0.05x + 0.08(10,000 - x) = 680

February 17, 2026

Understanding the Equation: Solving 0.05x + 0.08(10,000 - x) = 680

When faced with a linear equation like 0.05x + 0.08(10,000 - x) = 680, it might seem daunting at first—but breaking it down step by step makes solving it straightforward. Whether you're a student learning algebra or a professional working through real-world financial calculations, understanding how to solve equations systematically is essential.

What is the Equation?

The equation 0.05x + 0.08(10,000 - x) = 680 models a weighted average scenario, commonly used in finance, cost analysis, or profit modeling. It balances two different rates applied to parts of a total quantity.

In this case:- 0.05x represents 5% of a quantity x- 0.08(10,000 - x) represents 8% applied to the remaining (10,000 - x)

Their sum equals 680, likely modeling a total weighted cost or return.

Step 1: Expand the Expression

Start by distributing 0.08 across the parenthesis:

\[0.05x + 0.08 \ imes 10,000 - 0.08x = 680\]

Calculate \(0.08 \ imes 10,000 = 800\), so the equation becomes:

\[0.05x + 800 - 0.08x = 680\]

Step 2: Combine Like Terms

Combine the x terms:

\[(0.05 - 0.08)x + 800 = 680 \-0.03x + 800 = 680\]

Step 3: Isolate the Variable

Subtract 800 from both sides:

\[-0.03x = 680 - 800 \-0.03x = -120\]

Now divide both sides by \(-0.03\):

\[x = \frac{-120}{-0.03} = 4,000\]

Step 4: Interpret the Solution

The solution \(x = 4,000\) means:

40% of the total (10,000) is 4,000- 60% of the total (6,000) applies the 8% rate

Verify:

\[0.05 \ imes 4,000 = 200 \0.08 \ imes 6,000 = 480 \200 + 480 = 680\]

The equation balances correctly.

Why This Equation Matters

This type of equation arises in many practical situations:

Finance: Calculating weighted average costs or returns on mixed investments- Business: Determining break-even points when different margins apply to portions of sales- Engineering: Balancing material usage or efficiency across components

Modeling such scenarios with linear equations allows precise decision-making.

Final Thoughts

Solving 0.05x + 0.08(10,000 - x) = 680 involves basic algebraic techniques—expanding, combining like terms, and isolating variables. This process strengthens numerical reasoning and problem-solving skills useful across fields.

Try it yourself: Practice with different totals and percentages to master the skill. Understanding these fundamentals empowers you to tackle complex real-world problems with confidence.

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