Calculate Average: Compare {4,5,10,5,1,5} vs {4,5,10,5,1}

Q: Why do both groups have the same average when the sums are different?

The first group has sum 30 with 6 numbers, so average = $ \frac{30}{6} = 5 $. The second group has sum 25 with 5 numbers, so average = $ \frac{25}{5} = 5 $. Same average!

Q: What happens when I remove a number that equals the average?

When you remove a number that equals the current average (like removing the 5), the average stays the same! This is because removing an "average" value doesn't change the balance.

Q: How do I know which sign to use between the averages?

Calculate both averages completely first. Then compare the final decimal values: if left average > right average, use >; if left

Q: Can I just look at the numbers to guess the answer?

No! Always calculate the actual averages. Even though one group has more numbers, the averages might still be equal or the smaller group might have a higher average.

Q: What if I get different averages? How do I double-check?

Recount the elements in each groupRecalculate each sum carefullyDivide sum by count for each groupCompare the final decimal results

Average Comparison with Missing Elements

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

$4,5,10,5,1,5\stackrel{?}{=}4,5,10,5,1$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:14 First, find the average for each group. Choose the right sign.

00:20 To get the average, divide the total sum by the number of items. Simple, right?

00:36 We'll use the same steps to find the average for the next group.

00:46 Now, let's calculate each average and compare them. Ready?

01:46 And that's how we solve this problem. Great job!

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 (?):

$4,5,10,5,1,5\stackrel{?}{=}4,5,10,5,1$

Step-by-step solution

To solve this problem, let's calculate the average for each group of numbers:

Step 1: Calculate the average of the first group $4, 5, 10, 5, 1, 5$ :
- Sum of numbers: $4 + 5 + 10 + 5 + 1 + 5 = 30$
- Number of elements: $6$
- Average: $\frac{30}{6} = 5$
Step 2: Calculate the average of the second group $4, 5, 10, 5, 1$ :
- Sum of numbers: $4 + 5 + 10 + 5 + 1 = 25$
- Number of elements: $5$
- Average: $\frac{25}{5} = 5$

Both groups have an average of $5$ , so the correct sign to place between them is =.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Rule: Average equals sum divided by count of elements
Technique: Calculate sum: 4+5+10+5+1+5=30, then 30÷6=5
Check: Verify both averages match before comparing: 5=5 ✓

Common Mistakes

Avoid these frequent errors

Comparing sums instead of averages
Don't compare 30 vs 25 and conclude 30>25! This ignores the different counts (6 vs 5 elements). Always divide each sum by its count to find the true average before comparing.

Practice Quiz

Test your knowledge with interactive questions

Calculate the average of

$$ 10 $$ , $$ 10 $$ , $$ 10 $$ , and $$ 10 $$ .

FAQ

Everything you need to know about this question

Why do both groups have the same average when the sums are different?

The first group has sum 30 with 6 numbers, so average = $\frac{30}{6} = 5$ . The second group has sum 25 with 5 numbers, so average = $\frac{25}{5} = 5$ . Same average!

What happens when I remove a number that equals the average?

When you remove a number that equals the current average (like removing the 5), the average stays the same! This is because removing an "average" value doesn't change the balance.

How do I know which sign to use between the averages?

Calculate both averages completely first. Then compare the final decimal values: if left average > right average, use >; if left < right, use <; if equal, use =.

Can I just look at the numbers to guess the answer?

No! Always calculate the actual averages. Even though one group has more numbers, the averages might still be equal or the smaller group might have a higher average.

What if I get different averages? How do I double-check?

Recount the elements in each group
Recalculate each sum carefully
Divide sum by count for each group
Compare the final decimal results

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