Compare Averages: Determining if 12,16 = 10,20,25,5,10

Q: Do I need to compare every number in both groups?

No! You only compare the averages of each group. Calculate each average separately, then use =, to compare those two final numbers.

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

Absolutely! Groups can have completely different individual numbers but still end up with identical averages. That's exactly what happens in this problem!

Q: How do I avoid calculation errors when adding many numbers?

Add numbers systematically: $ 10 + 20 = 30 $, then $ 30 + 25 = 55 $, then $ 55 + 5 = 60 $, finally $ 60 + 10 = 70 $. Double-check by adding in a different order!

Q: What does the symbol between the groups mean?

The ? symbol means you need to determine the correct comparison sign. Calculate both averages, then replace ? with =, based on which average is larger (or if they're equal).

Average Comparison with Multiple Data Sets

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

$12,16\stackrel{?}{=}10,20,25,5,10$

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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

01:11 Let's compare and match the sign

01:15 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 (?):

$12,16\stackrel{?}{=}10,20,25,5,10$

Step-by-step solution

To solve this problem, we'll determine the average of both groups and compare the results:

Step 1: Calculate the average of the first group $(12, 16)$ .
The sum of the first group is $12 + 16 = 28$ .
There are 2 numbers, so the average is $\frac{28}{2} = 14$ .
Step 2: Calculate the average of the second group $(10, 20, 25, 5, 10)$ .
The sum of the second group is $10 + 20 + 25 + 5 + 10 = 70$ .
There are 5 numbers, so the average is $\frac{70}{5} = 14$ .
Step 3: Compare the averages.
Both groups have an average of 14; thus, the correct sign is $=$ .

Since both averages are equal, the conclusion is that the groups have the same average.
Therefore, the solution to the problem is =.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Definition: Average equals sum of values divided by count
Technique: Calculate $\frac{12+16}{2} = 14$ and $\frac{10+20+25+5+10}{5} = 14$
Check: Both averages equal 14, so 12,16 = 10,20,25,5,10 ✓

Common Mistakes

Avoid these frequent errors

Comparing individual numbers instead of averages
Don't compare 12 vs 10 or 16 vs 20 directly = wrong conclusion! This ignores the averaging process entirely. Always calculate the complete average of each group first, then compare those final results.

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

Do I need to compare every number in both groups?

No! You only compare the averages of each group. Calculate each average separately, then use =, <, or > to compare those two final numbers.

What if the groups have different numbers of values?

That's completely fine! The first group has 2 numbers, the second has 5 numbers. Just make sure to count correctly when dividing by the total number of values in each group.

Can two groups with different numbers have the same average?

Absolutely! Groups can have completely different individual numbers but still end up with identical averages. That's exactly what happens in this problem!

How do I avoid calculation errors when adding many numbers?

Add numbers systematically: $10 + 20 = 30$ , then $30 + 25 = 55$ , then $55 + 5 = 60$ , finally $60 + 10 = 70$ . Double-check by adding in a different order!

What does the symbol between the groups mean?

The ? symbol means you need to determine the correct comparison sign. Calculate both averages, then replace ? with =, <, or > based on which average is larger (or if they're equal).

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