Finding the Total Number of Items in a Group Using Mean

This problem involves using the concept of mean (average) to determine the total number of items in a group when information about the mean of subsets of the group is provided. The mean of a set of numbers is calculated as the sum of the numbers divided by the count of the numbers.

Let's break down the given information and set up equations based on the definition of the mean.

Understanding the Given Information

• The mean of the 5 smallest numbers from a group is 15.
• The mean of all the numbers of the group taken together is 17.
• The mean of the numbers leaving the smallest five out is 18.25.

Setting Up the Equations

Let \( N \) be the total number of numbers in the group that we want to find.

Let \( S_{\text{total}} \) be the sum of all \( N \) numbers in the group.

Let \( S_{\text{small\_5}} \) be the sum of the 5 smallest numbers.

Let \( S_{\text{rest}} \) be the sum of the numbers remaining after removing the smallest five. The number of remaining items is \( N - 5 \).

Based on the given means, we can write the following equations:

1. Mean of the 5 smallest numbers: \( 15 = \frac{S_{\text{small\_5}}}{5} \) From this, the sum of the 5 smallest numbers is: \( S_{\text{small\_5}} = 15 \times 5 = 75 \)
2. Mean of all \( N \) numbers: \( 17 = \frac{S_{\text{total}}}{N} \) From this, the sum of all numbers is: \( S_{\text{total}} = 17N \)
3. Mean of the remaining \( N-5 \) numbers: \( 18.25 = \frac{S_{\text{rest}}}{N-5} \) From this, the sum of the remaining numbers is: \( S_{\text{rest}} = 18.25 \times (N-5) \)

Relating the Sums

The sum of all numbers in the group is the sum of the 5 smallest numbers plus the sum of the remaining numbers.

\( S_{\text{total}} = S_{\text{small\_5}} + S_{\text{rest}} \)

Solving for the Total Number (N)

Now, substitute the expressions for \( S_{\text{total}} \), \( S_{\text{small\_5}} \), and \( S_{\text{rest}} \) from our equations into the relationship above:

\( 17N = 75 + 18.25 \times (N-5) \)

Let's solve this linear equation for \( N \):

\( 17N = 75 + 18.25N - 18.25 \times 5 \)

\( 17N = 75 + 18.25N - 91.25 \)

\( 17N = 18.25N - 16.25 \)

Now, rearrange the terms to isolate \( N \):

\( 16.25 = 18.25N - 17N \)

\( 16.25 = (18.25 - 17)N \)

\( 16.25 = 1.25N \)

To find \( N \), divide 16.25 by 1.25:

\( N = \frac{16.25}{1.25} \)

To make the division easier, we can multiply both the numerator and the denominator by 100 to remove decimals:

\( N = \frac{1625}{125} \)

Now, perform the division:

\( N = 13 \)

So, there were 13 numbers in the group in all.

Verification

Let's check if our answer \( N=13 \) is consistent with the given information.

• Total numbers \( N = 13 \).
• Smallest 5 numbers: sum = 75, mean = 75/5 = 15 (Correct).
• Numbers leaving the smallest five out: \( 13 - 5 = 8 \) numbers.
• Mean of these 8 numbers = 18.25. Their sum should be \( 18.25 \times 8 \). \( 18.25 \times 8 = 146 \)
• Sum of all numbers = sum of smallest 5 + sum of the rest \( S_{\text{total}} = 75 + 146 = 221 \)
• Mean of all 13 numbers = \( S_{\text{total}} / N = 221 / 13 \). \( 221 \div 13 = 17 \) (Correct).

The calculations match all the given conditions, confirming that the total number of numbers in the group is 13.

Revision Table: Key Concepts

Additional Information: Applying Mean Concepts

Problems involving means of subsets of a group are common in statistics and quantitative aptitude. The key to solving these problems is understanding that the sum of the values in a subset is the product of its mean and the number of values in that subset. Also, the sum of values in non-overlapping subsets that cover the whole group equals the total sum of the group.

In this problem, the group is divided into two non-overlapping parts: the 5 smallest numbers and the remaining numbers. The sum of these two parts gives the sum of the entire group. Setting up equations based on the given mean values and the definition \( \text{Sum} = \text{Mean} \times \text{Count} \) allows us to form a single equation with one unknown (the total count, \( N \)) which can then be solved.

This approach can be generalized to problems with more subsets, as long as enough information is given to set up a system of equations that can be solved for the unknown quantities, such as the total count or the mean of a particular subset.