Compare Group Averages: Is (5,5,5) = (20,0,0,0)?

Q: Why don't I just compare the biggest numbers in each group?

Because we're comparing averages, not individual values! The average considers all numbers equally, so 20 in the second group gets balanced out by the three zeros.

Q: Can groups with different numbers of terms have the same average?

Absolutely! The number of terms doesn't affect whether averages can be equal. What matters is the sum divided by count for each group.

Q: How do I avoid calculation errors when finding averages?

Always double-check your addition first, then your division. Write out: Sum ÷ Count = Average. For example: $ 20 ÷ 4 = 5 $

Q: What does the equal sign mean when comparing groups?

The = sign means the averages are identical. Even though the individual numbers look very different, both groups have the exact same average value.

Average Calculations with Group Comparisons

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

$5,5,5\stackrel{?}{=}20,0,0,0$

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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

00:06 To calculate the average, we need to divide the sum by the number of occurrences

00:27 This is the average, now let's use the same method for the second group

00:59 Let's compare and match the sign

01:05 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 (?):

$5,5,5\stackrel{?}{=}20,0,0,0$

Step-by-step solution

To solve this problem, we will follow these steps:

Step 1: Calculate the average of the first group (5, 5, 5).
Step 2: Calculate the average of the second group (20, 0, 0, 0).
Step 3: Compare the two averages to choose the appropriate sign.

Now, let's work through each step:

Step 1: Calculate the average of Group 1.
The sum of Group 1 is $5 + 5 + 5 = 15$ .
The number of terms in Group 1 is 3.
Thus, the average of Group 1 is $\frac{15}{3} = 5$ .

Step 2: Calculate the average of Group 2.
The sum of Group 2 is $20 + 0 + 0 + 0 = 20$ .
The number of terms in Group 2 is 4.
Thus, the average of Group 2 is $\frac{20}{4} = 5$ .

Step 3: Compare the averages.
The average of Group 1 is 5, and the average of Group 2 is also 5.
Since the averages are equal, the appropriate mathematical sign is '='.

Therefore, the correct comparison is $5, 5, 5 \stackrel{=}{=} 20, 0, 0, 0$ .

The appropriate sign is =.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Formula: Average equals sum divided by number of terms
Technique: Calculate $\frac{15}{3} = 5$ and $\frac{20}{4} = 5$ separately
Check: Verify both groups have same average: 5 = 5 ✓

Common Mistakes

Avoid these frequent errors

Comparing sums instead of averages
Don't compare 15 versus 20 directly = wrong conclusion! The sums are different but that doesn't matter for averages. Always divide each sum by its count of terms to find the actual averages before comparing.

Practice Quiz

Test your knowledge with interactive questions

Calculate the average of $$ 11 $$ and $$ 7 $$ .

FAQ

Everything you need to know about this question

Why don't I just compare the biggest numbers in each group?

Because we're comparing averages, not individual values! The average considers all numbers equally, so 20 in the second group gets balanced out by the three zeros.

Can groups with different numbers of terms have the same average?

Absolutely! The number of terms doesn't affect whether averages can be equal. What matters is the sum divided by count for each group.

What if one group has all the same numbers?

That makes it easier! When all numbers are the same (like 5, 5, 5), the average is just that number. No calculation needed!

How do I avoid calculation errors when finding averages?

Always double-check your addition first, then your division. Write out: Sum ÷ Count = Average. For example: $20 ÷ 4 = 5$

What does the equal sign mean when comparing groups?

The = sign means the averages are identical. Even though the individual numbers look very different, both groups have the exact same average value.

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