To solve for the numbers, we start by setting up the given information:
We are provided with four numbers: 127, [tex]\( x \)[/tex], [tex]\( x + 1 \)[/tex], and [tex]\( 2x \)[/tex]. We also know that the mean of these four numbers is 180.
The formula for the mean of four numbers is given by:
[tex]\[
\text{Mean} = \frac{\text{Sum of the numbers}}{4}
\][/tex]
Inserting the specific numbers into this formula, we have:
[tex]\[
180 = \frac{127 + x + (x + 1) + 2x}{4}
\][/tex]
First, combine the terms in the numerator:
[tex]\[
127 + x + x + 1 + 2x = 127 + 4x + 1
\][/tex]
Simplify further:
[tex]\[
127 + 4x + 1 = 128 + 4x
\][/tex]
Now, substitute this back into the mean equation:
[tex]\[
180 = \frac{128 + 4x}{4}
\][/tex]
To eliminate the denominator, multiply both sides by 4:
[tex]\[
180 \times 4 = 128 + 4x
\][/tex]
[tex]\[
720 = 128 + 4x
\][/tex]
Next, isolate the term with [tex]\( x \)[/tex] by subtracting 128 from both sides:
[tex]\[
720 - 128 = 4x
\][/tex]
[tex]\[
592 = 4x
\][/tex]
Finally, solve for [tex]\( x \)[/tex] by dividing both sides by 4:
[tex]\[
x = \frac{592}{4}
\][/tex]
[tex]\[
x = 148
\][/tex]
Now that we have [tex]\( x = 148 \)[/tex], we can find the other numbers:
1. [tex]\( x = 148 \)[/tex]
2. [tex]\( x + 1 = 148 + 1 = 149 \)[/tex]
3. [tex]\( 2x = 2 \times 148 = 296 \)[/tex]
Thus, the four numbers are:
- 127
- 148
- 149
- 296
