How many liters of a 25% saline solution must be added to 3 liters of a 10% saline solution to...

1. TRANSLATE the problem information

• Given information:
  • 3 liters of 10% saline solution (existing)
  • x liters of 25% saline solution (to be added)
  • Final mixture should be 15% saline solution
  • Need to find: value of x

2. INFER the key principle

• In mixture problems, the amount of pure substance is conserved
• Pure salt before mixing = Pure salt after mixing
• Each solution contributes pure salt based on: Volume × Concentration

3. TRANSLATE each component into math

• Pure salt from 10% solution: \(3 \times 0.10 = 0.30\) liters
• Pure salt from 25% solution: \(\mathrm{x} \times 0.25 = 0.25\mathrm{x}\) liters
• Pure salt in final mixture: \((3 + \mathrm{x}) \times 0.15 = 0.15(3 + \mathrm{x})\) liters

4. Set up the conservation equation

• Total pure salt before = Total pure salt after
• \(0.30 + 0.25\mathrm{x} = 0.15(3 + \mathrm{x})\)

5. SIMPLIFY the equation

• Distribute the right side: \(0.30 + 0.25\mathrm{x} = 0.45 + 0.15\mathrm{x}\)
• Move x terms to left, constants to right: \(0.25\mathrm{x} - 0.15\mathrm{x} = 0.45 - 0.30\)
• Combine like terms: \(0.10\mathrm{x} = 0.15\)
• Solve for x: \(\mathrm{x} = 0.15 \div 0.10 = 1.5\)

Answer: 1.5 liters (can also be written as 3/2 liters)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students incorrectly set up the mixture equation by confusing what quantities should be equal.

Many students write something like: \(3(0.10) + \mathrm{x}(0.25) = (3 + \mathrm{x})(0.15)\) but then think this means the concentrations add up, or they set up ratios incorrectly. Some try to work with the percentages directly without converting to decimals first, leading to equations like \(3(10) + \mathrm{x}(25) = (3 + \mathrm{x})(15)\).

This leads to confusion and either abandoning the systematic approach or getting nonsensical answers.

Second Most Common Error:

Poor SIMPLIFY execution: Students set up the correct equation but make algebraic mistakes when distributing or combining decimal terms.

For example, they might incorrectly distribute \(0.15(3 + \mathrm{x})\) as \(0.15 + 0.15\mathrm{x}\) instead of \(0.45 + 0.15\mathrm{x}\), or make sign errors when moving terms across the equals sign. These calculation errors lead to wrong final answers.

The Bottom Line:

This problem requires students to translate a real-world scenario into the mathematical principle of conservation - the total amount of pure substance stays constant during mixing. Students who struggle with the abstract setup often get lost before they can even begin solving.