Solve 0.2(1-s/500)=0.10-200(1-s/1000)/100s-1000

$$0.2 ( 1 - \frac { s } { 500 } ) = 0.10 - \frac { 200 ( 1 - \frac { s } { 1000 } ) } { 100 s - 1000 }$$

Solve for s

$$s = \frac{5 \sqrt{3001} + 255}{2} \approx 264.453459248$$

$$s=\frac{255-5\sqrt{3001}}{2}\approx -9.453459248$$

Steps Using the Quadratic Formula

Variable s cannot be equal to 10 since division by zero is not defined. Multiply both sides of the equation by 500\left(s-10\right), the least common multiple of 500,100s-1000.

$$0.2\left(1-\frac{s}{500}\right)\times 500\left(s-10\right)=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Multiply 0.2 and 500 to get 100.

$$100\left(1-\frac{s}{500}\right)\left(s-10\right)=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Use the distributive property to multiply 100 by 1-\frac{s}{500}.

$$\left(100+100\left(-\frac{s}{500}\right)\right)\left(s-10\right)=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Cancel out 500, the greatest common factor in 100 and 500.

$$\left(100+\frac{s}{-5}\right)\left(s-10\right)=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Use the distributive property to multiply 100+\frac{s}{-5} by s-10.

$$100s-1000+\frac{s}{-5}s-10\times \frac{s}{-5}=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Express \frac{s}{-5}s as a single fraction.

$$100s-1000+\frac{ss}{-5}-10\times \frac{s}{-5}=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Cancel out -5, the greatest common factor in 10 and -5.

$$100s-1000+\frac{ss}{-5}-2s=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Combine 100s and -2s to get 98s.

$$98s-1000+\frac{ss}{-5}=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Multiply s and s to get s^{2}.

$$98s-1000+\frac{s^{2}}{-5}=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Multiply 500 and 0.1 to get 50.

$$98s-1000+\frac{s^{2}}{-5}=50\left(s-10\right)-5\times 200\left(1-\frac{s}{1000}\right)$$

Use the distributive property to multiply 50 by s-10.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-5\times 200\left(1-\frac{s}{1000}\right)$$

Multiply -5 and 200 to get -1000.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000\left(1-\frac{s}{1000}\right)$$

Use the distributive property to multiply -1000 by 1-\frac{s}{1000}.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000-1000\left(-\frac{s}{1000}\right)$$

Multiply -1000 and -1 to get 1000.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000+1000\times \frac{s}{1000}$$

Express 1000\times \frac{s}{1000} as a single fraction.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000+\frac{1000s}{1000}$$

Cancel out 1000 and 1000.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000+s$$

Subtract 1000 from -500 to get -1500.

$$98s-1000+\frac{s^{2}}{-5}=50s-1500+s$$

Combine 50s and s to get 51s.

$$98s-1000+\frac{s^{2}}{-5}=51s-1500$$

Subtract 51s from both sides.

$$98s-1000+\frac{s^{2}}{-5}-51s=-1500$$

Combine 98s and -51s to get 47s.

$$47s-1000+\frac{s^{2}}{-5}=-1500$$

Add 1500 to both sides.

$$47s-1000+\frac{s^{2}}{-5}+1500=0$$

Add -1000 and 1500 to get 500.

$$47s+500+\frac{s^{2}}{-5}=0$$

Multiply both sides of the equation by -5.

$$-235s-2500+s^{2}=0$$

All equations of the form ax^{2}+bx+c=0 can be solved using the quadratic formula: \frac{-b±\sqrt{b^{2}-4ac}}{2a}. The quadratic formula gives two solutions, one when ± is addition and one when it is subtraction.

$$s^{2}-235s-2500=0$$

This equation is in standard form: ax^{2}+bx+c=0. Substitute 1 for a, -235 for b, and -2500 for c in the quadratic formula, \frac{-b±\sqrt{b^{2}-4ac}}{2a}.

$$s=\frac{-\left(-235\right)±\sqrt{\left(-235\right)^{2}-4\left(-2500\right)}}{2}$$

Square -235.

$$s=\frac{-\left(-235\right)±\sqrt{55225-4\left(-2500\right)}}{2}$$

Multiply -4 times -2500.

$$s=\frac{-\left(-235\right)±\sqrt{55225+10000}}{2}$$

Add 55225 to 10000.

$$s=\frac{-\left(-235\right)±\sqrt{65225}}{2}$$

Take the square root of 65225.

$$s=\frac{-\left(-235\right)±5\sqrt{2609}}{2}$$

The opposite of -235 is 235.

$$s=\frac{235±5\sqrt{2609}}{2}$$

Now solve the equation s=\frac{235±5\sqrt{2609}}{2} when ± is plus. Add 235 to 5\sqrt{2609}.

$$s=\frac{5\sqrt{2609}+235}{2}$$

Now solve the equation s=\frac{235±5\sqrt{2609}}{2} when ± is minus. Subtract 5\sqrt{2609} from 235.

$$s=\frac{235-5\sqrt{2609}}{2}$$

The equation is now solved.

$$s=\frac{5\sqrt{2609}+235}{2} s=\frac{235-5\sqrt{2609}}{2}$$

Steps for Completing the Square

Variable s cannot be equal to 10 since division by zero is not defined. Multiply both sides of the equation by 500\left(s-10\right), the least common multiple of 500,100s-1000.

$$0.2\left(1-\frac{s}{500}\right)\times 500\left(s-10\right)=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Multiply 0.2 and 500 to get 100.

$$100\left(1-\frac{s}{500}\right)\left(s-10\right)=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Use the distributive property to multiply 100 by 1-\frac{s}{500}.

$$\left(100+100\left(-\frac{s}{500}\right)\right)\left(s-10\right)=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Cancel out 500, the greatest common factor in 100 and 500.

$$\left(100+\frac{s}{-5}\right)\left(s-10\right)=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Use the distributive property to multiply 100+\frac{s}{-5} by s-10.

$$100s-1000+\frac{s}{-5}s-10\times \frac{s}{-5}=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Express \frac{s}{-5}s as a single fraction.

$$100s-1000+\frac{ss}{-5}-10\times \frac{s}{-5}=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Cancel out -5, the greatest common factor in 10 and -5.

$$100s-1000+\frac{ss}{-5}-2s=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Combine 100s and -2s to get 98s.

$$98s-1000+\frac{ss}{-5}=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Multiply s and s to get s^{2}.

$$98s-1000+\frac{s^{2}}{-5}=500\left(s-10\right)\times 0.1-5\times 200\left(1-\frac{s}{1000}\right)$$

Multiply 500 and 0.1 to get 50.

$$98s-1000+\frac{s^{2}}{-5}=50\left(s-10\right)-5\times 200\left(1-\frac{s}{1000}\right)$$

Use the distributive property to multiply 50 by s-10.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-5\times 200\left(1-\frac{s}{1000}\right)$$

Multiply -5 and 200 to get -1000.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000\left(1-\frac{s}{1000}\right)$$

Use the distributive property to multiply -1000 by 1-\frac{s}{1000}.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000-1000\left(-\frac{s}{1000}\right)$$

Multiply -1000 and -1 to get 1000.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000+1000\times \frac{s}{1000}$$

Express 1000\times \frac{s}{1000} as a single fraction.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000+\frac{1000s}{1000}$$

Cancel out 1000 and 1000.

$$98s-1000+\frac{s^{2}}{-5}=50s-500-1000+s$$

Subtract 1000 from -500 to get -1500.

$$98s-1000+\frac{s^{2}}{-5}=50s-1500+s$$

Combine 50s and s to get 51s.

$$98s-1000+\frac{s^{2}}{-5}=51s-1500$$

Subtract 51s from both sides.

$$98s-1000+\frac{s^{2}}{-5}-51s=-1500$$

Combine 98s and -51s to get 47s.

$$47s-1000+\frac{s^{2}}{-5}=-1500$$

Add 1000 to both sides.

$$47s+\frac{s^{2}}{-5}=-1500+1000$$

Add -1500 and 1000 to get -500.

$$47s+\frac{s^{2}}{-5}=-500$$

Multiply both sides of the equation by -5.

$$-235s+s^{2}=2500$$

Quadratic equations such as this one can be solved by completing the square. In order to complete the square, the equation must first be in the form x^{2}+bx=c.

$$s^{2}-235s=2500$$

Divide -235, the coefficient of the x term, by 2 to get -\frac{235}{2}. Then add the square of -\frac{235}{2} to both sides of the equation. This step makes the left hand side of the equation a perfect square.

$$s^{2}-235s+\left(-\frac{235}{2}\right)^{2}=2500+\left(-\frac{235}{2}\right)^{2}$$

Square -\frac{235}{2} by squaring both the numerator and the denominator of the fraction.

$$s^{2}-235s+\frac{55225}{4}=2500+\frac{55225}{4}$$

Add 2500 to \frac{55225}{4}.

$$s^{2}-235s+\frac{55225}{4}=\frac{65225}{4}$$

Factor s^{2}-235s+\frac{55225}{4}. In general, when x^{2}+bx+c is a perfect square, it can always be factored as \left(x+\frac{b}{2}\right)^{2}.

$$\left(s-\frac{235}{2}\right)^{2}=\frac{65225}{4}$$

Take the square root of both sides of the equation.

$$\sqrt{\left(s-\frac{235}{2}\right)^{2}}=\sqrt{\frac{65225}{4}}$$

Simplify.

$$s-\frac{235}{2}=\frac{5\sqrt{2609}}{2} s-\frac{235}{2}=-\frac{5\sqrt{2609}}{2}$$

Add \frac{235}{2} to both sides of the equation.

$$s=\frac{5\sqrt{2609}+235}{2} s=\frac{235-5\sqrt{2609}}{2}$$