What value of t is the solution to the equation \(0.8\mathrm{t} - 0.46 = 8(\mathrm{t} - 0.001) + 1.9\)?

1. TRANSLATE the problem information

• Given equation: \(0.8\mathrm{t} - 0.46 = 8(\mathrm{t} - 0.001) + 1.9\)
• Need to find: The value of t that makes this equation true

2. INFER the solution approach

• The right side has parentheses, so distribute first
• Then collect like terms to isolate t
• This is a standard linear equation solving process

3. SIMPLIFY by applying the distributive property

• Right side: \(8(\mathrm{t} - 0.001) + 1.9 = 8\mathrm{t} - 0.008 + 1.9\)
• Combine the constants: \(-0.008 + 1.9 = 1.892\)
• New equation: \(0.8\mathrm{t} - 0.46 = 8\mathrm{t} + 1.892\)

4. SIMPLIFY by collecting like terms

• Subtract 0.8t from both sides: \(-0.46 = 7.2\mathrm{t} + 1.892\)
• Subtract 1.892 from both sides: \(-0.46 - 1.892 = 7.2\mathrm{t}\)
• Calculate: \(-2.352 = 7.2\mathrm{t}\)

5. SIMPLIFY to solve for t

• Divide both sides by 7.2: \(\mathrm{t} = -2.352/7.2\) (use calculator)
• \(\mathrm{t} = -0.32666... \approx -0.3267\)

Answer: \(-0.3267\) (also acceptable: \(-49/150, -0.326, -0.327, -.3266, -.3267\))

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak SIMPLIFY execution: Students make arithmetic errors with the decimal calculations, especially when computing \(-0.46 - 1.892 = -2.352\) or the final division \(-2.352 ÷ 7.2\).

These calculation mistakes lead to incorrect values of t, causing them to second-guess their work or select an incorrect approximation if given multiple choice options.

Second Most Common Error:

Poor INFER reasoning about order of operations: Students might try to move terms before distributing, leading to incorrect manipulation like \(0.8\mathrm{t} - 8\mathrm{t} = 0.001 + 1.9 + 0.46\), completely missing the distributive step.

This leads to confusion about how to handle the parentheses and causes them to get stuck early in the solution process.

The Bottom Line:

This problem tests careful algebraic manipulation with decimals. The multiple decimal operations create many opportunities for arithmetic errors, while the parentheses require strategic thinking about order of operations.