Solve 0.6x - 402.8 = 5x/9 - 360 for x.

Step-by-step solution

1. 1

   Gather the $x$ terms on the left and the constants on the right.

   $0.6x-\frac{5x}{9}=402.8-360=42.8$

   Both sides are now single quantities, and the whole problem reduces to finding the coefficient on the left.

2. 2

   Put the two coefficients over a common denominator. $0.6=\frac35=\frac{27}{45}$ and $\frac59=\frac{25}{45}$, so

   $0.6-\frac59=\frac{27-25}{45}=\frac{2}{45}$

   The two coefficients are nearly equal — their difference is only about $0.0444$ — which is exactly why the solution will turn out to be large.

3. 3

   Write the reduced equation.

   $\frac{2}{45}x=42.8$

4. 4

   Solve exactly, avoiding decimals. Convert $42.8=\frac{214}{5}$ and multiply by the reciprocal $\frac{45}{2}$:

   $x=\frac{214}{5}\cdot\frac{45}{2}=214\cdot\frac{9}{2}=107\cdot 9=963$

   The answer is an exact integer, so the source numbers were chosen deliberately.

5. 5

   Check each side separately. Left: $0.6(963)-402.8=577.8-402.8=175$. Right: $\frac{5(963)}{9}-360=535-360=175$. Both give $175$, so $x=963$.