To solve this problem, we start with the information provided. We know that the number of people who ate meatballs (which is given as 65) is 11 less than four-fifths of the total number of people at the party (which we will denote as \( x \)).
Let's break down the statement mathematically:
- Four-fifths of the total number of people is represented as \( \frac{4}{5}x \).
- Eleven less than that value is represented as \( \frac{4}{5}x - 11 \).
According to the problem, this is equal to the number of people who ate meatballs, which is 65. Therefore, we can set up the following equation:
\[ \frac{4}{5}x - 11 = 65 \]
Now, to solve for \( x \):
- Add 11 to both sides of the equation:
\[ \frac{4}{5}x - 11 + 11 = 65 + 11 \] \[ \frac{4}{5}x = 76 \]
- Next, multiply both sides by the reciprocal of \( \frac{4}{5} \) to solve for \( x \):
\[ x = 76 \cdot \frac{5}{4} \]
- Calculate \( 76 \cdot \frac{5}{4} \):
\[ x = 76 \cdot 1.25 \] \[ x = 95 \]
Thus, the total number of people at the party is \( x = 95 \).
One-Step Equation
To write a one-step equation that has the same solution, we can isolate \( x \) in the original equation. Since we have:
\[ \frac{4}{5}x = 76 \]
We can rewrite it in a one-step form by multiplying both sides by \( \frac{5}{4} \):
\[ x = 76 \cdot \frac{5}{4} \]
This is the one-step equation, but if you prefer a more standard form to show this as a simple equation:
\[ x = 95 \]
This gives us a clear, straightforward solution for the number of people at the party.