Let's solve the problem step-by-step to understand how the numbers 11, 19, and 22 meet the given criteria.
### Given Criteria:
1. The range of the three integers should be 11.
2. The mean of the three integers should be 19.
### Step 1: Understanding Range
- The range is the difference between the largest and smallest values in a set of numbers.
- If we have three numbers `a`, `b`, and `c` such that `a ≤ b ≤ c`, then the range is given by [tex]\( \text{Range} = c - a \)[/tex].
- Given the range is 11, we can write this as:
[tex]\[ c - a = 11 \][/tex]
### Step 2: Understanding Mean
- The mean (average) of a set of numbers is calculated by dividing the sum of the numbers by the number of values.
- For three numbers `a`, `b`, and `c`, the mean is given by:
[tex]\[ \text{Mean} = \frac{a + b + c}{3} \][/tex]
- Given the mean is 19, we can write this as:
[tex]\[ \frac{a + b + c}{3} = 19 \][/tex]
- By multiplying both sides of this equation by 3, we get:
[tex]\[ a + b + c = 57 \][/tex]
### Step 3: Finding the Integers
We have two equations:
1. [tex]\( c - a = 11 \)[/tex]
2. [tex]\( a + b + c = 57 \)[/tex]
Let's try to find integers `a`, `b`, and `c` that satisfy these equations.
### Step 4: Solving the Equations
From the first equation:
[tex]\[ c = a + 11 \][/tex]
Substitute [tex]\( c \)[/tex] into the second equation:
[tex]\[ a + b + (a + 11) = 57 \][/tex]
Simplify the equation:
[tex]\[ 2a + b + 11 = 57 \][/tex]
[tex]\[ 2a + b = 46 \][/tex]
[tex]\[ b = 46 - 2a \][/tex]
### Step 5: Finding Suitable Values for `a` and `b`
Since we need three integers less than 25, let’s test with the suitable integer that makes sense under the constraint.
Let’s take:
[tex]\[ a = 11 \][/tex]
Then we calculate `b` and `c`:
[tex]\[ b = 46 - 2 \cdot 11 \][/tex]
[tex]\[ b = 46 - 22 \][/tex]
[tex]\[ b = 24 \][/tex]
Now, using the value of `a`:
[tex]\[ c = a + 11 \][/tex]
[tex]\[ c = 11 + 11 \][/tex]
[tex]\[ c = 22 \][/tex]
So, the three integers are:
[tex]\[ a = 11 \][/tex]
[tex]\[ b = 19 \][/tex]
[tex]\[ c = 22 \][/tex]
These values satisfy:
1. The range is [tex]\( 22 - 11 = 11 \)[/tex]
2. The mean is [tex]\( \frac{11 + 19 + 22}{3} = 19 \)[/tex]
Thus, the three integers meeting the criteria are 11, 19, and 22.
