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Have you ever mixed two different juices to create a new flavor? Or combined two types of rice to get a better price? Such everyday actions involve mixtures. A mixture is formed when two or more substances are combined without any chemical change. Understanding mixtures helps us solve many practical problems, such as finding the concentration of a solution or the price of a combined product.
When dealing with mixtures, especially those involving different concentrations or prices, it can be tricky to find the exact ratio or the mean value of the mixture. This is where the alligation method comes in handy. It is a simple and systematic way to solve mixture problems quickly and accurately.
In this chapter, we will explore the basic ideas of mixtures, learn the alligation rule, and solve various problems step-by-step. By the end, you will be confident in handling mixture problems often seen in competitive exams.
The alligation method is a technique used to find the ratio in which two or more ingredients at different prices or concentrations must be mixed to obtain a mixture of a desired mean price or concentration.
Imagine you have two solutions: one stronger and one weaker. When mixed, the resulting solution has a concentration somewhere between the two. The alligation method helps us find the exact ratio of mixing.
Let's define some terms:
This method uses a simple table to find the ratio of quantities to be mixed:
| Higher Price/Concentration (H) | Mean Price/Concentration (M) | Lower Price/Concentration (L) |
|---|---|---|
| H | M | L |
| Difference between H and M, and M and L | ||
To apply the rule, subtract the mean from the higher value and the lower value from the mean. These differences give the ratio of the quantities to be mixed.
Sometimes, you know the quantities and prices or concentrations of the components mixed, and you want to find the mean value of the mixture. This is done using the weighted average formula:
The formula to calculate the mean price or concentration \( M \) is:
where:
Step 1: Identify quantities and concentrations.
Quantity \( Q_1 = 3 \) liters, concentration \( P_1 = 10\% \)
Quantity \( Q_2 = 2 \) liters, concentration \( P_2 = 20\% \)
Step 2: Use the mean concentration formula:
\[ M = \frac{(3 \times 10) + (2 \times 20)}{3 + 2} = \frac{30 + 40}{5} = \frac{70}{5} = 14\% \]
Answer: The concentration of the mixture is 14%.
Step 1: Let the quantity of 15% solution be \( x \) liters.
Then, quantity of 25% solution = \( 20 - x \) liters.
Step 2: Use the mean concentration formula:
\[ 20 = \frac{(x \times 15) + ((20 - x) \times 25)}{20} \]
Multiply both sides by 20:
\[ 400 = 15x + 500 - 25x \]
Simplify:
\[ 400 = 500 - 10x \implies 10x = 500 - 400 = 100 \implies x = 10 \]
Answer: 10 liters of 15% solution must be mixed with 10 liters of 25% solution.
Step 1: Identify values:
Step 2: Apply alligation rule:
\[ \text{Ratio} = \frac{H - M}{M - L} = \frac{60 - 50}{50 - 40} = \frac{10}{10} = 1:1 \]
Answer: The two varieties should be mixed in the ratio 1:1.
Step 1: Let the quantity of 10% solution = quantity of 30% solution = \( x \) liters.
Quantity of 20% solution = \( 60 - 2x \) liters.
Step 2: Use the mean concentration formula:
\[ 18 = \frac{(x \times 10) + ((60 - 2x) \times 20) + (x \times 30)}{60} \]
Multiply both sides by 60:
\[ 1080 = 10x + 20(60 - 2x) + 30x \]
Expand:
\[ 1080 = 10x + 1200 - 40x + 30x = 1200 + (10x - 40x + 30x) = 1200 + 0x = 1200 \]
Notice the \( x \) terms cancel out, so:
\[ 1080 = 1200 \]
This is impossible, indicating no solution with the given mean concentration and equal quantities of 10% and 30% solutions.
Answer: No such mixture is possible under the given conditions.
Step 1: Calculate the cost price (CP) of the mixture.
\[ \text{Total cost} = (20 \times 30) + (30 \times 40) = 600 + 1200 = 1800 \text{ INR} \]
Total quantity = 20 + 30 = 50 kg
Cost price per kg:
\[ \frac{1800}{50} = 36 \text{ INR/kg} \]
Step 2: Selling price (SP) per kg = INR 38
Step 3: Calculate profit per kg:
\[ \text{Profit} = SP - CP = 38 - 36 = 2 \text{ INR/kg} \]
Total profit:
\[ 2 \times 50 = 100 \text{ INR} \]
Step 4: Calculate profit percentage:
\[ \text{Profit \%} = \frac{\text{Profit}}{\text{Cost Price}} \times 100 = \frac{2}{36} \times 100 \approx 5.56\% \]
Answer: The shopkeeper makes approximately 5.56% profit.
When to use: When mixing two components with known prices or concentrations to find the ratio.
When to use: To avoid sign errors and incorrect ratio calculations.
When to use: When dealing with concentration problems involving percentages.
When to use: To maintain consistency and avoid invalid results.
When to use: When the problem provides quantities and individual values but asks for the mean.
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