Login
OR
Don't worry. We'll fix this for you!
Mixture problems are a common and practical class of arithmetic problems where two or more quantities are combined to form a new mixture. These problems often involve combining substances or items that have different attributes such as cost, concentration, or quality. For example, mixing two types of milk with different fat percentages, or combining sugars bought at different prices, are typical mixture problems.
In real life, mixtures appear in many contexts: whether it's in food preparation, pharmaceuticals, alloys in metallurgy, or trade and commerce. Understanding how to solve these problems helps us determine important characteristics like the cost of the final mixture, concentration of a solution, or the ratio in which two substances must be mixed.
Key components in mixture problems include:
In this chapter, we will explore various methods to solve mixture problems - starting from basic concepts to complex multi-component mixtures. We will focus strongly on the Alligation Method and Weighted Average approaches, illustrating concepts with clear examples using metric units and Indian Rupees (INR).
The Alligation Method is a shortcut technique used to find the ratio in which two quantities, each having different prices or concentrations, should be mixed to achieve a mixture of a desired price or concentration.
It is especially helpful because it avoids the need to set up and solve algebraic equations every time.
| Higher Price/Concentration (C₂) | Lower Price/Concentration (C₁) | |
|---|---|---|
| C₂ | ✖ | C₁ |
| (C₂ - Cₘ) | (Cₘ - C₁) | |
| Quantity from C₁ | Quantity from C₂ |
Here,
The ratio of quantities of the two ingredients to be mixed is given by:
Important: Always subtract the lower from the mean and the mean from the higher correctly to avoid sign errors.
The Weighted Average method is another way to understand mixtures. It calculates the resultant value (price or concentration) after mixing known quantities of different ingredients based on their individual values.
It reflects the idea that the contribution of each component to the final value depends on both its amount and its individual cost or concentration.
The formula for the weighted average price or concentration \( P \) when mixing two different quantities is:
Similarly, for concentrations (like percentage of fat, purity, etc.), the formula is:
Two milk solutions have fat contents of 4% and 6%. In what ratio should these be mixed to get a mixture with 5% fat?
Step 1: Identify the quantities in the alligation table:
Step 2: Apply alligation rule:
Ratio = \(\frac{6 - 5}{5 - 4} = \frac{1}{1} = 1:1\)
Answer: The two milk solutions should be mixed in equal ratio 1:1 to get 5% fat content.
A grocer mixes 5 kg of sugar at Rs.40/kg with 7 kg of sugar at Rs.38/kg. What is the cost price (per kg) of the mixture?
Step 1: Identify given values:
Step 2: Apply weighted average price formula:
\[ P = \frac{(5 \times 40) + (7 \times 38)}{5 + 7} = \frac{200 + 266}{12} = \frac{466}{12} = 38.83 \]
Answer: The cost price of the mixture is Rs.38.83 per kg.
A shopkeeper mixes 10 kg of rice costing Rs.30/kg, 15 kg costing Rs.35/kg and some quantity of rice costing Rs.40/kg. If the cost price of the mixture is Rs.34/kg, find the quantity of the third type of rice.
Step 1: Let the quantity of third rice be \(x\) kg at Rs.40/kg.
Step 2: Write the weighted average equation:
\[ \frac{(10 \times 30) + (15 \times 35) + (x \times 40)}{10 + 15 + x} = 34 \]
Step 3: Calculate numerator terms:
\[ 300 + 525 + 40x = 34(25 + x) \]
Step 4: Expand right side:
\[ 825 + 40x = 850 + 34x \]
Step 5: Bring like terms together:
\[ 40x - 34x = 850 - 825 \Rightarrow 6x = 25 \Rightarrow x = \frac{25}{6} = 4.17 \text{ kg (approx.)} \]
Answer: The shopkeeper mixed approximately 4.17 kg of rice costing Rs.40/kg.
3 liters of a 10% acid solution is mixed with 5 liters of a 20% acid solution. What is the percentage concentration of the mixture?
Step 1: Identify given values:
Step 2: Use weighted average concentration formula:
\[ C = \frac{(3 \times 10) + (5 \times 20)}{3 + 5} = \frac{30 + 100}{8} = \frac{130}{8} = 16.25\% \]
Answer: The mixture has a concentration of 16.25% acid.
A merchant mixes 40 kg of sugar at Rs.50/kg with 60 kg sugar at Rs.55/kg. If the mixture is sold at Rs.54/kg, find the profit or loss percentage.
Step 1: Calculate cost price (CP) per kg of mixture:
\[ CP = \frac{(40 \times 50) + (60 \times 55)}{40 + 60} = \frac{2000 + 3300}{100} = \frac{5300}{100} = 53 \, \text{Rs./kg} \]
Step 2: Given selling price (SP) is Rs.54/kg.
Step 3: Calculate profit per kg:
\[ \text{Profit} = SP - CP = 54 - 53 = Rs.1 \]
Step 4: Calculate profit percentage:
\[ \text{Profit %} = \frac{1}{53} \times 100 \approx 1.89\% \]
Answer: The merchant makes a profit of approximately 1.89% on the mixture.
When to use: Whenever mixing two ingredients with known costs or concentrations and a desired mean value.
When to use: In every mixture problem to avoid unit mismatch errors.
When to use: Complex mixture problems involving multiple components.
When to use: After finishing problem to confirm accuracy.
When to use: Always for problem clarity involving money.
Progress tracking is paywalled — subscribe to mark subtopics as understood and save your streak.
Go to practice →
Your Score
Your Rank
| Rank | Name | Score |
|---|
