Q.A solution is obtained by mixing 300 g of 25% solution and 400 g of 40% solution by mass. Calculate the mass percentage of the resulting solution.

The key idea here is Mass Percentage, which defines the concentration of a solution as the mass of solute divided by the total mass of the solution, multiplied by 100.

First, calculate the mass of solute in each initial solution:

Mass of solute in the 25% solution $= 0.25 \times 300 \text{ g} = 75 \text{ g}$.

Mass of solute in the 40% solution $= 0.40 \times 400 \text{ g} = 160 \text{ g}$.

Next, determine the total mass of solute and the total mass of the resulting solution:

Total mass of solute $= 75 \text{ g} + 160 \text{ g} = 235 \text{ g}$.

Total mass of solution $= 300 \text{ g} + 400 \text{ g} = 700 \text{ g}$. …

When we talk about the "mass percentage" of a solution, we're essentially describing how much of the dissolved substance (the solute) is present relative to the total mass of the solution. It's a way to express concentration. Imagine you have a glass of sugar water; a 25% sugar solution means that for every 100 grams of that solution, 25 grams are sugar, and the remaining 75 grams are water.

When two solutions are mixed, the total amount of solute simply adds up, and the total amount of the solution also adds up. The key idea is that mass is conserved. We can't create or destroy solute or solvent by mixing them. Therefore, to find the new mass percentage, we need to find the total mass of solute in the final mixture and the total mass of the final solution.

Here's how we approach this problem:

1. Understand Mass Percentage

   The mass percentage of a solution is defined as:

   $\text{Mass Percentage} = \frac{\text{Mass of Solute}}{\text{Mass of Solution}} \times 100\%$

   This formula will be used at the end to calculate the final concentration.

2. Calculate the mass of solute in the first solution

   The first solution has a mass of 300 g and is 25% by mass. This means 25% of its total mass is the solute.

   Mass of solute in first solution $= 25\%$ of 300 g

   Mass of solute in first solution $= \frac{25}{100} \times 300 \text{ g}$

   Mass of solute in first solution $= 0.25 \times 300 \text{ g} = 75 \text{ g}$

3. Calculate the mass of solute in the second solution

   The second solution has a mass of 400 g and is 40% by mass.

   Mass of solute in second solution $= 40\%$ of 400 g

   Mass of solute in second solution $= \frac{40}{100} \times 400 \text{ g}$

   Mass of solute in second solution $= 0.40 \times 400 \text{ g} = 160 \text{ g}$

4. Calculate the total mass of solute in the resulting solution

   When the two solutions are mixed, the total amount of solute is the sum of the solute from each solution.

   Total mass of solute $= \text{Mass of solute in first solution} + \text{Mass of solute in second solution}$ …

Method: Weighted Average (Alligation / Mixture Rule)

This is the standard method for combining two solutions of the same substance — the final concentration is the weighted average of the individual concentrations, weighted by their masses.

Steps

Step 1: Identify the given data

• Mass of first solution: $m_1 = 300 \text{ g}$
• Mass percentage of first solution: $p_1 = 25\% = 0.25$
• Mass of second solution: $m_2 = 400 \text{ g}$
• Mass percentage of second solution: $p_2 = 40\% = 0.40$

Step 2: Find the total mass of solute in each solution

• Solute in first solution:

  $\text{Solute}_1 = m_1 \times p_1 = 300 \times 0.25 = 75 \text{ g}$

• Solute in second solution:

  $\text{Solute}_2 = m_2 \times p_2 = 400 \times 0.40 = 160 \text{ g}$

Step 3: Find total mass of solution and total mass of solute

• Total mass of solution:

  $m_{\text{total}} = 300 + 400 = 700 \text{ g}$

• Total mass of solute:

  $\text{Solute}_{\text{total}} = 75 + 160 = 235 \text{ g}$

Step 4: Calculate the mass percentage of the resulting solution

$\text{Mass percentage} = \frac{\text{Total solute}}{\text{Total solution}} \times 100$ …

🧠 The Core Concept First

Mass percentage tells us how many grams of solute are present in 100 g of solution.

When mixing two solutions, the total solute adds up, and the total solution mass adds up.

✗ Mistake #1: Averaging the Percentages Directly

What students do:

They take the average of 25% and 40%:

Why it’s wrong:

The two solutions have different masses (300 g and 400 g). A simple average ignores the weight of each solution — it treats 300 g and 400 g as equally important.

✓ How to avoid:

Always find the actual mass of solute from each solution first, then find the total solute and total solution mass.

✗ Mistake #2: Confusing Mass of Solute with Mass of Solution

What students do:

They treat the percentage as if it’s the mass of solute directly. For example, they might write:

• 25% of 300 g = 75 g ✓ (correct)
• But then they forget to add the 400 g solution mass.

Why it’s wrong:

They might calculate:

This just gives back the original — they didn’t include the second solution at all.

✓ How to avoid:

Write down two separate steps:

1. Solute from each solution

   • Solution 1: $25\%$ of $300\ \text{g} = 0.25 \times 300 = 75\ \text{g}$
   • Solution 2: $40\%$ of $400\ \text{g} = 0.40 \times 400 = 160\ \text{g}$

2. Total solute = $75 + 160 = 235\ \text{g}$

   Total solution = $300 + 400 = 700\ \text{g}$

✗ Mistake #3: Forgetting to Multiply by 100 …