Solve the equation for [tex] y [/tex]:

[tex]\[ 640 : \left( \frac{2y}{5} \right) \cdot y = 640 \][/tex]

Note: Ensure the colon symbol (:) denotes division and the multiplication is clear within the context of the equation.



Answer :

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]