Solve for x: 0.8x + (10000 - x)(0.21) = 10000(0.25). Give the exact value as a fraction and a decimal.

Step-by-step solution

1. 1

   Read the structure: this is a weighted-average (mixture) equation. An amount $x$ carrying rate $0.8$ is blended with the remaining $10000-x$ carrying rate $0.21$, and the blend must average $0.25$ over the whole $10000$. Recognising the shape tells you in advance that the answer must lie between $0$ and $10000$, and closer to $0$ because $0.25$ is much nearer $0.21$ than $0.8$.

2. 2

   Evaluate the right-hand side and distribute on the left. First $10000\times 0.25=2500$. Then

   $0.21(10000-x)=2100-0.21x,$

   so the equation is

   $0.8x+2100-0.21x=2500.$

   Distributing over the whole bracket, including the $-x$, is the step to be careful with.

3. 3

   Combine like terms. The two $x$ terms merge:

   $0.8x-0.21x=0.59x,$

   leaving

   $0.59x+2100=2500.$

   The coefficient $0.59$ is the spread between the two rates, $0.80-0.21$ — a useful check on the arithmetic.

4. 4

   Isolate $x$ and keep the answer exact. Subtracting $2100$ gives $0.59x=400$, and dividing:

   $x=\frac{400}{0.59}=\frac{40000}{59}\approx 677.966.$

   Multiplying numerator and denominator by $100$ turns the decimal division into an exact fraction; $40000/59$ does not reduce because $59$ is prime and does not divide $40000$.

5. 5

   Check the blend. With $x=\tfrac{40000}{59}$ the other part is $10000-\tfrac{40000}{59}=\tfrac{550000}{59}$, and

   $0.8\cdot\frac{40000}{59}+0.21\cdot\frac{550000}{59}=\frac{32000+115500}{59}=\frac{147500}{59}=2500 ✓.$

   The result also passes the plausibility test from step 1: $678$ is a small slice of $10000$, as expected.