Calculate Average to Compare: Are (7,8,4,3,6,5) = (3,7,8,5)?

Q: Why can't I just compare the sums of the two groups?

Because the groups have different numbers of values! The first group has 6 numbers while the second has only 4. Comparing sums would be unfair - like comparing total points scored by teams with different numbers of players.

Q: What if both groups had the same number of values?

Then comparing sums would work! When groups have equal sizes, the group with the larger sum automatically has the larger average. But here we have 6 vs 4 values, so we must calculate averages.

Q: How do I remember which symbol to use?

Think of it like a number line: 5.5 is to the left of 5.75, so 5.5

Q: Can I use decimals or should I keep fractions?

Either works! $ \frac{33}{6} = 5.5 $ and $ \frac{23}{4} = 5.75 $ are easier to compare as decimals, but you could also compare $ \frac{11}{2} $ vs $ \frac{23}{4} $ as fractions.

Q: What if I get the same average for both groups?

Then you'd use the equals sign (=)! This means both groups have identical averages even if they contain different individual numbers or have different sizes.

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

00:03 To calculate the average, we need to divide the sum by the number of occurrences

00:53 This is the average, now let's use the same method for the second group

01:32 Let's compare and match the sign

01:38 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

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

$7,8,4,3,6,5\stackrel{?}{=}3,7,8,5$

2

Step-by-step solution

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

Step 1: Calculate the average of the first group $(7, 8, 4, 3, 6, 5)$ .
Step 2: Calculate the average of the second group $(3, 7, 8, 5)$ .
Step 3: Compare the two averages.

Let's complete each step in detail:

Step 1: Calculate the average of the first group.

The numbers in the first group are $7, 8, 4, 3, 6, 5$ . First, find the sum:

$7 + 8 + 4 + 3 + 6 + 5 = 33$ .

There are 6 numbers in this group, so the average is:

$\text{Average} = \frac{33}{6} = 5.5$ .

Step 2: Calculate the average of the second group.

The numbers in the second group are $3, 7, 8, 5$ . First, find the sum:

$3 + 7 + 8 + 5 = 23$ .

There are 4 numbers in this group, so the average is:

$\text{Average} = \frac{23}{4} = 5.75$ .

Step 3: Compare the two averages.

The average of the first group is $5.5$ and the average of the second group is $5.75$ .

Therefore, $5.5$ is less than $5.75$ .

We choose the sign $<$ indicating that the average of the first group is less than the average of the second group.

The correct answer is $<$ .

3

Final Answer

<

Key Points to Remember

Essential concepts to master this topic

Formula: Average equals sum of values divided by count
Technique: First group: $\frac{33}{6} = 5.5$ , second group: $\frac{23}{4} = 5.75$
Check: Compare final averages: 5.5 < 5.75, so first group < second group ✓

Common Mistakes

Avoid these frequent errors

Comparing sums instead of averages
Don't compare 33 vs 23 directly = wrong conclusion! The first sum is larger, but groups have different sizes. Always divide each sum by its count to get the true average before comparing.

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 can't I just compare the sums of the two groups?

+

Because the groups have different numbers of values! The first group has 6 numbers while the second has only 4. Comparing sums would be unfair - like comparing total points scored by teams with different numbers of players.

What if both groups had the same number of values?

+

Then comparing sums would work! When groups have equal sizes, the group with the larger sum automatically has the larger average. But here we have 6 vs 4 values, so we must calculate averages.

How do I remember which symbol to use?

+

Think of it like a number line: 5.5 is to the left of 5.75, so 5.5 < 5.75. The smaller average gets the '<' symbol pointing toward the larger one.

Can I use decimals or should I keep fractions?

+

Either works! $\frac{33}{6} = 5.5$ and $\frac{23}{4} = 5.75$ are easier to compare as decimals, but you could also compare $\frac{11}{2}$ vs $\frac{23}{4}$ as fractions.

What if I get the same average for both groups?

+

Then you'd use the equals sign (=)! This means both groups have identical averages even if they contain different individual numbers or have different sizes.

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Calculate Average to Compare: Are (7,8,4,3,6,5) = (3,7,8,5)?

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