To solve the problem, let's break down the given sentence and translate it into a mathematical equation step by step.
### Given Sentence
"The quotient of 7 and a number is 10 less than 3 times the number."
### Step-by-Step Translation into an Equation
1. Identify the components:
- Quotient of 7 and a number: This can be written as [tex]\(\frac{7}{n}\)[/tex], where [tex]\(n\)[/tex] represents the unknown number.
- Is: This indicates that the two sides of the equation are equal, so we use the [tex]\(=\)[/tex] sign.
- 10 less than 3 times the number:
- 3 times the number is written as [tex]\(3n\)[/tex].
- 10 less than this is represented as [tex]\(3n - 10\)[/tex].
2. Combine these components into an equation:
- The sentence states that the quotient of 7 and a number is equal to 10 less than 3 times the number.
- Therefore, you can write:
[tex]\[
\frac{7}{n} = 3n - 10
\][/tex]
### Verification with Provided Options
Let's compare this equation with the given options to identify the correct one:
1. Option 1: [tex]\(\frac{7}{n} = 3(n - 10)\)[/tex]
- Expanding the right-hand side: [tex]\(3(n - 10) = 3n - 30\)[/tex], which is not equal to [tex]\(3n - 10\)[/tex].
2. Option 2: [tex]\(\frac{7}{n} = 10 - 3n\)[/tex]
- This is not in the correct form. It suggests that the quotient is equal to 10 minus 3 times the number, which is different from the original problem statement.
3. Option 3: [tex]\(\frac{n}{7} = 10 - 3n\)[/tex]
- The left-hand side is the quotient of the number and 7, not 7 and the number. This does not match the problem statement.
4. Option 4: [tex]\(\frac{7}{n} = 3n - 10\)[/tex]
- This matches our translated equation exactly.
### Conclusion
The equation that correctly represents the given number sentence is:
[tex]\[
\frac{7}{n} = 3n - 10
\][/tex]
Thus, the correct option is the fourth one.
