Weighted averages: word problems
Key Notes :
🎯 What is a Weighted Average?
- A weighted average gives different values more importance (weight) based on how frequently or significantly they occur.
The formula is:
Weighted Average = ∑(value×weight) / ∑weights
✅ Example: If a test counts for 60% of your grade and a project counts for 40%, the average isn’t a simple mean — the test carries more “weight.”
🔍 Understanding Weights
- Weights are percentages, frequencies, or any numbers representing importance.
- Total weight should add up to 100% (if dealing with percentages) or the total number of items.
🛠️Types of Weighted Average Problems
- Grades or scores: Different assignments/tests have different weights.
- Mixture problems: Two or more substances with different prices or concentrations are mixed.
- Speed or rate problems: Traveling different distances at different speeds.
- Financial averages: Calculating average prices, investments, or returns with different amounts invested.
🧠 Solving Word Problems Step-by-Step
- Identify the values and weights — What’s being averaged? What are the weights?
- Set up the equation — Multiply each value by its weight, then divide the total by the sum of the weights.
- Solve the equation — Simplify carefully to avoid mistakes.
- Check the answer — Does the result make sense?
Learn with an example
🗼 Daniel, a chemist, has 2 litres of a solution that contains 15% hydrochloric acid (HCl), plus unlimited 25% HCl solution. For a particular experiment, however, he needs to prepare some that contains 24% HCl. How much 25% HCl solution should be added to the 15% HCl solution to get the 24% HCl solution needed for the experiment?
🗼 Write your answer as a whole number or as a decimal rounded to the nearest tenth.______ litres
Step 1: Define a variable and make a table.
Let x represent the quantity of 15% HCl solution. Now make a table.
Start by filling in the first two columns with what you know.
| Concentration | Amount of HCl solution (litres) | Amount of HCl (litres) | |
| 15% HCl solution | 0.15 | 2 | |
| 25% HCl solution | 0.25 | x | |
| 24% HCl solution | 0.24 |
Next, finish filling in the second column. Since there are 2 litres of 15% HCl solution and x litres of 25% HCl solution, there are a total of 2 + x litres of 24% HCl solution. Fill in 2 + x.
| Concentration | Amount of HCl solution (litres) | Amount of HCl (litres) | |
| 15% HCl solution | 0.15 | 2 | |
| 25% HCl solution | 0.25 | x | |
| 24% HCl solution | 0.24 | 2 + x |
Finally, multiply across to fill in the third column.
| Concentration | Amount of HCl solution (litres) | Amount of HCl (litres) | |
| 15% HCl solution | 0.15 | 2 | 0.15(2) |
| 25% HCl solution | 0.25 | x | 0.25x |
| 24% HCl solution | 0.24 | 2 + x | 0.24(2 + x) |
Step 2: Write an equation and solve.
Using the information in the table, write an equation.
| amount of HCl in 15% HCl solution | plus | amount of HCl in 25% HCl solution | equals | amount of HCl in 24% HCl solution |
| 0.15(2) | + | 0.25x | = | 0.24(2 + x) |
Solve the equation.
| 0.15(2) + 0.25x | = | 0.24(2 + x) | |
| 0.3 + 0.25x | = | 0.48 + 0.24x | Multiply |
| 0.3 + 0.01x | = | 0.48 | Subtract 0.24x from both sides |
| 0.01x | = | 0.18 | Subtract 0.3 from both sides |
| x | = | 18 | Divide both sides by 0.01 |
To get the 24% HCl solution needed for the experiment, Daniel should add 18 litres of the 25% HCl solution.
🗼 The manager at a health foods store mixes a unique superfruit juice cocktail that costs ₹1,063.10 per litre to make. The cocktail includes mixed fruit juice and açai berry juice, which cost ₹344.15 per litre and ₹2,740.65 per litre, respectively. The manager has already opened 50 litres of the mixed fruit juice. How many litres of the açai berry juice does he need to add?
🗼 Write your answer as a whole number or as a decimal rounded to the nearest tenth.______ litres
Step 1: Define a variable and make a table.
Let x represent the quantity of mixed fruit juice. Now make a table.
Start by filling in the first two columns with what you know.
| Price per litre | Number of litres | Total price | |
| Mixed fruit juice | 344.15 | 50 | |
| Açai berry juice | 2,740.65 | x | |
| Superfruit juice cocktail | 1,063.1 |
Next, finish filling in the second column. Since there are 50 litres of mixed fruit juice and x litres of açai berry juice, there are a total of 50 + x litres of superfruit juice cocktail. Fill in 50 + x.
| Price per litre | Number of litres | Total price | |
| Mixed fruit juice | 344.15 | 50 | |
| Açai berry juice | 2,740.65 | x | |
| Superfruit juice cocktail | 1,063.1 | 50 + x |
Finally, multiply across to fill in the third column.
| Price per litre | Number of litres | Total price | |
| Mixed fruit juice | 344.15 | 50 | 344.15(50) |
| Açai berry juice | 2,740.65 | x | 2,740.65x |
| Superfruit juice cocktail | 1,063.1 | 50 + x | 1,063.1(50 + x) |
Step 2: Write an equation and solve.
Using the information in the table, write an equation.
| total price of mixed fruit juice | plus | total price of açai berry juice | equals | total price of superfruit juice cocktail |
| 344.15(50) | + | 2,740.65x | = | 1,063.1(50 + x) |
Solve the equation.
| 344.15(50) + 2,740.65x | = | 1,063.1(50 + x) | |
| 17,207.5 + 2,740.65x | = | 53,155 + 1,063.1x | Multiply |
| 17,207.5 + 1,677.55x | = | 53,155 | Subtract 1,063.1x from both sides |
| 1,677.55x | = | 35,947.5 | Subtract 17,207.5 from both sides |
| x | ≈ | 21.4 | Divide both sides by 1,677.55 |
The manager needs to add about 21.4 litres of the açai berry juice.
🗼 For a big cleaning job, Ayana wants to mix some ammonia-based cleaners to get a certain concentration of ammonia. She has 8 litres of cleaner that is 15% ammonia. She also has a large supply of cleaner that contains 20% ammonia. How many litres of the 20%-ammonia cleaner does Ayana need to add to the 15%-ammonia cleaner to obtain a batch that contains 18% ammonia?
🗼 Write your answer as a whole number or as a decimal rounded to the nearest tenth._____ litres
Step 1: Define a variable and make a table.
Let x represent the quantity of 15% cleaner. Now make a table.
Start by filling in the first two columns with what you know.
| Concentration | Amount of cleaner (litres) | Amount of ammonia (litres) | |
| 15% cleaner | 0.15 | 8 | |
| 20% cleaner | 0.2 | x | |
| 18% cleaner | 0.18 |
Next, finish filling in the second column. Since there are 8 litres of 15% cleaner and x litres of 20% cleaner, there are a total of 8 + x litres of 18% cleaner. Fill in 8 + x.
| Concentration | Amount of cleaner (litres) | Amount of ammonia (litres) | |
| 15% cleaner | 0.15 | 8 | |
| 20% cleaner | 0.2 | x | |
| 18% cleaner | 0.18 | 8 + x |
Finally, multiply across to fill in the third column.
| Concentration | Amount of cleaner (litres) | Amount of ammonia (litres) | |
| 15% cleaner | 0.15 | 8 | 0.15(8) |
| 20% cleaner | 0.2 | x | 0.2x |
| 18% cleaner | 0.18 | 8 + x | 0.18(8 + x) |
Step 2: Write an equation and solve.
Using the information in the table, write an equation.
| amount of ammonia in 15% cleaner | plus | amount of ammonia in 20% cleaner | equals | amount of ammonia in 18% cleaner |
| 0.15(8) | + | 0.2x | = | 0.18(8 + x) |
Solve the equation.
| 0.15(8) + 0.2x | = | 0.18(8 + x) | |
| 1.2 + 0.2x | = | 1.44 + 0.18x | Multiply |
| 1.2 + 0.02x | = | 1.44 | Subtract 0.18x from both sides |
| 0.02x | = | 0.24 | Subtract 1.2 from both sides |
| x | = | 12 | Divide both sides by 0.02 |
Ayana needs to add 12 litres of the 20%-ammonia cleaner.
Let’s practice!🖊️
