Calculating Weighted Average: Analyzing Roller Coaster Speeds

Weighted Average with Percentage Weights

A company is making an advertisement for a new roller coaster and they want to know what the weighted average of its travel speed is.

km/hWeight20%15%5%35%100401306015755%20%

What is the average speed of the roller coaster?

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1

Understand the problem

A company is making an advertisement for a new roller coaster and they want to know what the weighted average of its travel speed is.

km/hWeight20%15%5%35%100401306015755%20%

What is the average speed of the roller coaster?

2

Step-by-step solution

To solve this problem, we'll compute the weighted average speed of the roller coaster using the given weights and speeds:

  • Step 1: Convert the weights from percentages to decimal form:

    • 20% becomes 0.20

    • 15% becomes 0.15

    • 5% becomes 0.05

    • 35% becomes 0.35

    • 5% becomes 0.05

    • 20% becomes 0.20

  • Step 2: Calculate each term by multiplying the speed by its corresponding weight:

  • 100×0.20=20,40×0.15=6,130×0.05=6.5,60×0.35=21,15×0.05=0.75,75×0.20=15. \begin{aligned} 100 \times 0.20 &= 20, \\ 40 \times 0.15 &= 6, \\ 130 \times 0.05 &= 6.5, \\ 60 \times 0.35 &= 21, \\ 15 \times 0.05 &= 0.75, \\ 75 \times 0.20 &= 15. \end{aligned}

  • Step 3: Sum the results of these products:

  • 20+6+6.5+21+0.75+15=69.25 20 + 6 + 6.5 + 21 + 0.75 + 15 = 69.25

  • Step 4: Since the weights sum to 1, the weighted average speed is:

Therefore, the average speed of the roller coaster is 69.25 km/h.

3

Final Answer

69.25 km/h

Key Points to Remember

Essential concepts to master this topic
  • Formula: Weighted average equals sum of (value × weight)
  • Technique: Convert percentages to decimals: 20% = 0.20, then multiply
  • Check: All weights must sum to 100% or 1.0 ✓

Common Mistakes

Avoid these frequent errors
  • Adding values without considering weights
    Don't just add all speeds and divide by 6 = 70 km/h! This ignores that some speeds occur more frequently than others. Always multiply each speed by its weight first, then sum the products.

Practice Quiz

Test your knowledge with interactive questions

Norbert buys some new clothes.

When he gets home, he decides to work out how much each outfit cost him on average.

PriceOutfit4 T-shirts2 pairs of shorts3 pairs of pants2 sweaters45$50$80$100$210$1 coat

What answer should he come up with?

FAQ

Everything you need to know about this question

Why can't I just find the regular average of all the speeds?

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A regular average treats all values equally, but here each speed has a different weight (importance). The roller coaster spends 35% of time at 60 km/h but only 5% at 130 km/h!

Do I need to convert percentages to decimals?

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Yes! Always convert percentages to decimals when calculating. 20% becomes 0.20, 15% becomes 0.15, etc. This makes multiplication much easier.

What if my weights don't add up to 100%?

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Check your data carefully! In weighted average problems, weights must total 100% (or 1.0 as decimals). If they don't, there's likely an error in the problem setup.

How do I know which number is the weight and which is the value?

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Look at the context and units. Here, percentages (%) are weights showing how much time is spent at each speed, while km/h values are what we're averaging.

Is there a shortcut for this calculation?

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Not really! You must calculate each speed × weight term separately, then add them all up. Each step matters for getting the right weighted average.

Can I check my answer somehow?

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Yes! Your answer should be between the lowest and highest values (15-130 km/h). Also, it should be closer to values with higher weights - here, closer to 60 km/h since it has 35% weight.

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