speed is the sum of their individual speeds. In the context of the couriers, imagine two cars driving towards each other on a straight road. If one car travels at 50 miles per hour and another at 30 miles per hour, their relative speed is 80 miles per hour. This is because the distance between them reduces by 80 miles every hour.

In our problem, the courier from Rome travels at approximately 35.71 miles per day, while the courier from Venice travels at about 27.78 miles per day. So their relative speed when moving towards one another is:

• Relative Speed = 35.71 + 27.78 = 63.49 miles per day

Using their relative speed, we can determine how quickly they close the gap between each other until they meet.

Distance Calculation

Understanding how to calculate distance is crucial in everyday problems like this one. The concept revolves around the basic formula of distance, which is the product of speed and time:

• Distance = Speed × Time

If we know how fast someone is traveling and how long they've been traveling, we can easily find out how far they've gone. For instance, in this problem, after we calculated their meeting time, we used each courier's speed to determine how far each traveled to get there.

For example, the courier from Rome:

• His speed = 35.71 miles per day
• Time traveled = 3.94 days
• Distance traveled = 35.71 × 3.94 ≈ 140.79 miles

Knowing how far both couriers have gone helps verify their meeting point along the route.

Time Calculation

Time calculation is a key part in solving problems where multiple moving elements are involved. In this exercise, we wanted to know how long it would take two couriers to meet halfway. The formula to find time when you know distance and speed is:

• Time = Distance ÷ Speed

We had the total distance between Rome and Venice, which was 250 miles. By combining their speeds to find relative speed, we calculated how long it would take them to meet:

• Relative speed = 63.49 miles per day
• Time to meet = 250 miles ÷ 63.49 ≈ 3.94 days

This calculated time confirmed that in just under 4 days, the couriers would be approximately at halfway, meeting each other in their respective journeys.