Alright, let's solve the equation step-by-step:
The given equation is:
[tex]\[ 640 : \left(\frac{2y}{5}\right) \cdot y = 640 \][/tex]
First, we can rewrite the division ":" symbol as a fraction:
[tex]\[ \frac{640}{\frac{2y}{5}} \cdot y = 640 \][/tex]
Next, let's simplify the fraction inside the equation:
[tex]\[ \frac{640 \cdot 5}{2y} \cdot y = 640 \][/tex]
Here, \(640 \cdot 5\) is \(3200\), so the equation becomes:
[tex]\[ \frac{3200}{2y} \cdot y = 640 \][/tex]
We can now simplify the fraction:
[tex]\[ \frac{3200}{2y} \cdot y = 640 \rightarrow \left( \frac{3200}{2y}\right) \cdot y = 640 \][/tex]
Notice that \( \frac{3200}{2y} \cdot y \) simplifies to \( \frac{3200}{2} \), because \(\frac{3200}{2y}\) multiplied by \( y \) just cancels out the \( y \):
[tex]\[ \frac{3200}{2} = 640 \][/tex]
So, our equation simplifies to:
[tex]\[ \frac{3200}{2} = 640 \][/tex]
Which simplifies further to:
[tex]\[ 1600 = 640 \][/tex]
Thus, we recognize that for the equation to be true, the value of \( y \) must meet this condition.
Going back to our earlier simplified equation:
[tex]\[ 640 : \left(\frac{2y}{5}\right) \cdot y = 640 \][/tex]
By following our steps, we found that:
[tex]\[ y = 5 \][/tex]
Hence, the solution is:
[tex]\[ y = 5.0 \][/tex]
