Problem 82 - Algebra 1

Question

Evaluate the expression. If you ride a bicycle 5 miles per hour, how many miles will you ride in 45 minutes?

Step-by-Step Solution

Answer

If you ride a bicycle at a speed of 5 miles per hour, you will cover a distance of 3.75 miles in 45 minutes.

Step 1: Convert Minutes into Hours.

In order to make the units consistent with the rate, convert 45 minutes into hours. Since 1 hour is equivalent to 60 minutes, divide 45 minutes by 60 to get the equivalent hour which is equal to \( \frac{45}{60} = 0.75 \) hours.

Step 2: Multiply the rate by time.

Find the total distance travelled by multiplying the rate by the time. Use the formula for speed which is distance = speed * time. Substituting the given values, you get: \( distance = 5 miles/hour * 0.75 hours \)

Step 3: Calculation of the Distance

Now calculate the total distance by multiplying 5 by 0.75. The result is 3.75 miles.

Key Concepts

Unit Conversion

Understanding how to convert units is crucial when working with expressions that involve measurements. In our example, the bicyclist's rate is given in miles per hour, but the time is given in minutes. To accurately evaluate the expression, we need these units to match.

Why is this necessary? Because our formula for distance relies on consistent units to make sense. If you multiply 'miles per hour' by 'hours', the units 'hours' will cancel out, leaving you with 'miles', which is the unit for distance we’re looking for. However, if the time was left in minutes, our answer would have been in 'mile minutes', which doesn't make sense for distance.

To convert minutes to hours, you simply divide the number of minutes by 60, since there are 60 minutes in an hour. This method can be applied across different unit conversions, always referring to the standard values: 60 minutes in an hour, 24 hours in a day, 100 centimeters in a meter, and so on.

Rate and Time Multiplication

The formula for calculating distance based on rate and time is one of the most fundamental concepts in physics and everyday life: 'Distance' equals 'Rate' times 'Time'. What this means is simple: how far you travel depends on how fast you're going (rate) and how long you're moving (time).

To apply this, you need to know two of the three variables. In this case, we're given the rate (5 miles per hour) and the time (after conversion, 0.75 hours). When we multiply these two, we can find the third variable: the distance traveled.

This formula works consistently across various contexts whether you are calculating the distance of a car trip, the stretch a plane covers, or even the distance a light beam travels in a certain time at the speed of light. Remembering that 'rate' is a measure of 'distance per unit of time' helps visualize why the multiplication of rate by time gives us a measure of distance.

Distance Calculation

Calculating distance is straightforward once units are consistent and you have the rate and time. Applying the distance formula to our exercise, we multiply the bicyclist’s speed, 5 miles per hour, by the time spent riding, 0.75 hours. The multiplication of these values gives us the distance.

It’s worth noting that understanding the context of the problem can also influence how you calculate distance. For example, traveling in a straight line versus following a winding path can result in different distances, even with the same rate and time. However, for our problem's purposes, we're assuming a constant rate and a direct route.

Calculation of the distance is essential in many fields beyond just exercise problems. It plays a critical role in logistics, travel planning, and in science to study the motion of objects. So, mastering distance calculations can be quite valuable.