Solve for x: x = 1,000,000 × 12% + (4,500,000 − 1,000,000 − 100,000 − 50,000 − x − 1,000,000) × 10%.

Step-by-step solution

1. 1

   Collapse the constants in the bracket.

   $4{,}500{,}000-1{,}000{,}000-100{,}000-50{,}000-1{,}000{,}000=2{,}350{,}000,$

   so the bracket is $2{,}350{,}000-x$ and the equation reads

   $x=1{,}000{,}000(0.12)+0.10\bigl(2{,}350{,}000-x\bigr).$

2. 2

   Expand both products. With $1{,}000{,}000(0.12)=120{,}000$ and $0.10(2{,}350{,}000-x)=235{,}000-0.10x$,

   $x=120{,}000+235{,}000-0.10x=355{,}000-0.10x.$

   The unknown appears on both sides because $x$ is subtracted from the pool before the $10\%$ share of it is taken.

3. 3

   Move the $x$-term across.

   $x+0.10x=355{,}000\;\Longrightarrow\;1.10x=355{,}000.$

4. 4

   Divide and simplify to an exact fraction.

   $x=\frac{355{,}000}{1.10}=\frac{3{,}550{,}000}{11}=322{,}727.2727\ldots\approx 322{,}727.27.$

5. 5

   Verify and compare. Substituting exactly: $2{,}350{,}000-\tfrac{3{,}550{,}000}{11}=\tfrac{22{,}300{,}000}{11}$, and $120{,}000+0.10\cdot\tfrac{22{,}300{,}000}{11}=\tfrac{3{,}550{,}000}{11}$ ✓. Applying an extra $80\%$ factor to the whole bracket would change the divisor from $1.10$ to $1.08$ and lower the answer to $262{,}962.96$ — a $19\%$ drop from a factor that looks like it should only cost $20\%$ of the total.