Determine the Boat's Speed Over 50 km After an Initial 40-Minute Journey

Q: Why isn't the speed for the second segment different from 45 km/h?

This happens when the first segment's speed exactly equals the average speed! Since the first 30 km was covered at 45 km/h, the remaining 50 km must also be at 45 km/h to maintain the overall average.

Q: How do I convert 40 minutes to hours?

Divide by 60: $ \frac{40}{60} = \frac{2}{3} $ hours. Remember there are 60 minutes in 1 hour, so always divide minutes by 60 to get hours.

Q: What if the first segment speed was different from the average?

Then the second segment would need a different speed to compensate! If the first segment is slower than average, the second must be faster, and vice versa.

Q: How do I subtract fractions like 16/9 - 2/3?

Convert to the same denominator first: $ \frac{2}{3} = \frac{6}{9} $, so $ \frac{16}{9} - \frac{6}{9} = \frac{10}{9} $. Always find a common denominator before adding or subtracting fractions.

Q: Can I solve this without finding total time?

Not easily! The total time method is the most reliable approach for multi-segment average speed problems. It ensures you account for the overall average correctly.

Step-by-step video solution

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Step-by-step written solution

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1

Understand the problem

The average speed of a boat is 45 km/h.

At the beginning of the voyage, the boat covers a distance of 30 km in 40 minutes.

What was the speed of the boat over the next 50 km?

2

Step-by-step solution

To address this problem, let's follow these steps:

Step 1: Calculate the speed for the first segment of the journey.
Step 2: Use the overall average speed to find the total time required for the entire voyage.
Step 3: Determine the time available for the second segment and calculate the required speed.

Step 1: For the first segment, the distance is $30$ km, and the time taken is $40$ minutes, which is $\frac{40}{60} = \frac{2}{3}$ hours. Thus, the speed for the first segment is:

\text{Speed}_1 = \frac{30 \text{ km}}{\frac{2}{3} \text{ hours}} = 45 \text{ km/h}

Step 2: Since the average speed for the entire trip is $45$ km/h, let's find the total time required for both segments. The total distance is $30 + 50 = 80$ km. Thus, the total time required is:

\text{Total Time} = \frac{80 \text{ km}}{45 \text{ km/h}} = \frac{16}{9} \text{ hours}

Step 3: The first segment took $\frac{2}{3}$ hours. Hence, the time available for the second segment of $50$ km is:

\text{Time}_2 = \frac{16}{9} - \frac{2}{3} = \frac{16}{9} - \frac{6}{9} = \frac{10}{9} \text{ hours}

Therefore, the speed required for the second segment is:

\text{Speed}_2 = \frac{50 \text{ km}}{\frac{10}{9} \text{ hours}} = 45 \text{ km/h}

Thus, the speed of the boat over the next 50 km is $45$ km/h.

3

Final Answer

$45$ km/h

Key Points to Remember

Essential concepts to master this topic

Concept: Average speed applies to entire journey, not individual segments
Method: Find total time first: $\frac{80\text{ km}}{45\text{ km/h}} = \frac{16}{9}\text{ hours}$
Verify: Check both segments average to given speed: $\frac{30+50}{\frac{2}{3}+\frac{10}{9}} = 45\text{ km/h}$ ✓

Common Mistakes

Avoid these frequent errors

Assuming each segment has the same speed as average
Don't think the boat travels 45 km/h for each segment just because the average is 45 km/h = wrong understanding of averages! Different segments can have different speeds while maintaining the same overall average. Always calculate total time first, then find remaining time for unknown segments.

Practice Quiz

Test your knowledge with interactive questions

What is the average speed according to the data?

FAQ

Everything you need to know about this question

Why isn't the speed for the second segment different from 45 km/h?

+

This happens when the first segment's speed exactly equals the average speed! Since the first 30 km was covered at 45 km/h, the remaining 50 km must also be at 45 km/h to maintain the overall average.

How do I convert 40 minutes to hours?

+

Divide by 60: $\frac{40}{60} = \frac{2}{3}$ hours. Remember there are 60 minutes in 1 hour, so always divide minutes by 60 to get hours.

What if the first segment speed was different from the average?

+

Then the second segment would need a different speed to compensate! If the first segment is slower than average, the second must be faster, and vice versa.

How do I subtract fractions like 16/9 - 2/3?

+

Convert to the same denominator first: $\frac{2}{3} = \frac{6}{9}$ , so $\frac{16}{9} - \frac{6}{9} = \frac{10}{9}$ . Always find a common denominator before adding or subtracting fractions.

Can I solve this without finding total time?

+

Not easily! The total time method is the most reliable approach for multi-segment average speed problems. It ensures you account for the overall average correctly.

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Determine the Boat's Speed Over 50 km After an Initial 40-Minute Journey

Average Speed with Multi-Segment Journeys

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Step-by-step video solution

Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why isn't the speed for the second segment different from 45 km/h?

How do I convert 40 minutes to hours?

What if the first segment speed was different from the average?

How do I subtract fractions like 16/9 - 2/3?

Can I solve this without finding total time?

🌟 Unlock Your Math Potential

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