If you’re preparing for competitive exams like SSC, Bank PO, UPSC, or CAT, Mixtures and Alligations is a must-know topic in quantitative aptitude. It combines logic with math and can be a real time-saver if you understand it well. This guide explains everything — from basic concepts to smart shortcuts and solved practice questions — all designed to help you become a master of this topic.
What Are Mixtures and Alligations?
Let’s break it down:
- A mixture is created when two or more substances are combined together, like mixing milk and water, or two types of rice.
- Alligation is a rule used to find the correct ratio in which different ingredients should be mixed to achieve a mixture of a desired value, cost, or concentration.
The topic usually includes questions such as:
- Mixing two components and finding the average value.
- Replacing part of a mixture and finding the final composition.
- Using the Alligation rule to quickly determine mixing ratios.
Read this also :Â Understanding the 2020 Impact Factor of Nature Chemistry: Insights & Analysis
Types of Mixture Problems
There are different scenarios you might come across:
1. Simple Mixtures
Two or more components are mixed directly.
Example: Mixing sugar and jaggery.
2. Compound Mixtures
Two already-prepared mixtures are mixed together.
Example: Mixing two saltwater solutions with different salt concentrations.
3. Replacement Type
A portion of the mixture is removed and replaced with another substance.
Example: Taking out some milk from a milk-water mixture and replacing it with water.
Understanding the Alligation Method
The Alligation method is a shortcut trick that helps in solving mixture questions faster.
The Alligation Cross Rule
This is how it looks:
Cost of Higher Quantity \
> Mean Value
Cost of Lower Quantity /
Then subtract diagonally to find the ratio:High – Mean:Mean – Low\text{High – Mean} : \text{Mean – Low}High – Mean:Mean – Low
Read this also :Â Nature Chemistry Editor: Shaping the Future of Chemical Research
Real-Life Example:
Suppose pure milk costs ₹30/liter and water is free (₹0/liter). If you want a mixture worth ₹18/liter, what should be the mixing ratio?
Milk (₹30) \
> ₹18
Water (₹0) /
Cross subtraction:
Milk = 18 – 0 = 18
Water = 30 – 18 = 12
Ratio (Milk : Water) = 18:12 = 3:2
This means, to get the desired value, mix milk and water in a 3:2 ratio.
Key Formula for Replacement Questions
When part of the mixture is taken out and replaced multiple times:Final Quantity of Original Component=Q×(1−RQ)n\text{Final Quantity of Original Component} = Q \times \left(1 – \frac{R}{Q}\right)^nFinal Quantity of Original Component=Q×(1−QR​)n
Where:
- Q = Total volume
- R = Volume replaced each time
- n = Number of times the replacement is done
Example:
You have 20 liters of milk. You remove 5 liters and add 5 liters of water. This is done 2 times.=20×(1−520)2=20×0.752=20×0.5625=11.25 liters of milk left= 20 \times \left(1 – \frac{5}{20}\right)^2 = 20 \times 0.75^2 = 20 \times 0.5625 = 11.25 \text{ liters of milk left}=20×(1−205​)2=20×0.752=20×0.5625=11.25 liters of milk left
Read this also :Â From Molecules to Materials: Insights from Nature Chemistry Communications
Profit & Loss in Mixtures
Shopkeepers often mix two types of goods and sell them at a single price. You can calculate the profit using:
Profit % = (Selling Price – Average Cost Price) / Average Cost Price × 100
Important Shortcuts & Tricks
- Use the Alligation Cross Rule for quick ratio calculation.
- Apply the replacement formula when the same quantity is removed and replaced more than once.
- Simplify ratios and fractions as much as possible to save time.
- Use average formula when components are added directly without price or concentration changes.
Practice Questions (With Solutions)
Q1:
In what ratio should sugar costing ₹50/kg be mixed with sugar costing ₹30/kg to get a mixture costing ₹40/kg?
Solution:
Costlier = 50, Cheaper = 30, Mean = 40
Cross Difference:
50 – 40 = 10
40 – 30 = 10
Ratio = 10:10 = 1:1
Q2:
A 20-liter solution contains milk and water in a 3:2 ratio. 5 liters are taken out and replaced with water. What is the final ratio?
Step-by-step:
- Initial milk = 3/5 × 20 = 12 L
- Initial water = 8 L
- Removed = 5 L → Milk = 3/5 × 5 = 3 L
- Water removed = 2 L
- New milk = 12 – 3 = 9 L
- New water = 8 – 2 + 5 = 11 L
Final ratio = 9:11
Q3:
A seller mixes two varieties of wheat: ₹60/kg and ₹40/kg in a 3:2 ratio. He sells the mix at ₹55/kg. Find the profit %.
Solution:
- Cost Price = (3×60 + 2×40) / 5 = (180 + 80)/5 = ₹52/kg
- Selling Price = ₹55
- Profit = 55 – 52 = ₹3
- Profit % = (3/52) × 100 ≈ 5.77%
Read this also :Â Nature Chemistry ISSN: Comprehensive Journal Overview, Impact, and Research Scope
Why You Should Master Mixtures and Alligations
Here’s why this topic is high-value for competitive exams:
- Quick to solve once you know the tricks.
- Scoring topic that can boost your overall marks.
- Frequently asked in quantitative aptitude sections.
Final Tips to Excel in Mixtures and Alligations
- Understand the core idea, don’t just memorize formulas.
- Solve 10–15 questions daily to build speed and confidence.
- Attempt previous year questions to understand patterns.
- Keep a formula sheet for revision during exam time.
Read this also : Nature’s Lab: How Chemistry Shapes Life and the Environment
Conclusion
Mixtures and Alligations is not a tough topic once you grasp the basics and apply the right shortcuts. It’s all about logical thinking and applying the cross-rule or replacement formula in the right context. Whether you’re preparing for government jobs, MBA entrances, or banking exams, mastering this topic will give you an edge over others.
Keep practicing regularly, follow the tips above, and soon you’ll be solving even complex mixture problems within seconds. Remember, practice makes perfect — and with Mixtures and Alligations, that couldn’t be more true!https://unstop.com/blog/mixture-and-alligation
