Calculate Jonathan's Average Speed: Varying Speeds Over 3-Hour Cycling Competition

Q: Why can't I just average all the speeds together?

Because Jonathan spent different amounts of time at each speed! He rode 2 hours at 24 km/h but only 15 minutes at 32 km/h. The longer times should have more influence on the average.

Q: How do I convert minutes to hours?

Divide minutes by 60! So 15 minutes = 15 ÷ 60 = 0.25 hours and 30 minutes = 30 ÷ 60 = 0.5 hours. Always use the same time units throughout.

Q: Do I need to convert the final answer to a mixed number?

Not necessarily! $ 25.888... $ km/h is perfectly acceptable. You could also write it as $ 25\frac{8}{9} $ km/h, but decimal form is more common for speed problems.

Q: What if I get confused with all the time segments?

Make a table! List each segment with its speed, time, and calculated distance. This helps you stay organized and avoid missing any segments.

Q: How do I check if my total time is correct?

Add up all time intervals: 30 minutes + 2 hours + 15 minutes + 1 hour. Convert to hours: 0.5 + 2 + 0.25 + 1 = 3.75 hours total.

Q: Why is the answer less than most of the individual speeds?

Because Jonathan spent the longest time (2 hours) going at his slowest sustained speed of 24 km/h. This pulls the average down, even though he had brief periods at higher speeds.

Average Speed with Multiple Time Intervals

Jonathan is reviewing his cycling records from his last competition.

During the first half hour, he rode at a speed of 28 km/h.

The following two hours, he rode at a speed of 24 km/h, then 15 minutes downhill at a speed of 32 km/h, before continuing for another hour at a speed of 27 km/h.

What was his average speed?

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Understand the problem

Step-by-step solution

To solve this problem, we'll follow these steps:

Step 1: Calculate the distance for each segment.
Step 2: Sum all distances for the total distance.
Step 3: Sum all time intervals for the total time.
Step 4: Use the average speed formula to find the solution.

Now, let's work through each step:
Step 1: Calculate the distance for each segment.
- First 30 minutes (0.5 hours) at 28 km/h: $\text{Distance} = 28 \times 0.5 = 14$ km.
- Next 2 hours at 24 km/h: $\text{Distance} = 24 \times 2 = 48$ km.
- Next 15 minutes (0.25 hours) at 32 km/h: $\text{Distance} = 32 \times 0.25 = 8$ km.
- Final 1 hour at 27 km/h: $\text{Distance} = 27 \times 1 = 27$ km.

Step 2: Total distance:
$\text{Total Distance} = 14 + 48 + 8 + 27 = 97$ km.

Step 3: Total time:
$\text{Total Time} = 0.5 + 2 + 0.25 + 1 = 3.75$ hours.

Step 4: Calculate the average speed:
$\text{Average Speed} = \frac{97}{3.75} \approx 25.888...$ km/h.

Therefore, the solution to the problem is 25.888....

Final Answer

25.888...

Key Points to Remember

Essential concepts to master this topic

Formula: Average speed equals total distance divided by total time
Technique: Calculate each segment: 28 × 0.5 = 14 km for first segment
Check: Total time should equal sum of all intervals: 0.5 + 2 + 0.25 + 1 = 3.75 hours ✓

Common Mistakes

Avoid these frequent errors

Taking the average of all speeds instead of using distance-time formula
Don't calculate (28 + 24 + 32 + 27) ÷ 4 = 27.75 km/h! This ignores that Jonathan spent different amounts of time at each speed. Always find total distance first, then divide by total time.

Practice Quiz

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What is the average speed according to the data?

FAQ

Everything you need to know about this question

Why can't I just average all the speeds together?

Because Jonathan spent different amounts of time at each speed! He rode 2 hours at 24 km/h but only 15 minutes at 32 km/h. The longer times should have more influence on the average.

How do I convert minutes to hours?

Divide minutes by 60! So 15 minutes = 15 ÷ 60 = 0.25 hours and 30 minutes = 30 ÷ 60 = 0.5 hours. Always use the same time units throughout.

Do I need to convert the final answer to a mixed number?

Not necessarily! $25.888...$ km/h is perfectly acceptable. You could also write it as $25\frac{8}{9}$ km/h, but decimal form is more common for speed problems.

What if I get confused with all the time segments?

Make a table! List each segment with its speed, time, and calculated distance. This helps you stay organized and avoid missing any segments.

How do I check if my total time is correct?

Add up all time intervals: 30 minutes + 2 hours + 15 minutes + 1 hour. Convert to hours: 0.5 + 2 + 0.25 + 1 = 3.75 hours total.

Why is the answer less than most of the individual speeds?

Because Jonathan spent the longest time (2 hours) going at his slowest sustained speed of 24 km/h. This pulls the average down, even though he had brief periods at higher speeds.

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