Three-fifths of the members of the Spanish club are girls. There are a total of 30 girls in the Spanish club.
Which statements can be used to solve for [tex]$x$[/tex], the total number of members in the Spanish club? Select three options.
[tex]$\frac{3}{5} x=30$[/tex]
[tex]$\frac{3}{5}=\frac{x}{30}$[/tex]
[tex]$\left(\frac{5}{3}\right) \frac{3}{5} x=30\left(\frac{3}{5}\right)$[/tex]
[tex]$\left(\frac{5}{3}\right) \frac{3}{5} x=30\left(\frac{5}{3}\right)$[/tex]
[tex]$x=50$[/tex]
Answer (2)
Set up the equation representing the problem: 5 3 x = 30 .
Multiply both sides of the equation by 3 5 to isolate x : ( 3 5 ) 5 3 x = 30 ( 3 5 ) .
Simplify the equation to find x = 50 .
The correct statements are: 5 3 x = 30 , ( 3 5 ) 5 3 x = 30 ( 3 5 ) , and x = 50 .
5 3 x = 30 , ( 3 5 ) 5 3 x = 30 ( 3 5 ) , x = 50
Explanation
Understanding the Problem Let's analyze the problem. We know that three-fifths of the members of the Spanish club are girls, and there are 30 girls in the club. We want to find the total number of members, which we'll call x .
Setting up the Equation We can set up an equation to represent this situation. Since 5 3 of the members are girls, we can write the equation: 5 3 x = 30 This equation states that three-fifths of the total number of members ( x ) is equal to the number of girls (30).
Isolating x To solve for x , we need to isolate it. We can do this by multiplying both sides of the equation by the reciprocal of 5 3 , which is 3 5 : ( 3 5 ) 5 3 x = 30 ( 3 5 )
Solving for x Now, let's simplify the equation: ( 3 5 ) 5 3 x = 30 ( 3 5 ) x = 30 ( 3 5 ) x = 3 30 × 5 x = 3 150 x = 50 So, the total number of members in the Spanish club is 50.
Identifying Correct Statements Now, let's check which of the given statements can be used to solve for x :
5 3 x = 30 - This is the equation we set up, so it's correct.
5 3 = 30 x - This is incorrect. It states that three-fifths is equal to the ratio of the total members to the number of girls, which is not what we want.
( 3 5 ) 5 3 x = 30 ( 5 3 ) - This is incorrect. It multiplies both sides of the equation by 3 5 on the left side, but multiplies by 5 3 on the right side, which is not the correct way to isolate x .
( 3 5 ) 5 3 x = 30 ( 3 5 ) - This is correct. It multiplies both sides of the equation by 3 5 , which is the correct step to isolate x .
x = 50 - This is the solution we found, so it's correct.
Examples
Imagine you're baking a cake and the recipe says that three-fifths of the flour should be all-purpose flour, and you know you need 30 cups of all-purpose flour. To find out the total amount of flour needed for the recipe, you can use the same math we used here. Setting up the equation $\frac{3}{5}x = 30$ helps you determine the total amount of flour, $x$, needed for your cake. This kind of proportional reasoning is useful in many real-life situations, from cooking to mixing chemicals in a lab!
The correct statements to solve for the total number of members, x , in the Spanish club are: 5 3 x = 30 , ( 3 5 ) 5 3 x = 30 ( 3 5 ) , and x = 50 . After setting up the equation and isolating x , we find that x = 50 .
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