Solve for X: 600 = 2210 - (2000/180)(5/3)X.

Step-by-step solution

1. 1

   Combine the two fractions into a single coefficient. Reduce the first fraction by dividing top and bottom by $20$, then multiply:

   $\frac{2000}{180}=\frac{100}{9},\qquad \frac{100}{9}\cdot\frac53=\frac{500}{27}.$

   Simplifying before multiplying keeps the numbers small; multiplying first would give $\frac{10000}{540}$, the same value with more to reduce.

2. 2

   Predict the sign of the answer. The equation reads $600=2210-\frac{500}{27}X$. Since $600$ is less than $2210$, the subtracted term must be positive, so $X>0$. (In the companion problem where the left side is $2400$, the same reasoning forces $X<0$.)

3. 3

   Isolate the $X$ term. Subtracting $2210$ from both sides:

   $600-2210=-\frac{500}{27}X\;\Longrightarrow\;-1610=-\frac{500}{27}X.$

   Both sides are negative, which is consistent with the sign prediction.

4. 4

   Divide by the fractional coefficient. Multiplying by the reciprocal $-\tfrac{27}{500}$:

   $X=(-1610)\left(-\frac{27}{500}\right)=\frac{43470}{500}=\frac{4347}{50}=86.94.$

   The fraction reduces by $10$; since $4347=3^{4}\times 53.67\ldots$ shares no factor with $50=2\times5^{2}$, $\tfrac{4347}{50}$ is in lowest terms.

5. 5

   Substitute back to confirm. With $X=\tfrac{4347}{50}$,

   $\frac{500}{27}\times\frac{4347}{50}=1610,\qquad 2210-1610=600 ✓.$

   The positive answer matches the sign prediction from step 2.