Understanding Ratio Comparisons for the GMAT

Ratios are a fundamental concept tested repeatedly on the GMAT quantitative section. Whether you are solving a direct comparison question or working through complex word problems, your ability to understand and manipulate ratios can make a significant difference to your score. This article will cover the foundational knowledge you need to confidently compare ratios, helping you build a solid quantitative reasoning base. We will cover what ratios are, how to interpret different ratio formats, basic methods for comparison, common mistakes, and practice problems to solidify your understanding.

What Is a Ratio?

A ratio is a way to express the relationship between two quantities. It tells you how many times one quantity contains or is contained within the other. For example, if there are 3 apples and 5 oranges, the ratio of apples to oranges is 3 to 5, or 3:5. This ratio shows that for every 3 apples, there are 5 oranges.

Ratios can represent part-to-part relationships, part-to-whole relationships, or rates, such as speed or efficiency. On the GMAT, you might encounter ratios describing people, money, distances, mixtures, or other contexts. Understanding how to read and compare these ratios is critical for solving a wide variety of quantitative problems.

Different Formats of Ratios

Ratios on the GMAT may be presented in several formats. It is important to be comfortable working with all of these and converting between them quickly.

• Colon format, such as 2:3, which reads as “2 to 3”.
  
  
• Fraction format, such as 2/3, which represents the same ratio as 2:3.
  
  
• Decimal format, such as 0.666…, which is the decimal equivalent of 2 divided by 3.

Being able to switch effortlessly between these forms will allow you to choose the most convenient representation for comparison or calculation.

Why Comparing Ratios Is Important on the GMAT

Many GMAT questions require you to determine which ratio is larger or smaller. Sometimes this is straightforward, but often it is embedded in word problems or data sufficiency questions that test your reasoning skills. For example, questions involving mixtures, speed, work rates, or populations often hinge on comparing ratios correctly.

Since the GMAT is a timed exam, knowing efficient techniques for comparing ratios will save you valuable minutes and reduce the risk of mistakes.

Basic Methods to Compare Ratios

When you want to compare two ratios to find which is larger or smaller, there are several methods you can use. We will discuss two of the most common and useful approaches: converting ratios to decimals and cross multiplication.

Converting Ratios to Decimals

One straightforward way to compare two ratios is to convert each to a decimal by dividing the first term by the second. For example, to compare 3:4 and 5:7, divide 3 by 4 to get 0.75 and divide 5 by 7 to get approximately 0.714.

Since 0.75 is greater than 0.714, the ratio 3:4 is larger than 5:7.

This method is easy to understand and works well for simple comparisons. However, it requires a calculator or mental division, which can be slow and may lead to rounding errors on the GMAT.

Cross Multiplication: The Preferred GMAT Technique

Cross multiplication is a fast, reliable method to compare two ratios without converting them to decimals. Consider two ratios a:b and c:d. To compare them, cross multiply and compare the products: compute a × d and b × c.

If a × d is greater than b × c, then the ratio a:b is larger. If a × d is less than b × c, then a:b is smaller.

For example, compare 3:4 and 5:7 using cross multiplication:

• Calculate 3 × 7 = 21
  
  
• Calculate 4 × 5 = 20

Since 21 > 20, 3:4 is larger.

Cross multiplication avoids decimals and rounding errors, making it especially effective for the GMAT where accuracy and speed are critical.

Practice Question 1

Compare 7:9 and 4:5.

Cross multiply:

• 7 × 5 = 35
  
  
• 9 × 4 = 36

Since 35 < 36, 7:9 is less than 4:5.

Simplifying Ratios Before Comparison

Before comparing ratios, always check if they can be simplified by dividing both terms by their greatest common divisor. Simplification makes the comparison clearer and easier.

For example, compare 10:15 and 6:9.

• Simplify 10:15 by dividing both by 5 → 2:3
  
  
• Simplify 6:9 by dividing both by 3 → 2:3

Both ratios simplify to 2:3, meaning they are equal.

Simplifying ratios prevents mistakes and helps identify equivalent ratios.

Common Pitfalls When Comparing Ratios

Many test takers make simple mistakes when comparing ratios. Being aware of these pitfalls can save you from losing easy points.

• Not simplifying ratios before comparing can lead to confusion.
  
  

• Confusing ratios with fractions can cause errors, especially if the context is not considered.
  
  

• Comparing ratios with different units without converting units first will give meaningless results.
  
  

• Assuming the larger numerator means a larger ratio without considering the denominator.
  
  

• Forgetting to cross multiply or incorrectly performing the multiplication steps.

By staying vigilant about these common errors, you can approach ratio problems with confidence.

Practice Question 2

Which is larger: 12:25 or 13:27?

Cross multiply:

• 12 × 27 = 324
  
  
• 25 × 13 = 325

Since 324 < 325, 12:25 is less than 13:27.

Applying Ratio Comparisons in Word Problems

Often, ratios are part of larger word problems on the GMAT. Understanding how to interpret ratios within a real-world context is essential.

For example, a question might state that the ratio of men to women in a company is 3:5, and the ratio of women to children is 4:7. You might be asked to find the ratio of men to children.

To solve, first write down the given ratios:

Men : Women = 3 : 5
Women : Children = 4 : 7

To compare or combine these, the common term “women” must be the same in both ratios. Multiply the ratios to equalize women:

• Men : Women = 3 : 5
  
  
• Women : Children = 4 : 7

Multiply the first ratio by 4 (to make women = 20) and the second ratio by 5 (to make women = 20):

Men : Women = 3 × 4 : 5 × 4 = 12 : 20
Women : Children = 4 × 5 : 7 × 5 = 20 : 35

Now, Men : Children = 12 : 35.

This process is essential for handling complex ratio problems on the GMAT.

Practice Question 3

If the ratio of A to B is 7:9 and the ratio of B to C is 5:11, what is the ratio of A to C?

First, express the two ratios with a common middle term:

A : B = 7 : 9
B : C = 5 : 11

Find the least common multiple of 9 and 5, which is 45. Adjust the ratios:

Multiply the first ratio by 5: 7 × 5 : 9 × 5 = 35 : 45
Multiply the second ratio by 9: 5 × 9 : 11 × 9 = 45 : 99

Now, A : C = 35 : 99.

you have learned the basics of ratios, how to interpret different ratio formats, and two effective methods for comparing ratios — decimal conversion and cross multiplication. You also saw why simplifying ratios is important and how to avoid common mistakes. Additionally, we introduced how ratios are applied in real-world contexts and multi-step problems.

Mastering these fundamentals is crucial for the GMAT quantitative section. The next article will delve into more complex ratio problems, including those involving variables, multiple ratios, and applications in mixtures, work rates, and speed problems.

By practicing these foundational concepts, you will be well-prepared to tackle the challenging ratio questions that appear on the GMAT.

Advanced Techniques and Complex Problems 

Building on the foundational knowledge from, this article will explore more advanced techniques for comparing ratios on the GMAT. Many ratio problems in the exam go beyond simple comparisons and involve variables, multiple connected ratios, mixtures, work, and speed problems. Understanding these advanced concepts will elevate your problem-solving skills and allow you to handle the full spectrum of ratio questions efficiently.

Working with Variables in Ratios

On the GMAT, you often encounter ratios expressed with variables instead of explicit numbers. For example, a problem may say the ratio of x to y is 3:5 or that a:b = m:n.

When working with variables, the same principles of ratio comparison apply, but you need to treat variables carefully.

For instance, if a:b = 3:4, this means there is some common multiplier k such that:

a = 3k
b = 4k

The value of k is unknown but fixed. Ratios express relative quantities, so you cannot assign random values to variables unless additional information is provided.

If a question asks whether a > b given a:b = 3:4, you know that 3k < 4k for all positive k, so a < b.

Always pay attention to any conditions restricting variable values (e.g., positive, integer) to avoid incorrect assumptions.

Combining Multiple Ratios Involving Variables

When given multiple ratios involving variables, you often need to combine them into a single ratio. For example, if a:b = 2:3 and b:c = 5:7, find the ratio a:c.

To combine, you must make the middle terms (b) equal:

a : b = 2 : 3
b : c = 5 : 7

Find the least common multiple of 3 and 5, which is 15. Adjust the ratios accordingly:

Multiply the first ratio by 5 → a : b = 10 : 15
Multiply the second ratio by 3 → b : c = 15 : 21

Now combine:

a : b : c = 10 : 15 : 21

Therefore, a:c = 10:21.

This technique is crucial for complex ratio problems that appear on the GMAT.

Practice Question 1

If the ratio of x to y is 4:7 and the ratio of y to z is 3:5, find the ratio of x to z.

Step 1: Make the middle terms equal:

y in first ratio = 7
y in second ratio = 3

LCM of 7 and 3 is 21.

Multiply the first ratio by 3 → x:y = 12:21
Multiply the second ratio by 7 → y:z = 21:35

Step 2: Combine ratios:

x : y : z = 12 : 21 : 35

Therefore, x:z = 12:35.

Ratios in Mixture Problems

Mixture problems are a common type of ratio question on the GMAT. These problems involve mixing two or more components with known ratios or percentages and finding the resulting ratio or quantity.

Example:

A solution contains acid and water in the ratio 3:7. If 10 liters of acid are added to 40 liters of water, what is the new ratio of acid to water?

Step 1: Understand the initial quantities.

• The initial ratio of acid to water is 3:7.
  
  
• The total volume corresponds to 3 parts acid and 7 parts water.
  
  
• The amount of water is given as 40 liters, which corresponds to 7 parts.

Step 2: Find the value of one part.

• 1 part = 40 liters / 7 ≈ 5.71 liters.

Step 3: Calculate the initial amount of acid.

• Acid = 3 parts × 5.71 ≈ 17.14 liters.

Step 4: Add 10 liters of acid.

• New acid amount = 17.14 + 10 = 27.14 liters.
  
  
• Water remains 40 liters.

Step 5: Find the new ratio.

• Acid : Water = 27.14 : 40.

Step 6: Simplify if possible.

• Divide both by the smaller number: 27.14/27.14 : 40/27.14 ≈ 1 : 1.47.

So the new ratio is approximately 1:1.47.

This stepwise method is essential for GMAT mixture problems involving ratios.

Practice Question 2

A juice contains orange and apple juices in the ratio 5:3. If 15 liters of apple juice are added to 40 liters of orange juice, what is the new ratio of orange to apple juice?

Step 1: Identify given information.

• Original ratio: Orange : Apple = 5 : 3.
  
  
• Orange juice = 40 liters.
  
  
• Apple juice added = 15 liters.

Step 2: Find the value of one part.

• Orange juice corresponds to 5 parts.
  
  
• 1 part = 40 liters / 5 = 8 liters.

Step 3: Calculate original apple juice amount.

• Apple juice = 3 parts × 8 = 24 liters.

Step 4: Add 15 liters apple juice.

• New apple juice = 24 + 15 = 39 liters.

Step 5: Write new ratio.

• Orange : Apple = 40 : 39.

Step 6: Simplify ratio if possible.

• 40 and 39 share no common factors, so ratio stays 40:39.

The new ratio is 40:39.

Ratios in Work and Speed Problems

Work and speed problems on the GMAT often rely on ratios to compare rates, times, and productivity.

For example, if Worker A completes a job in 6 hours and Worker B in 9 hours, the ratio of their work rates is key to solving problems about combined work.

Calculating Work Rate Ratios

The work rate is typically the reciprocal of the time taken to complete the job.

• Work rate of A = 1 job / 6 hours = 1/6 jobs per hour.
  
  
• Work rate of B = 1 job / 9 hours = 1/9 jobs per hour.

The ratio of A’s work rate to B’s work rate is:

(1/6) : (1/9) = 1/6 ÷ 1/9 = 1/6 × 9/1 = 9/6 = 3/2.

Thus, A works 3/2 times faster than B.

Practice Question 3

Two machines, X and Y, can produce 100 units in 5 hours and 4 hours respectively. What is the ratio of their production rates?

Step 1: Calculate production rates.

• Rate of X = 100 units / 5 hours = 20 units per hour.
  
  
• Rate of Y = 100 units / 4 hours = 25 units per hour.

Step 2: Find the ratio of X to Y.

• 20 : 25 = 4 : 5 after simplification.

So, the ratio of production rates is 4:5.

Combining Ratios in Work Problems

If two workers work together, their combined rate is the sum of their individual rates.

For example, with rates 1/6 and 1/9 jobs per hour, combined rate:

1/6 + 1/9 = 3/18 + 2/18 = 5/18 jobs per hour.

The time taken to complete one job together is the reciprocal:

Time = 1 / (5/18) = 18/5 = 3.6 hours.

Practice Question 4

Worker A takes 8 hours to complete a task, and worker B takes 12 hours. How long will they take working together?

Step 1: Calculate individual work rates.

• A’s rate = 1/8 job/hour.
  
  
• B’s rate = 1/12 job/hour.

Step 2: Add rates.

• Combined rate = 1/8 + 1/12 = 3/24 + 2/24 = 5/24 jobs/hour.

Step 3: Calculate combined time.

• Time = 1 ÷ (5/24) = 24/5 = 4.8 hours.

They will complete the task in 4.8 hours working together.

Speed and Distance Ratio Problems

Speed, distance, and time problems often rely on ratio concepts. Remember the fundamental relationship:

Distance = Speed × Time.

If you know two of these quantities, you can find the third.

Practice Question 5

Car A travels at 60 mph and Car B at 75 mph. What is the ratio of the time taken by Car A to Car B to cover the same distance?

Step 1: Let the distance be D.

Step 2: Calculate time for each car.

• Time for A = D / 60
  
  
• Time for B = D / 75

Step 3: Write the ratio of times.

• (D/60) : (D/75) = 1/60 ÷ 1/75 = 75/60 = 5/4.

Therefore, Car A takes 5/4 times as long as Car B.

Practice Question 6

Two trains travel the same distance. Train X travels at 80 km/h, Train Y at 100 km/h. What is the ratio of the speeds, and what is the ratio of the times taken?

Speed ratio: 80 : 100 = 4 : 5.

Time ratio = reciprocal of speed ratio = 5 : 4.

we have explored advanced techniques to compare ratios on the GMAT, including working with variables, combining multiple ratios, and applying ratio comparisons in mixture, work, and speed problems. These problem types are common in the GMAT quantitative section and require a nuanced understanding of ratios and their applications.

Mastering these advanced methods will boost your confidence and accuracy on the exam.

The final part of this series will focus on common pitfalls to avoid, strategic tips for time management, and extensive practice problems with detailed solutions to help you cement your understanding and perform at your best on test day.

Comparing Ratios on the GMAT: Common Pitfalls, Strategic Tips, and Practice Problems

It will highlight common pitfalls to avoid, provide strategic tips for efficient problem-solving, and present a variety of practice problems with detailed explanations. By mastering these aspects, you can significantly enhance your accuracy and speed when tackling ratio questions on the GMAT.

Common Pitfalls in Ratio Problems

Even experienced test takers can stumble on ratio problems due to some common traps. Recognizing these pitfalls can prevent costly mistakes.

Confusing Ratios with Fractions or Percentages

Ratios express a relationship between quantities, not standalone values. For example, a ratio of 3:5 means for every 3 units of one quantity, there are 5 units of another. It is not equivalent to the fraction 3/5 unless specifically stated.

Avoid directly converting ratios to fractions without understanding context. If the question involves part-to-whole comparisons, fractions or percentages might be appropriate; otherwise, maintain the ratio form.

Assuming the Actual Values of Variables

Ratios represent relative magnitudes. The absolute values of quantities are often unknown. Assigning random values to variables in ratios without justification can lead to incorrect conclusions.

For example, if a:b = 2:3, you cannot conclude a = 2 and b = 3 unless additional information is provided. Instead, think in terms of multiples or a common scaling factor.

Ignoring Units or Mixing Units

When dealing with quantities such as speed, distance, time, or mixtures, always check that units are consistent before forming or comparing ratios.

For instance, comparing speeds given in km/h with distances in miles without converting units can yield incorrect results. Similarly, mixing liters and milliliters in mixture problems requires unit conversion first.

Overlooking Simplification Opportunities

Simplifying ratios at every step can prevent arithmetic errors and make comparison easier. For example, 20:30 simplifies to 2:3.

Some students leave ratios unsimplified, making subsequent steps more complex and error-prone.

Strategic Tips for Comparing Ratios on the GMAT

When comparing ratios expressed as fractions or when dealing with multiple ratios, converting to a common denominator or scaling to a common term facilitates direct comparison.

Example: To compare 2:5 and 3:7, rewrite both as fractions with a common denominator:

2/5 = 14/35
3/7 = 15/35

Since 14/35 < 15/35, 2:5 < 3:7.

Cross-Multiply to Compare Ratios

When comparing two ratios a:b and c:d, cross-multiplication helps determine which ratio is larger without calculating decimal values:

Compare a × d with b × c.

If a × d > b × c, then a:b > c:d.

This method is quick and avoids calculator dependence.

Use Variables and Scaling to Represent Ratios

If the ratio of A to B is m:n, express quantities as A = m × k and B = n × k, where k is a positive multiplier.

This representation helps when adding or subtracting quantities or combining multiple ratios.

Draw Diagrams or Use Tables

Visual aids like bar models, pie charts, or tables clarify ratio problems, especially mixture or work problems.

For example, use a bar divided into parts corresponding to the ratio to visualize quantities.

Practice Back-Solving and Estimation

Sometimes, plugging in numbers consistent with the ratio or estimating approximate values quickly narrows answer choices.

Back-solving can confirm if a proposed answer fits the problem’s conditions.

Practice Problems with Solutions

Problem 1: Comparing Ratios with Variables

If the ratio of x to y is 5:8 and y to z is 7:12, what is the ratio of x to z?

Solution:

Step 1: Make the middle terms equal.

• y in the first ratio = 8
  
  
• y in the second ratio = 7

LCM of 8 and 7 is 56.

Multiply the first ratio by 7 → x:y = 35:56
Multiply the second ratio by 8 → y:z = 56:96

Step 2: Combine ratios:

x : y : z = 35 : 56 : 96

Therefore, x:z = 35:96.

Problem 2: Mixture Problem

A tank contains a solution of alcohol and water in the ratio 3:5. If 12 liters of alcohol are added without changing the amount of water, what is the new ratio of alcohol to water?

Solution:

Step 1: Assume the original parts correspond to quantities.

• Alcohol : Water = 3 : 5.

Step 2: Let one part equal k liters. So, alcohol = 3k, water = 5k.

Step 3: Add 12 liters of alcohol.

• New alcohol quantity = 3k + 12.

Step 4: New ratio is (3k + 12) : 5k.

Step 5: The ratio can be simplified or left in terms of k if k is known.

If no value of k is given, this is the simplified form.

Problem 3: Work Rate Problem

Worker A completes a job in 10 hours. Worker B completes the same job in 15 hours. How long will it take them working together?

Solution:

Step 1: Calculate individual rates.

• A’s rate = 1/10 job/hour.
  
  
• B’s rate = 1/15 job/hour.

Step 2: Add rates.

• Combined rate = 1/10 + 1/15 = 3/30 + 2/30 = 5/30 = 1/6 job/hour.

Step 3: Calculate time to complete together.

• Time = 1 ÷ (1/6) = 6 hours.

Problem 4: Speed and Time Ratio

A cyclist travels 48 miles in 3 hours. Another cyclist travels the same distance in 4 hours. What is the ratio of their speeds and the ratio of their times?

Solution:

Step 1: Calculate speeds.

• Cyclist 1 speed = 48 / 3 = 16 mph.
  
  
• Cyclist 2 speed = 48 / 4 = 12 mph.

Step 2: Speed ratio = 16:12 = 4:3.

Step 3: Time ratio = reciprocal of speed ratio = 3:4.

Time Management Tips for Ratio Problems on the GMAT

Identify Ratio Questions Early

Ratio problems have recognizable keywords like “ratio,” “proportion,” “parts,” or references to relative quantities. Flag these questions to allocate focused time.

Simplify and Eliminate

Simplify ratios at the start and eliminate answer choices that do not match the simplified form.

Use Shortcut Methods

Apply cross-multiplication or common denominator techniques to compare ratios quickly instead of converting to decimals.

Skip and Return if Stuck

Ratio problems can become time-consuming. Skip difficult ones and return if time permits. Prioritize easier questions first.

This article series has comprehensively covered comparing ratios on the GMAT—from basic definitions and simple comparisons to advanced applications involving variables, mixtures, work, and speed. It highlighted common pitfalls and strategic approaches to help you tackle ratio problems with confidence and efficiency.

Ratio questions test your understanding of relative quantities and require precision and flexibility in problem-solving. With consistent practice and application of the strategies discussed, you will improve your speed, accuracy, and overall GMAT performance.

Conclusion

Understanding and comparing ratios is a fundamental skill that can significantly impact your performance on the GMAT quantitative section. Ratios express the relationship between quantities rather than absolute values, which means grasping their relative nature is crucial. Approaching ratio problems with clarity and strategy enables you to solve a wide variety of questions involving proportions, mixtures, rates, and more.

Successful ratio problem-solving hinges on several key principles: simplifying ratios whenever possible, using variables and scaling factors to represent quantities, and employing techniques such as cross-multiplication or finding common denominators to make comparisons more straightforward. Additionally, visual tools like tables and diagrams can help conceptualize complex problems and reduce confusion.

It is equally important to avoid common pitfalls such as confusing ratios with fractions, neglecting unit consistency, or making unwarranted assumptions about absolute values. By staying attentive to these traps and applying efficient problem-solving methods, you can enhance both your accuracy and speed.

Time management also plays a vital role during the GMAT. Recognizing ratio problems early, simplifying and eliminating options quickly, and knowing when to skip and return to a challenging question can save precious minutes and boost your overall score.

Regular practice with diverse ratio problems deepens your understanding and builds intuition, turning what might initially seem complex into a familiar and manageable part of the test. Ratios are more than just numbers — they are a language describing how quantities relate. Mastering this language empowers you to confidently navigate the GMAT and strengthens your quantitative reasoning skills for future academic and professional endeavors.

By integrating these concepts and strategies into your preparation, you equip yourself with a powerful toolkit to tackle ratio questions effectively, maximize your performance, and move closer to your target GMAT score.