Hello! In our last lesson, we explored the concept of weighted averages and saw how the deviation method can be a powerful shortcut for solving problems involving grouped data and mixtures. We ended with a promise to look at an even more direct, visual shortcut for mixture problems.
Today, we will deliver on that promise by diving into the Alligation Cross Method. The learning outcome for this session is to solve mixture and alligation problems using the alligation cross method. This technique is a cornerstone for quantitative aptitude in exams like the SSC CGL because it transforms complex weighted average calculations into a simple, graphical process. It's especially useful when you need to find the ratio in which two ingredients were mixed to achieve a specific mean value.
1. From Weighted Average to the Alligation Cross
In the previous lesson, we established that if you mix two ingredients, the final average price (or concentration) will be a weighted average. The alligation cross method is simply a visual representation of the same principle. It's a fantastic shortcut for a specific scenario:
- You know the value (price, concentration, etc.) of two ingredients.
- You know the final mean value of the mixture.
- You need to find the ratio of the quantities of the two ingredients.
The method is structured as a cross, which gives it its name.
Here’s how it works:
- Write the value of the cheaper ingredient on the top left (
c). - Write the value of the dearer ingredient on the top right (
d). - Write the mean value of the final mixture in the center (
m). - Subtract diagonally:
- Calculate
d - mand write the result on the bottom left. - Calculate
m - cand write the result on the bottom right.
- Calculate
- The two results at the bottom give you the ratio. The value under the 'cheaper' side corresponds to the quantity of the 'dearer' ingredient, and vice versa. However, the standard convention is to directly link the results as shown in the diagram:
(d - m)corresponds to the cheaper ingredient's quantity and(m - c)to the dearer's.
The final ratio is:
A crucial rule to remember, which we also saw with weighted averages, is that the mean price (m) must always lie between the cheaper (c) and dearer (d) prices.
The following video provides a very clear animated introduction to this "cheap and costly" trick.
Mixture and Alligation - Shortcuts & Tricks for 2026 Placement Tests, Job Interviews & Exams
This video from CareerRide clearly explains the basic mechanics of the alligation cross method.
Watch the first part of the video, from the beginning to the end of the trick explanation. The presenter uses an example of mixing wheat and rice to demonstrate how to set up the diagram and calculate the ratio of quantities.
2. What Does the Ratio Represent?
One of the most common points of confusion is understanding what the final ratio actually represents. Is it a ratio of prices, weights, or volumes? The answer is simple and logical.
The ratio you get from alligation is always the ratio of the base quantities over which the prices/percentages are defined.
The instructor in the next video explains this critical point very clearly.
SSC CGL Maths Class 2026 | Mixture and Alligation SSC CGL 2026 | Maths by Ravinder Sir
This segment from the SSC Wallah video clarifies what the result of the alligation method signifies.
Watch the instructor's explanation from this section. He explains that if your input values are in 'Rupees per kg', the resulting ratio will be of the quantities in 'kg'. If the values are 'km per hour', the ratio will be of the time in 'hours'. This is a fundamental concept to master.
3. Alligation in Action: Core Problem Types for SSC CGL
Now let's see how this method is applied to different types of problems you'll encounter in your exam.
Type 1: Simple Cost and Concentration Problems
This is the most direct application. You are given the costs of two items and the cost of the mixture, and you need to find the ratio.
SSC CGL Maths Class 2026 | Mixture and Alligation SSC CGL 2026 | Maths by Ravinder Sir
This video by Ravinder Sir provides two quick, foundational examples that are typical in SSC exams.
First, watch the solution to the acid solution problem from this segment. It shows how to apply the method to percentages. Next, see how he solves the coffee powder problem from this part. Notice the smart trick he uses: since you're finding a ratio, you can simplify the numbers by cancelling common zeros before you even start the calculation.
A common variation is mixing something of value with something of zero value, like milk with water. In such cases, the price of water is taken as 0.
Mixture and Alligation - Shortcuts & Tricks for 2026 Placement Tests, Job Interviews & Exams
This segment from CareerRide shows how to handle a problem where one of the ingredients has a value of zero.
Watch the sandalwood oil and water example from this clip. This demonstrates how to set up the alligation when one component, like water, has a cost of ₹0.
Type 2: Alligation with Profit and Loss
This is a very important question type for the SSC CGL. You will be given the cost prices of two ingredients, but for the mixture, you'll be given the selling price and a profit or loss percentage.
Remember: Alligation works only when all three values (cheaper, dearer, mean) are of the same type. You cannot mix cost price (CP) and selling price (SP) in the same diagram.
The correct approach is:
- Use the selling price and profit/loss percentage to calculate the mean cost price of the mixture.
- Use this mean cost price in the alligation diagram along with the cost prices of the ingredients.
SSC CGL Maths Class 2026 | Mixture and Alligation SSC CGL 2026 | Maths by Ravinder Sir
This video segment demonstrates exactly how to tackle this common exam pattern.
Watch carefully from this section. The instructor explains why you must first calculate the mean Cost Price from the given Selling Price and profit percentage before you can apply the alligation rule. This is a crucial step.
4. Advanced Techniques and Common Pitfalls
Mastering the basics is good, but for a competitive edge, you need to handle variations and avoid common traps.
Technique: Handling Fractions
Sometimes, concentrations are given as fractions. While you can do the subtraction with fractions, it's slow and prone to errors. A much faster way is to get rid of the denominators.
Shortcut: Find the Least Common Multiple (LCM) of all the denominators in your alligation setup. Multiply all three of your starting values (cheaper, dearer, mean) by this LCM. This will convert them into simple integers, making the subtraction trivial.
SSC CGL Maths Class 2026 | Mixture and Alligation SSC CGL 2026 | Maths by Ravinder Sir
This is a must-watch trick. It shows how to simplify a problem involving fractional concentrations.
Observe the method from this segment. The problem involves milk concentrations of 4/7, 2/5, and a mean of 1/2. Instead of subtracting fractions, the instructor multiplies everything by the LCM of (7, 5, 2), which is 70, turning the problem into a simple integer calculation.
Pitfall: Dealing with Loss
When a problem involves both a profit and a loss, you must be careful with your signs. A profit is a positive value, but a loss should be treated as a negative value.
SSC CGL Maths Class 2026 | Mixture and Alligation SSC CGL 2026 | Maths by Ravinder Sir
This part of the video highlights a common mistake students make when dealing with loss.
Pay close attention from this example. One item is sold at a 10% profit (+10), the other at a 5% loss (-5). The instructor shows how subtracting a negative number (7 - (-5)) becomes an addition (7 + 5 = 12). Getting this wrong will lead to an incorrect ratio.
Pitfall: The Mean Value Logic Check
Always remember the golden rule: the mean value must be between the cheaper and dearer values. This simple check can save you from picking the wrong answer when two options seem mathematically possible.
SSC CGL Maths Class 2026 | Mixture and Alligation SSC CGL 2026 | Maths by Ravinder Sir
This is an excellent example of using logic to eliminate incorrect choices.
Watch the problem from this segment. After setting up the alligation, the question becomes whether the missing value should be 11% or 19% to get the required difference. Since the mean is 15% and the other value is 10% (which is smaller), the missing value must be larger than 15%. Therefore, 19% is the only logical answer.
Conclusion
Today we have armed you with the Alligation Cross Method, a powerful and fast technique to solve mixture problems. It is a visual shortcut that builds directly on the principles of weighted averages we learned previously.
Key Takeaways:
- The Alligation Cross Method is used to find the ratio of quantities when mixing two ingredients with known values to get a mixture with a known mean value.
- The setup is crucial: Cheaper and Dearer values at the top, Mean value in the center. The ratio is found by cross-subtraction.
- The Mean value must always lie between the Cheaper and Dearer values.
- The resulting ratio is always of the base quantities (e.g., kg, litres).
- For problems involving profit/loss, always calculate the mean Cost Price first before applying alligation.
- Simplify problems with fractions by multiplying all terms by the LCM of the denominators.
- Remember to use a negative sign for loss to avoid calculation errors.
This lesson completes our module on the Foundations of Arithmetic. The concepts of profit, loss, and percentages that we touched upon today are a perfect lead-in to our next module, "Commercial Mathematics," where we will begin by formalizing the concepts of profit and loss calculation.