Compare Averages: Does 9,0 Equal the Mean of 9,0,0?

Q: Why do the averages change when I add more zeros?

Adding zeros doesn't change the sum (still 9), but it increases the count of numbers. Since average = sum ÷ count, more numbers in the denominator makes the average smaller!

Q: How do I remember which symbol to use for comparison?

Think of the symbol as a mouth eating the bigger number! Since 4.5 > 3, the mouth opens toward 4.5, giving us the > symbol.

Average Calculations with Different Sample Sizes

Calculate the average of each group and mark the appropriate sign:

$9,0\stackrel{?}{=}9,0,0$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's calculate the average for each group and pick the right sign.

00:14 To find the average, divide the total sum by how many times it appears.

00:27 That's the average! Now, let's apply the same steps for the second group.

00:53 Next, we'll compare and find the matching sign!

00:59 And there you have it! That's how we solve the problem.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Calculate the average of each group and mark the appropriate sign:

$9,0\stackrel{?}{=}9,0,0$

Step-by-step solution

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

Step 1: Calculate the average of each group.
Step 2: Compare the averages using the appropriate sign.

Now, let's work through each step:

Step 1: The first group is $(9,0)$ . To find the average:

$\text{Average of (9,0)} = \frac{9 + 0}{2} = \frac{9}{2} = 4.5$

Step 2: The second group is $(9,0,0)$ . To find the average:

$\text{Average of (9,0,0)} = \frac{9 + 0 + 0}{3} = \frac{9}{3} = 3$

Now, we compare the two averages:

$4.5$ compared to $3$ .

Since $4.5$ is greater than $3$ , we mark the appropriate sign as $\gt$ .

Therefore, the solution to the problem is $9,0 \gt 9,0,0$ .

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: Average equals sum divided by count of numbers
Technique: (9+0)/2 = 4.5 vs (9+0+0)/3 = 3
Check: Verify by multiplying: 4.5×2 = 9 and 3×3 = 9 ✓

Common Mistakes

Avoid these frequent errors

Using the same denominator for both groups
Don't divide by 2 for both groups = wrong averages! The second group has 3 numbers, not 2, so dividing by 2 gives 4.5 instead of 3. Always count the actual numbers in each group and use that as your denominator.

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 the averages change when I add more zeros?

Adding zeros doesn't change the sum (still 9), but it increases the count of numbers. Since average = sum ÷ count, more numbers in the denominator makes the average smaller!

Will adding any number of zeros always decrease the average?

Yes! When your original numbers include values greater than zero, adding zeros will always pull the average down because you're dividing the same sum by a larger count.

How do I remember which symbol to use for comparison?

Think of the symbol as a mouth eating the bigger number! Since 4.5 > 3, the mouth opens toward 4.5, giving us the > symbol.

What if both groups had the same count of numbers?

If both groups had the same number of values, then the group with the larger sum would have the larger average. The count wouldn't affect the comparison.

Can I solve this without calculating exact averages?

Yes! You can reason that since both groups sum to 9, but one has fewer numbers to divide by, that group will have the larger average without doing the division.

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