Finding Impossible Averages in the Sequence 10,11,12,13,14: No-Calculator Method

Q: Why can't I just calculate the average to solve this?

The question asks what cannot be the average, not what is the average. Since 9 is less than the smallest number (10), it's impossible for 9 to be the average of any subset of these numbers!

Q: Could 12 be the average of some of these numbers?

Yes! For example, $ \frac{11 + 13}{2} = 12 $. Since 12 falls within the range [10, 14], it's possible to select numbers from our set that average to 12.

Q: What if the question had a number larger than 14?

Any number greater than 14 would also be impossible! The average of any group of numbers must fall between the smallest and largest values in the original set.

Q: Does this rule work for any set of numbers?

Absolutely! This is a universal property of averages. No matter what numbers you have, their average will always be between the minimum and maximum values.

Q: How do I quickly identify the range?

Look for the smallest and largest numbers in your set. In consecutive integers like 10, 11, 12, 13, 14, it's easy: the range is [10, 14].

Range-Based Reasoning with Sequential Numbers

Without calculating, choose the number that cannot be the average of the following group of numbers:

$10,11,12,13,14$

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:09 Let's find the incorrect average without doing any calculations.

00:14 First, plot the numbers on a number line.

00:18 The average should be about in the middle of these numbers.

00:23 This option is good because it's around the center.

00:27 This one is too small, so it can't be the average.

00:32 This option is good because it's around the center.

00:35 This option is good because it's around the center.

00:39 This option is good because it's around the center.

00:46 And that's how you find the solution to this question.

Step-by-step written solution

Follow each step carefully to understand the complete solution

Understand the problem

Without calculating, choose the number that cannot be the average of the following group of numbers:

$10,11,12,13,14$

Step-by-step solution

To solve this problem, let us examine the range of the numbers given:

The smallest number in the set is $10$ , and the largest is $14$ . Hence, any average of these numbers must fall within this range, i.e., between 10 and 14.

Considering the choices:

$12$ is within the range $10$ to $14$ .
$9$ is outside of this range.
$13$ is within the range $10$ to $14$ .
$11$ is within the range $10$ to $14$ .

Thus, it is clear that the number that cannot be the average of $(10, 11, 12, 13, 14)$ is $9$ , as it does not lie within the required range.

Final Answer

Key Points to Remember

Essential concepts to master this topic

Range Rule: Average must lie between minimum and maximum values
Technique: Check if 9 falls between 10 and 14
Check: Any value outside [10, 14] cannot be the average ✓

Common Mistakes

Avoid these frequent errors

Calculating the actual average instead of using range logic
Don't compute (10+11+12+13+14)÷5 = 12 to solve this! This wastes time and misses the point. The question asks what CANNOT be the average, so any number outside the range [10, 14] is impossible. Always check the range first before calculating.

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 calculate the average to solve this?

The question asks what cannot be the average, not what is the average. Since 9 is less than the smallest number (10), it's impossible for 9 to be the average of any subset of these numbers!

Could 12 be the average of some of these numbers?

Yes! For example, $\frac{11 + 13}{2} = 12$ . Since 12 falls within the range [10, 14], it's possible to select numbers from our set that average to 12.

What if the question had a number larger than 14?

Any number greater than 14 would also be impossible! The average of any group of numbers must fall between the smallest and largest values in the original set.

Does this rule work for any set of numbers?

Absolutely! This is a universal property of averages. No matter what numbers you have, their average will always be between the minimum and maximum values.

How do I quickly identify the range?

Look for the smallest and largest numbers in your set. In consecutive integers like 10, 11, 12, 13, 14, it's easy: the range is [10, 14].

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