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In many real-life situations, we often combine two or more substances to form a mixture. This could be mixing different solutions in a laboratory, blending various grades of rice in a market, or combining metals to form an alloy. Understanding how quantities and concentrations combine is essential, especially in competitive exams where time and accuracy matter.
For example, if a shopkeeper mixes two types of rice priced differently to sell at a new price, or a chemist mixes two solutions of different strengths, knowing how to calculate the resulting concentration or price is crucial. This chapter will teach you how to solve such problems efficiently using the alligation method, a powerful shortcut technique.
All quantities will be considered in metric units (liters, kilograms, etc.) and prices in Indian Rupees (INR) to keep examples practical and relatable.
A mixture is a combination of two or more substances where each retains its own properties. Mixtures can be of different types:
When two substances with different concentrations or prices are mixed, the resulting mixture has a concentration or price somewhere between the two. This is because the final concentration depends on both the amount and concentration of each component.
To understand this better, consider the idea of a weighted average. Unlike a simple average, a weighted average takes into account the quantity of each component. For example, mixing 2 liters of 10% salt solution with 3 liters of 20% salt solution results in a concentration closer to 20% because the larger quantity has more influence.
In the above diagram, two solutions with different volumes and concentrations are mixed to form a new solution. The question often is: what is the concentration of the mixture, or in what ratio were the two solutions mixed?
The alligation method is a simple and quick way to find the ratio in which two substances of different concentrations or prices must be mixed to get a desired concentration or price. It avoids lengthy algebraic calculations.
Here is how the alligation method works step-by-step:
This can be visualized in the following alligation grid:
| Component | Concentration (or Price) | Difference from Mean |
|---|---|---|
| Component 1 | \( C_1 \) | Ratio of quantities = \(\frac{C_2 - C_m}{C_m - C_1}\) |
| Component 2 | \( C_2 \) | |
| Mixture | \( C_m \) |
Important: The mean concentration \( C_m \) must lie between \( C_1 \) and \( C_2 \) for the alligation method to work correctly.
Step 1: Identify the concentrations:
\( C_1 = 10\% \), \( C_2 = 30\% \), desired \( C_m = 20\% \)
Step 2: Calculate the differences:
\( d_1 = C_2 - C_m = 30 - 20 = 10 \)
\( d_2 = C_m - C_1 = 20 - 10 = 10 \)
Step 3: Find the ratio of quantities:
\[ \frac{Q_1}{Q_2} = \frac{d_1}{d_2} = \frac{10}{10} = 1 \]
Answer: The two solutions should be mixed in the ratio 1:1.
Step 1: Let the quantity of 5% solution be \( Q_1 \) liters, and 15% solution be \( Q_2 \) liters.
Step 2: Total volume:
\( Q_1 + Q_2 = 20 \)
Step 3: Use alligation to find the ratio:
\( C_1 = 5\%, C_2 = 15\%, C_m = 10\% \)
\[ d_1 = 15 - 10 = 5, \quad d_2 = 10 - 5 = 5 \]
Ratio \( Q_1 : Q_2 = 5 : 5 = 1 : 1 \)
Step 4: Since total is 20 liters, each quantity is:
\( Q_1 = Q_2 = \frac{20}{2} = 10 \) liters
Answer: Mix 10 liters of 5% solution and 10 liters of 15% solution.
Step 1: Calculate the cost price (CP) of the mixture:
\[ \text{Total cost} = (40 \times 20) + (60 \times 30) = 800 + 1800 = Rs.2600 \]
Step 2: Total quantity = 40 + 60 = 100 kg
Step 3: Cost price per kg:
\[ CP = \frac{2600}{100} = Rs.26 \]
Step 4: Selling price (SP) per kg = Rs.28
Step 5: Calculate profit percentage:
\[ \text{Profit\%} = \frac{SP - CP}{CP} \times 100 = \frac{28 - 26}{26} \times 100 = \frac{2}{26} \times 100 \approx 7.69\% \]
Answer: The shopkeeper makes approximately 7.69% profit.
Step 1: Let the quantity of 5% solution be \( Q \) liters, and 15% solution also \( Q \) liters.
Let the quantity of 10% solution be \( x \) liters.
Step 2: Total concentration equation using weighted average:
\[ \frac{5Q + 10x + 15Q}{Q + x + Q} = 12 \]
Simplify numerator and denominator:
\[ \frac{5Q + 15Q + 10x}{2Q + x} = 12 \implies \frac{20Q + 10x}{2Q + x} = 12 \]
Step 3: Cross-multiplied:
\[ 20Q + 10x = 12(2Q + x) = 24Q + 12x \]
Step 4: Rearranged terms:
\[ 20Q + 10x - 24Q - 12x = 0 \implies -4Q - 2x = 0 \]
Step 5: Simplify:
\[ 2x = -4Q \implies x = -2Q \]
This is impossible since quantity cannot be negative. Check if 12% lies between 5% and 15%.
Step 6: Since 12% lies between 10% and 15%, try mixing 10% and 15% first.
Alternatively, use alligation for 10% and 15% to get 12%:
\[ \frac{Q_{10}}{Q_{15}} = \frac{15 - 12}{12 - 10} = \frac{3}{2} = 3:2 \]
Given \( Q_5 = Q_{15} \), let \( Q_5 = Q_{15} = y \), and \( Q_{10} = \frac{3}{2} y \).
Answer: The ratio of quantities is \( Q_5 : Q_{10} : Q_{15} = 1 : \frac{3}{2} : 1 = 2 : 3 : 2 \).
Step 1: Let the quantity of Rs.200/kg tea be \( Q_1 \) kg, and Rs.250/kg tea be \( Q_2 \) kg.
Step 2: Total quantity:
\( Q_1 + Q_2 = 100 \)
Step 3: Use alligation to find ratio:
\( C_1 = 200, C_2 = 250, C_m = 230 \)
\[ d_1 = 250 - 230 = 20, \quad d_2 = 230 - 200 = 30 \]
Ratio \( Q_1 : Q_2 = 20 : 30 = 2 : 3 \)
Step 4: Find quantities:
\[ Q_1 = \frac{2}{5} \times 100 = 40 \text{ kg}, \quad Q_2 = \frac{3}{5} \times 100 = 60 \text{ kg} \]
Answer: Mix 40 kg of Rs.200/kg tea and 60 kg of Rs.250/kg tea.
When to use: When mixing two components with known concentrations or prices to find mixing ratio.
When to use: In all mixture and concentration problems involving percentages.
When to use: To validate if alligation method is applicable.
When to use: When dealing with mixtures of commodities bought at different prices.
When to use: During timed competitive exams for quick calculation.
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