Let the three numbers be \(a\), \(b\), and \(c\). To make it easier to work with the range and difference, let's assume the numbers are arranged in ascending order: \(a < b < c\).

Understanding the Given Information

We are given three pieces of information about this set of three numbers:

• The mean of the three numbers is 15.
• The range of this data set is 8.
• The difference between the two smallest numbers is 1.

Translating Information into Equations

Let's translate each piece of information into a mathematical equation based on our ordered numbers \(a, b, c\).

1. Mean is 15: The mean is the sum of the numbers divided by the count of numbers. $$ \text{Mean} = \frac{a + b + c}{3} $$ Given that the mean is 15, we have: $$ \frac{a + b + c}{3} = 15 $$ Multiplying both sides by 3 gives us the sum of the numbers: $$ a + b + c = 45 \quad \text{(Equation 1)} $$

2. Range is 8: The range of a data set is the difference between the greatest number and the smallest number. Since we ordered the numbers as \(a < b < c\), the smallest number is \(a\) and the greatest number is \(c\). $$ \text{Range} = c - a $$ Given that the range is 8, we have: $$ c - a = 8 \quad \text{(Equation 2)} $$

3. Difference between the two smallest numbers is 1: The two smallest numbers are \(a\) and \(b\). Since \(a < b\), the difference is \(b - a\). $$ b - a = 1 \quad \text{(Equation 3)} $$

Solving the System of Equations

Now we have a system of three linear equations with three variables:

1. \(a + b + c = 45\)
2. \(c - a = 8\)
3. \(b - a = 1\)

We can use substitution to solve this system. From Equation 3, we can express \(b\) in terms of \(a\):

From Equation 2, we can express \(c\) in terms of \(a\):

Now substitute these expressions for \(b\) and \(c\) into Equation 1:

Combine the like terms:

Subtract 9 from both sides:

Divide by 3 to find the value of \(a\):

Now that we have the value of the smallest number, \(a\), we can find the values of \(b\) and \(c\).

Using \(b = a + 1\):

Using \(c = a + 8\):

The three numbers are 12, 13, and 20. Let's quickly check if they satisfy the original conditions:

• Mean: \(\frac{12 + 13 + 20}{3} = \frac{45}{3} = 15\). (Correct)
• Range: Greatest number (20) - Smallest number (12) = \(20 - 12 = 8\). (Correct)
• Difference between the two smallest numbers (12 and 13): \(13 - 12 = 1\). (Correct)

The question asks for the greatest of the three numbers. The numbers are 12, 13, and 20. The greatest number is 20.

Conclusion

Based on the given mean, range, and difference between the two smallest numbers, the three numbers are 12, 13, and 20. The greatest number in this set is 20.

Revision Table: Key Concepts

Additional Information: Data Set Properties

Data sets can be described using various measures. The mean and range are just two of them. Other important measures include:

• Median: The middle value in a data set when arranged in order. If there's an even number of values, it's the average of the two middle values.
• Mode: The value that appears most frequently in a data set. A set can have one mode, many modes, or no mode.
• Standard Deviation: A measure of how spread out the numbers are from the mean. A low standard deviation means numbers are close to the mean, while a high standard deviation means numbers are more spread out.

Understanding mean, range, median, and mode helps in getting a complete picture of the distribution and central tendency of a data set.