Compare Averages: Determine if 30,30,30 Equals 30,30,30,30,30

Q: Why do both groups have the same average even though one has more numbers?

When all numbers in a group are identical, the average will always equal that number! It doesn't matter if you have 3 thirties or 5 thirties - the average is still 30.

Q: Do I really need to calculate if all numbers are the same?

While you can recognize the pattern, it's good practice to show your work! Calculate $ \frac{\text{sum}}{\text{count}} $ to demonstrate your understanding of averages.

Q: What if the numbers were different, like 30,30,35?

Then you'd get different averages! For example: $ \frac{30+30+35}{3} = \frac{95}{3} = 31.67 $. Identical values are the key to equal averages.

Q: How do I remember the average formula?

Think of it as "fair sharing" - if you shared all the values equally among everyone, how much would each person get? That's your average!

Q: Can averages ever be negative or fractions?

Absolutely! Averages can be any real number. If your data includes negatives or results in fractions, that's completely normal and mathematically correct.

Average Calculation with Identical Values

Calculate the average of each group and choose the appropriate sign (?):

$30,30,30\stackrel{?}{=}30,30,30,30,30$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Calculate the average of each group and choose the appropriate sign

00:05 To calculate the average, we need to divide the sum by the number of occurrences

00:23 This is the average, now let's use the same method for the second group

00:30 Let's compare and match the sign

00:34 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Calculate the average of each group and choose the appropriate sign (?):

$30,30,30\stackrel{?}{=}30,30,30,30,30$

Step-by-step solution

To solve this problem, follow these steps:

Step 1: Calculate the average for the first group of numbers: 30, 30, 30.
Step 2: Calculate the average for the second group of numbers: 30, 30, 30, 30, 30, 30.
Step 3: Compare the two averages using the appropriate sign.

Now, let's calculate each step:

Step 1: The first group is 30, 30, 30.
The sum is $30 + 30 + 30 = 90$ .
The count is 3.
The average is $\frac{90}{3} = 30$ .

Step 2: The second group is 30, 30, 30, 30, 30, 30.
The sum is $30 + 30 + 30 + 30 + 30 + 30 = 180$ .
The count is 6.
The average is $\frac{180}{6} = 30$ .

Step 3: Compare the averages:
First average = 30 and Second average = 30. Thus, they are equal.

Therefore, the appropriate mathematical sign between the two groups is $\boxed{=}$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Average Rule: Sum of all values divided by count of values
Technique: When all values are identical, average equals that value: $\frac{30 \times 3}{3} = 30$
Check: Verify both groups give same average: 30 and 30 are equal ✓

Common Mistakes

Avoid these frequent errors

Thinking more numbers always means larger average
Don't assume the group with more numbers (30,30,30,30,30) has a larger average = wrong comparison! The number of identical values doesn't change the average when all values are the same. Always calculate each average separately using sum ÷ count.

Practice Quiz

Test your knowledge with interactive questions

If the balls below are divided so that each column in the table contains an equal number, then how many balls will there be in each column?

FAQ

Everything you need to know about this question

Why do both groups have the same average even though one has more numbers?

When all numbers in a group are identical, the average will always equal that number! It doesn't matter if you have 3 thirties or 5 thirties - the average is still 30.

Do I really need to calculate if all numbers are the same?

While you can recognize the pattern, it's good practice to show your work! Calculate $\frac{\text{sum}}{\text{count}}$ to demonstrate your understanding of averages.

What if the numbers were different, like 30,30,35?

Then you'd get different averages! For example: $\frac{30+30+35}{3} = \frac{95}{3} = 31.67$ . Identical values are the key to equal averages.

How do I remember the average formula?

Think of it as "fair sharing" - if you shared all the values equally among everyone, how much would each person get? That's your average!

Can averages ever be negative or fractions?

Absolutely! Averages can be any real number. If your data includes negatives or results in fractions, that's completely normal and mathematically correct.

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