Solving $\frac{4}{3} = \frac{3.2}{y}$

Question

This is what I've tried.

$\frac{1}{3.2} \cdot \frac{4}{3} = \frac{3.2}{y} \cdot \frac{1}{3.2}$ to isolate $y$ via product and quotient identity.

When simplified this gives me $\frac{4}{9.6} = y$ so $y = 0.41\overline{6}$

However, the textbook gives an answer of $y=2.4$ and in the example they used the cross-product method which does follow, but what I did should also give me the same answer.

Answer 1

If you multiply $\dfrac{1}{3.2}$ both side: $$\dfrac{1}{3.2}\times\dfrac{4}{3}=\dfrac{3.2}{y}\times\dfrac{1}{3.2}\\\dfrac{4}{9.6}=\dfrac{1}{y}\\\dfrac{1}{2.4}=\dfrac{1}{y}\\y=\boxed{2.4}$$

Another way to do this is to use cross method: $$\dfrac{4}{3}=\dfrac{3.2}{y}\\ 4y=3.2\times 3 \\4y=9.6 \\ y=\boxed{2.4}$$

Answer 2

You arrive at $$\frac{4}{3\cdot 3.2}=\frac{1}{y}$$ so the correct result is the reciprocal of your result

Answer 3

When you got as far as $$ \frac{1}{y} = \frac{4}{3.2 \times 3} $$ (which is correct) you thought the left side was the same as $y$.