Let's break down the problem step-by-step to solve for [tex]\( x \)[/tex], the total amount that Isaiah spent at the store.
Isaiah spent \[tex]$19.60 on a gift, which was \(\frac{5}{7}\) of the total amount spent.
To find the total amount spent, we can set up the equation
\[ \frac{5}{7} x = 19.60 \]
Next, we can solve for \( x \).
1. Multiply both sides of the equation by the reciprocal of \(\frac{5}{7}\), which is \(\frac{7}{5}\):
\[ \left(\frac{5}{7}\right) x \cdot \left(\frac{7}{5}\right) = 19.60 \cdot \left(\frac{7}{5}\right) \]
Simplifying the left side of the equation:
\[ x = 19.60 \cdot \left(\frac{7}{5}\right) \]
2. Now, calculate the right side of the equation:
\[ x = 19.60 \cdot \left(\frac{7}{5}\right) \]
\[ x = 19.60 \cdot 1.4 \]
\[ x = 27.44 \]
Therefore, the total amount Isaiah spent is $[/tex]27.44.
Let's identify the correct statements from the options given:
1. [tex]\(\boxtimes \frac{5}{7}=\frac{x}{19.60}\)[/tex]
This option is incorrect. The correct relationship should be [tex]\(\frac{5}{7} x = 19.60\)[/tex].
2. [tex]\(\frac{5}{7} x=19.60\)[/tex]
This option is correct. It directly relates the fraction of the total amount spent to the given gift cost.
3. [tex]\(\frac{5}{7} \left(\frac{7}{5}\right) x=19.60 \left(\frac{7}{5}\right)\)[/tex]
This option is correct. It appropriately shows multiplying both sides by the reciprocal to solve for [tex]\( x \)[/tex].
4. [tex]\(\boxtimes \frac{5}{7}\left(\frac{7}{5}\right)=\frac{x}{19.80}\left(\frac{7}{5}\right)\)[/tex]
This option is incorrect. The equation has an incorrect transformation and incorrect value on the left and right sides.
5. [tex]\(x=27.44\)[/tex]
This option is correct. It provides the final solution for the total amount spent.
Thus, the three correct statements are:
- [tex]\(\frac{5}{7} x=19.60\)[/tex]
- [tex]\(\frac{5}{7} \left(\frac{7}{5}\right) x=19.60 \left(\frac{7}{5}\right)\)[/tex]
- [tex]\(x=27.44\)[/tex]
