A Boat running upstream takes 8 hours 48 minutes to cover a certain distance, While it takes 4 hours to cover the same distance running downstream. What is the ratio between the speed of the boat and speed of the water current respectively?

Let's denote:

• B = Speed of the boat in still water
• S = Speed of the water current

Upstream:

• Speed = B - S
• Time = 8 hours 48 minutes = 8 + 48/60 = 8.8 hours

Downstream:

• Speed = B + S
• Time = 4 hours

Since the distance traveled is the same in both directions, we can set up the following equation:

(B - S) * 8.8 = (B + S) * 4

Expanding and rearranging:

8.8B - 8.8S = 4B + 4S 4.8B = 12.8S B/S = 12.8 / 4.8 B/S = 8:3

Therefore, the ratio between the speed of the boat and the speed of the water current is 8:3. The boat is 8 times faster than the current in still water.

Rates and Ratios

Rates and ratios are mathematical concepts used to compare quantities and express relationships between them. They are often used in various fields, including mathematics, finance, science, and everyday life.

Rates:

A rate is a comparison of two quantities with different units. It is expressed as a ratio of one quantity to another, with specified units.

Formula: Rate = Quantity 1/Quantity 2

Example: If a car travels 150 miles in 3 hours, the rate of speed is 150 miles / 3 hours = 50 miles per hour.

Ratios:

A ratio is a comparison of two quantities by division. It represents the relative size of two quantities and is often expressed as a fraction, a colon, or using the word "to."

Formula: Ratio = Quantity 1 / Quantity 2

Example: If there are 4 red balls and 6 blue balls, the ratio of red balls to blue balls is 4 / 6 or 4:6.

Key Differences:

1. Units:

   • Rates involve quantities with different units.
   • Ratios involve quantities with the same units.

2. Purpose:

   • Rates are used to compare the measurements of different units in terms of one another, such as speed or efficiency.
   • Ratios are used to compare the sizes of two quantities with the same unit.

3. Representation:

   • Rates are typically represented as a fraction with units (e.g., miles per hour).
   • Ratios can be represented as a fraction, a colon, or using the word "to" (e.g., 2:3 or 2 to 3).

4. Application:

   • Rates are commonly used in problems involving speed, prices, and other measurements with different units.
   • Ratios are used in various contexts, such as comparing ingredients in a recipe, mixing substances, or expressing relationships between quantities.

Understanding rates and ratios is fundamental in solving problems that involve comparing quantities or understanding proportions in different contexts.