Solve for x: x = [1,000,000 × 12% + (4,500,000 − 1,000,000 − 100,000 − 50,000 − x − 1,000,000) × 10%] × 80%.

Step-by-step solution

1. 1

   Convert the percentages and collect the constants inside the inner bracket. Writing $12\%=0.12$, $10\%=0.10$, $80\%=0.80$, the constants inside the inner bracket give

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

   so that bracket is $2{,}350{,}000-x$. Note the unknown appears on both sides — the equation is self-referential, which is why it cannot simply be evaluated.

2. 2

   Simplify the outer bracket. Since $1{,}000{,}000\times 0.12=120{,}000$ and $0.10(2{,}350{,}000-x)=235{,}000-0.10x$,

   $x=\bigl[120{,}000+235{,}000-0.10x\bigr]\times 0.80=\bigl[355{,}000-0.10x\bigr]\times 0.80.$

3. 3

   Distribute the 80%.

   $x=355{,}000(0.80)-0.10x(0.80)=284{,}000-0.08x.$

   The coefficient $0.08$ is the product $0.10\times0.80$: the unknown is discounted twice on its way through the formula.

4. 4

   Gather the $x$-terms and divide.

   $x+0.08x=284{,}000\;\Longrightarrow\;1.08x=284{,}000\;\Longrightarrow\;x=\frac{284{,}000}{1.08}=\frac{7{,}100{,}000}{27}.$

5. 5

   Give the decimal value and check. $x=262{,}962.962962\ldots\approx 262{,}962.96$. Substituting the exact fraction back with rational arithmetic: the inner bracket is $2{,}350{,}000-\tfrac{7{,}100{,}000}{27}$, and the whole right-hand side evaluates to exactly $\tfrac{7{,}100{,}000}{27}$ — the equation balances exactly, not just to rounding.