To solve for [tex]\( x \)[/tex], the total number of members in the Spanish club, we need to use the given information and set up the appropriate equations. Here are the steps to determine which statements are useful for solving [tex]\( x \)[/tex]:
1. Given Information:
- [tex]\( \frac{3}{5} \)[/tex] of the members are girls.
- There are a total of 30 girls in the club.
2. Formulating the Equation:
- Let [tex]\( x \)[/tex] be the total number of members in the club.
- Since three-fifths of the members are girls and the number of girls is 30, we can set up the following equation:
[tex]\[
\frac{3}{5}x = 30
\][/tex]
- This equation can be used directly to solve for [tex]\( x \)[/tex].
3. Solving the Equation:
- To isolate [tex]\( x \)[/tex], multiply both sides by the reciprocal of [tex]\( \frac{3}{5} \)[/tex], which is [tex]\( \frac{5}{3} \)[/tex]:
[tex]\[
x = 30 \times \frac{5}{3}
\][/tex]
[tex]\[
x = 30 \times \frac{5}{3}
\][/tex]
[tex]\[
x = 50
\][/tex]
- Therefore, the total number of members in the club is 50.
4. Analyzing the Options:
- Option 1: [tex]\( \frac{3}{5} x = 30 \)[/tex]
- Correct. This is the direct equation we used to solve for [tex]\( x \)[/tex].
- Option 2: [tex]\( \frac{3}{5} = \frac{x}{30} \)[/tex]
- Incorrect. This equation is not equivalent to the original problem setup. It compares a fraction of the total members to the number of girls incorrectly.
- Option 3: [tex]\( \left(\frac{5}{3}\right) \frac{3}{5} x = 30 \left(\frac{3}{5}\right) \)[/tex]
- Incorrect. This equation incorrectly applies the scaling factor to both sides without maintaining equality.
- Option 4: [tex]\( \left(\frac{5}{3}\right) \frac{3}{5} x = 30 \left(\frac{5}{3}\right) \)[/tex]
- Correct. This isolates [tex]\( x \)[/tex] correctly by equating [tex]\( x \)[/tex] to [tex]\( 50 \)[/tex].
- Option 5: [tex]\( x = 50 \)[/tex]
- Correct. This is the direct solution to the equation after performing the operations.
5. Conclusion:
- The three correct statements that can be used to solve for [tex]\( x \)[/tex] are:
- [tex]\( \frac{3}{5} x = 30 \)[/tex]
- [tex]\( \left(\frac{5}{3}\right) \frac{3}{5} x = 30 \left(\frac{5}{3}\right) \)[/tex]
- [tex]\( x = 50 \)[/tex]
