Solve 2/5y5/3y=5/4

$$\frac{ 2 }{ 5y } \frac{ 5 }{ 3y } = \frac{ 5 }{ 4 }$$

Solve for y

$$y=\frac{2\sqrt{30}}{15}\approx 0.730296743$$

$$y=-\frac{2\sqrt{30}}{15}\approx -0.730296743$$

Steps by Finding Square Root

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 60y, the least common multiple of 5y,3y,4.

$$12\times 2\times \frac{5}{3y}=75y$$

Multiply 12 and 2 to get 24.

$$24\times \frac{5}{3y}=75y$$

Express 24\times \frac{5}{3y} as a single fraction.

$$\frac{24\times 5}{3y}=75y$$

Cancel out 3 in both numerator and denominator.

$$\frac{5\times 8}{y}=75y$$

Multiply 5 and 8 to get 40.

$$\frac{40}{y}=75y$$

Subtract 75y from both sides.

$$\frac{40}{y}-75y=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -75y times \frac{y}{y}.

$$\frac{40}{y}+\frac{-75yy}{y}=0$$

Since \frac{40}{y} and \frac{-75yy}{y} have the same denominator, add them by adding their numerators.

$$\frac{40-75yy}{y}=0$$

Do the multiplications in 40-75yy.

$$\frac{40-75y^{2}}{y}=0$$

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by y.

$$40-75y^{2}=0$$

Subtract 40 from both sides. Anything subtracted from zero gives its negation.

$$-75y^{2}=-40$$

Divide both sides by -75.

$$y^{2}=\frac{-40}{-75}$$

Reduce the fraction \frac{-40}{-75} to lowest terms by extracting and canceling out -5.

$$y^{2}=\frac{8}{15}$$

Take the square root of both sides of the equation.

$$y=\frac{2\sqrt{30}}{15} y=-\frac{2\sqrt{30}}{15}$$

Steps Using the Quadratic Formula

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by 60y, the least common multiple of 5y,3y,4.

$$12\times 2\times \frac{5}{3y}=75y$$

Multiply 12 and 2 to get 24.

$$24\times \frac{5}{3y}=75y$$

Express 24\times \frac{5}{3y} as a single fraction.

$$\frac{24\times 5}{3y}=75y$$

Cancel out 3 in both numerator and denominator.

$$\frac{5\times 8}{y}=75y$$

Multiply 5 and 8 to get 40.

$$\frac{40}{y}=75y$$

Subtract 75y from both sides.

$$\frac{40}{y}-75y=0$$

To add or subtract expressions, expand them to make their denominators the same. Multiply -75y times \frac{y}{y}.

$$\frac{40}{y}+\frac{-75yy}{y}=0$$

Since \frac{40}{y} and \frac{-75yy}{y} have the same denominator, add them by adding their numerators.

$$\frac{40-75yy}{y}=0$$

Do the multiplications in 40-75yy.

$$\frac{40-75y^{2}}{y}=0$$

Variable y cannot be equal to 0 since division by zero is not defined. Multiply both sides of the equation by y.

$$40-75y^{2}=0$$

Quadratic equations like this one, with an x^{2} term but no x term, can still be solved using the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}, once they are put in standard form: ax^{2}+bx+c=0.

$$-75y^{2}+40=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute -75 for a, 0 for b, and 40 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$y=\frac{0±\sqrt{0^{2}-4\left(-75\right)\times 40}}{2\left(-75\right)}$$

Square 0.

$$y=\frac{0±\sqrt{-4\left(-75\right)\times 40}}{2\left(-75\right)}$$

Multiply -4 times -75.

$$y=\frac{0±\sqrt{300\times 40}}{2\left(-75\right)}$$

Multiply 300 times 40.

$$y=\frac{0±\sqrt{12000}}{2\left(-75\right)}$$

Take the square root of 12000.

$$y=\frac{0±20\sqrt{30}}{2\left(-75\right)}$$

Multiply 2 times -75.

$$y=\frac{0±20\sqrt{30}}{-150}$$

Now solve the equation y=\frac{0±20\sqrt{30}}{-150} when ± is plus.

$$y=-\frac{2\sqrt{30}}{15}$$

Now solve the equation y=\frac{0±20\sqrt{30}}{-150} when ± is minus.

$$y=\frac{2\sqrt{30}}{15}$$

The equation is now solved.

$$y=-\frac{2\sqrt{30}}{15} y=\frac{2\sqrt{30}}{15}$$