Averages for 5th Grade: Logic and comprehension questions

To solve this problem, we'll first calculate the average of the original set of numbers and then calculate the average of the new set with the added number. Finally, we compare both averages to see if they differ.

• Step 1: Calculate the average of the original set (9, 6, 7, 2).

First, find the sum of the original numbers:
$9 + 6 + 7 + 2 = 24$

Next, compute the average by dividing the sum by the number of elements:
$\text{Average} = \frac{24}{4} = 6$

• Step 2: Calculate the average of the new set (9, 6, 7, 2, 6).

First, find the sum of the new numbers:
$9 + 6 + 7 + 2 + 6 = 30$

Now, compute the average by dividing the sum by the number of elements:
$\text{Average} = \frac{30}{5} = 6$

• Step 3: Compare the two averages.

Both averages are 6. Therefore, adding the number 6 to the group does not change the average.

Thus, the conclusion is that the average does not change, confirming:

No, the average will not change when adding the number 6 to this set of numbers.

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

• Step 1: Calculate the sum of the numbers in the group.
• Step 2: Apply the formula for the average.
• Step 3: Determine how adding a smaller number affects the average.

Now, let's work through each step:

Step 1: Calculate the sum of the given numbers.
The numbers are $5, 3, 7, 5$. Their sum is:

Step 2: Find the average.
The formula for the average of these numbers is:

Therefore, the average of the numbers $5, 3, 7, 5$ is $5$.

Step 3: Determine the effect of adding a smaller number.
If we add a number smaller than $5$ to the group, the new average will be calculated over five numbers with the new sum including this smaller number. Since this number is smaller than the current average, the overall average will decrease.

Hence, the correct answer to this problem is:
Average: 5

The average will decrease.

To solve this problem, we'll determine the initial average of the numbers provided and address the change in the average when a new number is added:

• Step 1: Calculate the initial average.
• Step 2: Add the new number to recalculate the average.
• Step 3: Compare the old and new averages.

Step 1: Calculate the initial average of 10, 12, and 8:
- First, find the sum of the numbers: $10 + 12 + 8 = 30$.
- There are 3 numbers, so divide the sum by 3 to get the average: $\frac{30}{3} = 10$.

Step 2: Add the number 11 (which is larger than the initial average) to the group and recalculate the average:
- The new sum is $30 + 11 = 41$ with a total of 4 numbers.
- The new average becomes $\frac{41}{4} = 10.25$.

Step 3: Compare the averages:
- The initial average is 10.
- The new average is 10.25, which is higher than 10.

Therefore, adding a number larger than the initial average results in an increased average. So, the average will increase when 11 is added.

The correct answer to the question is: Average 10, the average will increase.

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

• Step 1: Calculate the average of the original numbers.
• Step 2: Calculate the average of the increased numbers.
• Step 3: Compare the averages.

Now, let's work through each step:

Step 1: The original numbers are 10, 12, and 8.
Calculate their sum: $10 + 12 + 8 = 30$.
Divide the sum by the number of numbers: $\frac{30}{3} = 10$.
Therefore, the average of the original numbers is $10$.

Step 2: The increased numbers are 12, 14, and 10.
Calculate their sum: $12 + 14 + 10 = 36$.
Divide the sum by the number of numbers: $\frac{36}{3} = 12$.
Therefore, the average of the increased numbers is $12$.

Step 3: Compare the averages.
The average increased from $10$ to $12$, which is an increase by $2$.

Conclusion: Since the average increases by $2$, the answer is that the average of the new group of numbers is $12$, and it increases by $2$ from the original average.

Therefore, the correct answer is choice 1:
Average = 10
The average will increase by 2.

To solve this problem, let's follow these concise steps:

• Step 1: Calculate the current average of the given numbers 5, 7, and 3.

The sum of the numbers is $5 + 7 + 3 = 15$.
The count of numbers is 3.
Thus, the average is $\frac{15}{3} = 5$.

• Step 2: Set up an equation with an additional number, $x$, so that the average remains the same.

Adding a fourth number, $x$, gives us a new set of numbers: 5, 7, 3, and $x$.
The new average should still be 5, so:

$\frac{5 + 7 + 3 + x}{4} = 5$

• **Step 3: Solve the equation for $x$.

Calculate the left side: $5 + 7 + 3 + x = 15 + x$.
Set up the equation: $\frac{15 + x}{4} = 5$.

Multiply both sides by 4 to eliminate the fraction: $15 + x = 20$.
Subtract 15 from both sides: $x = 20 - 15$, so $x = 5$.

Hence, the number that can be added without changing the average is $x = 5$.

Therefore, the correct choice from the provided options is option 3:

$5$

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

• Step 1: Calculate the sum and average of the original numbers.
• Step 2: Add the number 10 to the original list and calculate the new sum and average.
• Step 3: Compare the two averages to determine if there is an increase, decrease, or no change.

Now, let's work through each step:

Step 1: Calculate the sum and average of the original numbers (2, 3, 4, 5).
Sum = $2 + 3 + 4 + 5 = 14$
Average = $\frac{14}{4} = 3.5$

Step 2: Include the number 10 in the group.
New sum = $14 + 10 = 24$
New average = $\frac{24}{5} = 4.8$

Step 3: Compare the averages.
Original average was 3.5, and the new average is 4.8.

The new average (4.8) is greater than the original average (3.5), which means the average will increase.

Therefore, the correct choice is: It will increase.

To solve this problem, we'll begin by calculating the average of the original group of numbers ($2, 3, 4, 5$):

1. Calculate the sum of the original group: $2 + 3 + 4 + 5 = 14$.

2. Determine the number of elements in the original group: $4$.

3. Calculate the average of the original group: $\frac{14}{4} = 3.5$.

Next, we'll calculate the average after adding the number $1$ to the group, forming the new group ($1, 2, 3, 4, 5$):

1. Calculate the sum of the new group: $1 + 2 + 3 + 4 + 5 = 15$.

2. Determine the number of elements in the new group: $5$.

3. Calculate the average of the new group: $\frac{15}{5} = 3$.

Finally, compare the two averages:

The average of the original group is $3.5$, while the average of the new group is $3$.

The average decreases when the number 1 is added to the group.

Therefore, the solution to the problem is that the average will decrease.

To solve this problem, let's calculate the average of the numbers before and after introducing an additional $7$.

Initially, we have four numbers: $7, 7, 7, 7$.

• Step 1: Calculate the sum of these numbers.
  $7 + 7 + 7 + 7 = 28$.
• Step 2: Calculate the initial average.
  $\frac{28}{4} = 7$.

Now, we add another $7$ to the group, resulting in five numbers: $7, 7, 7, 7, 7$.

• Step 3: Calculate the new sum.
  $28 + 7 = 35$.
• Step 4: Calculate the new average.
  $\frac{35}{5} = 7$.

Both the initial average and the new average are $7$. Therefore, the average remains unchanged.

Based on this analysis, the correct choice is:

It will remain the same.

Therefore, the solution to the problem is it will remain the same..

To solve this problem, we need to calculate the average of the numbers before and after removing one 10.

Initially, we have the numbers $10, 10, 10, 10$.

• Calculation of the initial average:
  • Sum of numbers: $10 + 10 + 10 + 10 = 40$
  • Number of numbers: 4
  • Average: $\frac{40}{4} = 10$

Now, we remove one of the 10s, leaving the numbers $10, 10, 10$.

• Calculation of the new average:
  • Sum of numbers: $10 + 10 + 10 = 30$
  • Number of numbers: 3
  • Average: $\frac{30}{3} = 10$

We see that in both cases, the average is 10. Thus, removing one 10 does not change the average of the numbers.

Therefore, the average will remain the same.

Hence, the correct choice is: It will remain the same.

To solve this problem, we need to follow these steps:

• Step 1: Determine the average of the initial set of numbers $(1, 2, 3, 11)$.
• Step 2: Determine the average of the set after removing 11 $(1, 2, 3)$.
• Step 3: Compare the averages to see the change—the average should be noted as decreasing or increasing or remaining the same.

Step 1: Calculate the average of the initial set $(1, 2, 3, 11)$:

The sum of these numbers is $1 + 2 + 3 + 11 = 17$.
There are 4 numbers in this set.
Thus, the average is $\frac{17}{4} = 4.25$.

Step 2: Calculate the average of the new set $(1, 2, 3)$:

The sum of these numbers is $1 + 2 + 3 = 6$.
There are 3 numbers in this set.
Thus, the average is $\frac{6}{3} = 2$.

Step 3: Compare the two averages:

The original average was $4.25$, and the new average is $2$. The average has decreased from $4.25$ to $2$.

Therefore, the solution to the problem is: It will decrease.

To solve this problem, we need to compare the average of the numbers $1, 2, 3, 11$ before and after removing the number 1.

Let's follow these steps:

• Step 1: Calculate the original average.

The original set of numbers is $1, 2, 3, 11$.
Calculate the sum of these numbers:

$1 + 2 + 3 + 11 = 17$.

There are 4 numbers in the original set, so the original average is:

$\text{Original Average} = \frac{17}{4} = 4.25$.

• Step 2: Calculate the new average after removing the number 1.

The new set of numbers is $2, 3, 11$.
Calculate the sum of these numbers:

$2 + 3 + 11 = 16$.

There are 3 numbers in the new set, so the new average is:

$\text{New Average} = \frac{16}{3} \approx 5.33$.

• Step 3: Compare the two averages.

Original average: $4.25$
New average: $5.33$

Since $5.33 > 4.25$, the average increases when we remove the number 1 from the group.

Therefore, the solution to the problem is It will increase.

To solve this problem, we should understand what it means when the average of five numbers is 7. The average is calculated by taking the sum of the numbers and dividing by the number of numbers. Given that the average is $7$, the equation can be written as:

$\frac{a + b + c + d + e}{5} = 7$

Multiplying both sides by $5$ gives:

$a + b + c + d + e = 35$

This equation tells us that the sum of the numbers $a, b, c, d, e$ is $35$. Now, we address whether at least one of these numbers must be $7$.

Consider the set of numbers $a = 1$, $b = 1$, $c = 1$, $d = 1$, and $e = 31$. The sum of these numbers is:

$1 + 1 + 1 + 1 + 31 = 35$

The average is:

$\frac{35}{5} = 7$

Here, none of the numbers are $7$, yet the average is still $7$. This example shows that it is not necessary for any of the numbers to be $7$ for the average to be $7$.

Therefore, the conclusion is that the number $7$ is not necessarily one of the numbers.

Thus, the correct choice is : No.