GRE Quant: Alligation Problems

Alligation is a method used to combine quantities with different values to reach a required average or final composition. This concept offers a simplified and direct way to handle situations involving weighted averages, proportions, and mixtures. By organizing values logically, it helps you decide how much of each component goes into the final result. On the GRE, alligation supports a variety of nuanced questions and word problems that test how accurately you connect individual values to an overall outcome.

The following video, part of our GRE online course, walks you through alligation with a focused and practical approach. It demonstrates how to set up the framework correctly, align given values with the target average, and determine quantities efficiently. The lesson then works through GRE-style problems so you see the method in action and understand how to apply it during practice. Use these learnings steadily across your GRE prep, practice exercises, full length GRE sectional mock tests, and full length GRE mock tests.

Comprehensive GRE Prep Online Course

When to Use Alligation

When to use Alligation - concept

The core concept involves determining the ratio in which two different concentrations of a substance must be combined to achieve a specific target concentration. This is a common application of the rule of alligation or weighted averages.

The specific problem presented asks for the ratio needed to mix a 30 percent alcohol solution with an 80 percent alcohol solution to result in a final mixture that is 50 percent alcohol.

Breaking Down the Method

To solve this, you calculate the difference between each starting concentration and the desired result:

The Difference for the Lower Concentration: The target (50 percent) minus the lower concentration (30 percent) equals 20 units.
The Difference for the Higher Concentration: The higher concentration (80 percent) minus the target (50 percent) equals 30 units.

Calculating the Ratio

The ratio of the quantities is inversely proportional to these differences.

The amount of the 30 percent solution corresponds to the difference from the higher side (80 – 50 = 30).
The amount of the 80 percent solution corresponds to the difference from the lower side (50 – 30 = 20).

Therefore, the solutions should be mixed in a ratio of 30 to 20, which simplifies to 3 to 2.

The Way Alligation Works

The way Alligation works

The concept of alligation is a mathematical method used to determine the relationship between the qualities of two individual components and the resulting mixture.

The process involves the following components:

Quality A and Quality B

These represent the individual characteristics (such as price, concentration, or percentage) of two different ingredients.

Result

This represents the desired mean or average quality achieved once the two components are mixed together.

Quantity A and Quantity B

These represent the specific amounts or ratios of each ingredient needed to achieve the target result.

The method works by calculating the cross-differences between the individual qualities and the average result. Specifically:

The difference between Quality B and the Result determines the proportion of Quantity A.
The difference between Quality A and the Result determines the proportion of Quantity B.

This visual framework allows for a quick calculation of the ratio in which two items must be mixed to produce a mixture of a specific value.

Alligation: Worked Example 1

Alligation – Worked example 1

To mix a 30% alcohol solution and an 80% alcohol solution to achieve a final 50% alcohol concentration, the alligation method is used.

By placing the target concentration (50) in the center and the two initial concentrations (30 and 80) at the top, the differences are calculated diagonally:

The difference between the higher concentration (80) and the target (50) is 30.
The difference between the target (50) and the lower concentration (30) is 20.

These differences represent the relative parts of each solution needed.

The resulting ratio of the 30% solution to the 80% solution is 30:20, which simplifies to a 3:2 ratio.

Alligation: Worked Example 2

Alligation – Worked example 2

Correct Answer: 7:1.

For a detailed explanation, please refer to the video presented earlier on this page.

For visual illustration, here is a snapshot from the video…

Alligation – example solution

Following is a concise, step-wise written explanation…

Question

In what ratio should balls with a weight of 10 grams each and those with a weight of 50 grams each be mixed to get an average weight of 15 grams for all balls?

Solution

We can solve this using the rule of allegation, which is a quick way to find ratios when mixing two items.

Subtract the smaller weight from the desired average: 15 – 10 = 5. This value represents the parts of the 50 gram balls.
Subtract the desired average from the larger weight: 50 – 15 = 35. This value represents the parts of the 10 gram balls.
Write the results as a ratio of 10 gram balls to 50 gram balls: 35 : 5.
Simplify the ratio by dividing both sides by 5: 35 / 5 = 7 and 5 / 5 = 1.

Correct Answer: 7 : 1

GRE-style Practice Question 1

GRE-style practice question 1

Correct Answer: 3 : 5.

For a detailed explanation, please refer to the video presented earlier on this page.

For visual illustration, here is a snapshot from the video…

practice question 1 - solution

Following is a concise, step-wise written explanation…

Question

Two qualities of sugar – $2 per lbs and $10 per lbs – are mixed to form a new quality of $7 per lbs. Find the ratio of the two qualities used.

2 : 5
5 : 2
3 : 5
5 : 3
5 : 8

Step-by-Step Solution

To solve this, we use the rule of alligation. It is a simple shortcut to find the ratio of ingredients in a mixture.

Find the difference between the higher price and the mean price: 10 – 7 = 3
Find the difference between the mean price and the lower price: 7 – 2 = 5
Arrange the differences to find the ratio: The ratio of the first quality to the second quality is the result of step 1 compared to the result of step 2. Ratio = 3 : 5

Correct Answer: 3 : 5

GRE-style Practice Question 2

GRE-style practice question 2

Correct Answer: 3:2.

For a detailed explanation, please refer to the video presented earlier on this page.

For visual illustration, here is a snapshot from the video…

practice question 2 - solution

Following is a concise, step-wise written explanation…

Question

Section A has Boys : Girls ratio 1:2, and section B has the ratio for the same as 3:1. The Boys : Girls ratio for the two sections combined is 1:1. What was the ratio of students in the two sections?

Step-by-Step Explanation

Identify the fraction of boys in each section.

In section A, the ratio is 1:2, so boys are 1 / (1 + 2) = 1/3 of the total.

In section B, the ratio is 3:1, so boys are 3 / (3 + 1) = 3/4 of the total.

In the combined group, the ratio is 1:1, so boys are 1 / (1 + 1) = 1/2 of the total.

Use the weighted average or allegation method.

Subtract the smaller value from the larger value to find the differences:

Difference 1 (Section B and Combined): 3/4 – 1/2 = 3/4 – 2/4 = 1/4.

Difference 2 (Combined and Section A): 1/2 – 1/3 = 3/6 – 2/6 = 1/6.

Find the ratio of students.

The ratio of total students in Section A to Section B is the ratio of these differences: Ratio = 1/4 : 1/6.

Multiply both sides by 12 to clear the fractions: 3 : 2.

Correct Answer: 3:2

Do Not Feel Obliged to Use Alligation

Do not feel obliged to use Alligation

Alligation is not a fundamentally new mathematical concept. It is a specific shortcut used to calculate weighted averages more efficiently and save time during problem-solving.

If the alligation method does not feel intuitive or easy to apply, it is perfectly acceptable to ignore it. You should feel comfortable continuing to use the standard weighted average method if that approach works better for you and yields more consistent results.

Key Takeaways

Alligation is a Tool: It serves as a visual or procedural simplification of weighted average algebra.
Efficiency vs. Intuition: While designed to be faster, the best method is always the one you understand and can apply accurately.
Method Substitution: You can always solve a mixture or average problem using the standard weighted average formula: (Weight1 * Value1 + Weight2 * Value2) / (Weight1 + Weight2).