The mean of the 5 smallest numbers from a group is 21 while the mean of all the numbers of the group taken togethe is 25. If the mean of the numbers, leaving the smallest five out, is 27.5, how many numbers were there in the group in all?

Options

A. 12
B. 13
C. 8
D. 14

Answer

To solve the problem step by step, we need to use the information given about the means of different groups of numbers.

### Step 1: Calculate the total of the 5 smallest numbers.
The mean of the 5 smallest numbers is given as 21.
To find the total of these 5 numbers, we use the formula for mean:

\[ \text{Mean} = \frac{\text{Total}}{\text{Number of items}} \]

So, the total of the 5 smallest numbers is:

\[ \text{Total of 5 smallest numbers} = \text{Mean} \times \text{Number of items} = 21 \times 5 = 105 \]

**Hint:** Remember that the mean is the total divided by the number of items. You can rearrange this formula to find the total.

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### Step 2: Calculate the total of all numbers in the group.
The mean of all the numbers in the group is given as 25. Let \( n \) be the total number of numbers in the group. Thus, the total of all numbers is:

\[ \text{Total of all numbers} = \text{Mean} \times n = 25n \]

**Hint:** The total of all numbers can be expressed in terms of the mean and the total count of numbers.

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### Step 3: Calculate the total of the numbers excluding the 5 smallest.
The mean of the numbers excluding the smallest five is given as 27.5. The number of remaining numbers is \( n - 5 \). Therefore, the total of the remaining numbers is:

\[ \text{Total of remaining numbers} = 27.5 \times (n - 5) \]

**Hint:** When calculating the total for a subset of numbers, remember to adjust the count accordingly.

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### Step 4: Set up the equation.
The total of all numbers can also be expressed as the sum of the total of the 5 smallest numbers and the total of the remaining numbers:

\[ 25n = 105 + 27.5(n - 5) \]

**Hint:** You can combine the totals from different groups to form an equation.

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### Step 5: Simplify the equation.
Now, let's simplify the equation:

\[ 25n = 105 + 27.5n - 137.5 \]

Combine like terms:

\[ 25n = 27.5n - 32.5 \]

Rearranging gives:

\[ 25n - 27.5n = -32.5 \]

\[ -2.5n = -32.5 \]

**Hint:** When rearranging equations, make sure to keep track of the signs.

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### Step 6: Solve for \( n \).
Now, divide both sides by -2.5:

\[ n = \frac{-32.5}{-2.5} = 13 \]

**Hint:** When dividing, ensure that you are dividing both sides of the equation by the same number.

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### Conclusion
The total number of numbers in the group is \( n = 13 \).

**Final Answer:** There are 13 numbers in the group.

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