Key Takeaways From the Blog
- Ratios compare two quantities; proportions show equality between two ratios.
- These concepts are vital for exams like CAT, GMAT, SSC, and banking.
- Mastering key formulas and practice questions boosts speed and accuracy.
- Practice includes both detailed solutions and MCQs for effective learning.
- Downloadable PDFs and resources are available for offline preparation.
- Real-world applications include sharing, mixtures, partnership, and age problems.
Introduction
Ratio and proportion are fundamental mathematical concepts used for comparison and for understanding different relationships. The ratio simply compares one quantity to another, while proportion compares equal ratios. It is used regularly in everyday living, for instance, in preparing a mixture of different substances, equal amounts of the products, and also in dividing costs among different individuals.
With regard to quantitive aptitudes, aptitude questions on ratio and proportion often form a part of various competitive exams, interviews, as well as academic tests. When these two concepts are mastered, not only do your question-solving skills improve, but you also become able to tackle things of the real world efficiently. With regard to ratio and proportion, if their basics are understood, solving ratio and proportion aptitude questions can become even easier.
What Are Ratio and Proportion?
A ratio is a way to compare two quantities by showing how many times one value contains or is contained within the other. For example, if you have 2 apples and 3 oranges, the ratio of apples to oranges is 2:3. Basic ratio and proportion questions help us understand the relative sizes of two or more quantities and are often written with a colon (:) or as a fraction.
A proportion is an equation that states two ratios are equal. For instance, if 2:3 = 4:6, then the two pairs of numbers are in proportion. Proportion questions and answers are useful for solving problems where you need to find an unknown value that maintains the same relationship as the given ratios. These concepts form the basis for many practical and mathematical applications, such as scaling recipes, dividing profits, or comparing speeds.
Why Are Ratio and Proportion Important?
Ratio and proportion questions are common in exams such as CAT, GMAT, GRE, SSC, banking, and more. They test your ability to:
- Analyze relationships between numbers
- Apply logical reasoning
- Perform quick calculations
- Solve word problems involving sharing, mixtures, partnerships, and more
- Practicing these types of ratio and proportion practice questions helps improve your speed and accuracy.
In order to answer ratio and proportion problems with appropriate accuracy and efficiency, it’s suggested that one uses a few sure-fire methods to get the job done properly and avoid errors. Toward that end, here are some useful tips to keep in mind:
- Ratio:
The ratio of two quantities a and b is written as a : b or a/b. It shows how many times one quantity is compared to another. - Proportion:
When two ratios are equal, they form a proportion. For example, a : b :: c : d means a/b = c/d. - Mean Proportion:
The mean proportional between two numbers a and b is √(a × b). - Continued Proportion:
If a, b, and c are in continued proportion, then a : b = b : c. - Dividing a Quantity in a Given Ratio:
If a sum S is divided between two people in the ratio m : n:
Share of first = (m / (m + n)) × S
Share of second = (n / (m + n)) × S - Simplifying Ratios:
Always reduce ratios to their simplest form by dividing both terms by their greatest common divisor. - Cross Multiplication in Proportion:
If a : b = c : d, then a × d = b × c.
Tips for Solving Ratio and Proportion Questions
To solve ratio and proportion questions accurately and efficiently, it’s helpful to use a few proven strategies. Applying these techniques can simplify complex problems and reduce the chances of making mistakes. Here are some practical tips to keep in mind:
- Express unknowns using variables: If the ratio is 2:3, let the quantities be 2x and 3x.
- Cross-multiply when working with proportions: If a:b = c:d, then a × d = b × c.
- Convert word problems into equations: Carefully read the problem, assign variables, and set up equations based on the given ratios and relationships.
- Simplify ratios when possible: Always reduce ratios to their simplest form for easier calculations.
- Practice common scenarios: Focus on partnership, mixtures, age, salary, and distribution problems, as these are frequently tested.
Practice Questions with Detailed Solutions on Ratio and Proportion
Here are practice questions for ratio and proportion, each with a detailed solution. These cover various types and difficulty levels to help you strengthen your understanding and problem-solving skills.
- Is the ratio 5:10 proportional to 1:2?
Solution:
5:10 simplifies to 1:2, so yes, they are proportional. - Divide 100 into two parts such that they are in the ratio 3:5.
Solution:
Let the parts be 3x and 5x.
3x + 5x = 100 → 8x = 100 → x = 12.5
Parts: 3 × 12.5 = 37.5 and 5 × 12.5 = 62.5 - If a:b = 5:9 and b:c = 7:4, find a:b:c.
Solution:
Make b equal:
a:b = 5×7 : 9×7 = 35:63
b:c = 7×9 : 4×9 = 63:36
So, a:b:c = 35:63:36 - A, B, and C share a profit of Rs. 200,000 in the ratio 2:3:5. What is B’s share?
Solution:
Total ratio = 2+3+5 = 10
B’s share = (3/10) × 200,000 = Rs. 60,000 - A mixture contains milk and water in the ratio 2:1. If the total volume is 60 liters, how much water should be added to make the ratio 1:2?
Solution:
Milk = (2/3)×60 = 40 liters
Water = (1/3)×60 = 20 liters
Let x be the water to add:
40 : (20 + x) = 1 : 2
40/(20 + x) = 1/2 → 2×40 = 20 + x → 80 = 20 + x → x = 60 liters - Divide Rs. 981 in the ratio 5:4.
Solution:
Total parts = 5 + 4 = 9
Each part = 981/9 = 109
Shares: 5×109 = 545, 4×109 = 436 - Find the mean proportional between 0.23 and 0.24.
Solution:
Mean proportional = √(0.23 × 0.24) = √0.0552 ≈ 0.235 - A bag contains 50p, 25p, and 10p coins in the ratio 2:5:3, amounting to Rs. 510. Find the number of coins of each type.
Solution:
Let common ratio = k
Number of 50p coins = 2k, 25p = 5k, 10p = 3k
Total value = 0.5×2k + 0.25×5k + 0.1×3k = 1k + 1.25k + 0.3k = 2.55k
2.55k = 510 → k = 200
Coins: 400 (50p), 1000 (25p), 600 (10p) - If x² + 6y² = 5xy, find x/y.
Solution:
Let x/y = t
t² + 6 = 5t → t² - 5t + 6 = 0 → (t-2)(t-3)=0
x/y = 2 or 3 - The ratio of the ages of two friends is 6:4. Five years ago, their ages were in the ratio 5:3. Find their present ages.
Solution:
Let present ages be 6x and 4x.
Five years ago: (6x-5):(4x-5)=5:3
(6x-5)/5 = (4x-5)/3
Cross-multiplied: 3(6x-5)=5(4x-5) → 18x-15=20x-25 → 2x=10 → x=5
Ages: 30 and 20 - The monthly salaries of two persons are in the ratio 3:5. If each gets an increment of Rs. 5,000, the new ratio becomes 29:45. Find their original salaries.
Solution:
Let salaries be 3x and 5x.
(3x + 5000)/(5x + 5000) = 29/45
Cross-multiplied: 45(3x+5000)=29(5x+5000)
135x+225000=145x+145000
10x=80000 → x=8000
Salaries: 24,000 and 40,000 - A sum of Rs. 1,500 is divided among A, B, and C in the ratio 2:3:5. How much does B get?
Solution:
Total parts = 2+3+5=10
B’s share = (3/10)×1500 = Rs. 450 - If a:b = 3:7 and b:c = 2:5, find a:b:c.
Solution:
Make b equal:
a:b = 3×2:7×2=6:14
b:c = 2×7:5×7=14:35
So, a:b:c = 6:14:35 - In a mixture of 60 liters, the ratio of milk to water is 2:1. If this ratio is to be 1:2, how much water must be added?
Solution:
Milk = 40L, Water = 20L
Let x be added: 40:(20+x)=1:2
40/(20+x)=1/2 → x=60L - If Rs. 432 is divided among A, B, and C such that 8×A’s share = 12×B’s share = 6×C’s share, find A’s share.
Solution:
Let A’s share = x, B’s = y, C’s = z
8x=12y=6z=k
x=k/8, y=k/12, z=k/6
Sum: k/8 + k/12 + k/6 = 432
LCM of 8,12,6=24
(3k+2k+4k)/24=432 → 9k/24=432 → k=1152
A’s share = 1152/8 = Rs. 144 - A bag contains coins in the ratio 2:3:5 (50p, 25p, 20p). If the total value is Rs. 550, find the number of each type of coin.
Solution:
Let coins be 2x, 3x, 5x
Total value: 0.5×2x + 0.25×3x + 0.2×5x = Rs. 550
(1x+0.75x+1x)=2.75x=550 → x=200
Coins: 400, 600, 1000 - If A and B invest Rs. 1,00,000 and Rs. 2,00,000 respectively in a business and profit at year-end is Rs. 30,000, find each one’s share.
Solution:
Investment ratio = 1:2
A’s share = (1/3)×30,000 = Rs. 10,000
B’s share = (2/3)×30,000 = Rs. 20,000 - A 10-foot pole casts an 8-foot shadow. Simultaneously, a tower casts a 40-foot shadow. How tall is the tower?
Solution:
Height/Shadow = Constant
10/8 = x/40 → x = (10/8)×40 = 50 feet - The ratio of marks obtained by Vinod and Basu is 6:5. If their sum is 275, find their individual marks.
Solution:
Total parts = 6+5=11
Vinod = (6/11)×275 = 150
Basu = (5/11)×275 = 125 - If a:b = 7:8 and b:c = 7:9, find a:b:c.
Solution:
a:b = 7:8 = 49:56
b:c = 7:9 = 56:72
a:b:c = 49:56:72 - If x:y = 5:4, find (x/y):(y/x).
Solution:
(x/y):(y/x) = (5/4):(4/5) = 25:16 - A sum of Rs. 981 is divided in the ratio 5:4. What is the smaller share?
Solution:
Total parts = 9
Smaller share = (4/9)×981 = Rs. 436 - If the ratio of two numbers is 4:7 and their sum is 44, find the numbers.
Solution:
4x+7x=44 → 11x=44 → x=4
Numbers: 16 and 28 - If the ratio of the present ages of A and B is 6:4, and five years ago it was 5:3, find their current ages.
Solution:
Let ages be 6x, 4x
Five years ago: (6x-5):(4x-5)=5:3
3(6x-5)=5(4x-5) → 18x-15=20x-25 → 2x=10 → x=5
Ages: 30 and 20 - Divide Rs. 1500 among A, B, C in the ratio 2:3:5.
Solution:
Total parts = 10
A: 300, B: 450, C: 750 - If the seats for Mathematics, Physics, and Biology in a school are in the ratio 5:7:8 and increased by 40%, 50%, and 75% respectively, what is the new ratio?
Solution:
Math: 5x×1.4=7x, Physics: 7x×1.5=10.5x, Biology: 8x×1.75=14x
Ratio: 7:10.5:14 = 14:21:28 - In a partnership, A, B, and C invest Rs. 3600, Rs. 4400, and Rs. 2800. A is a working partner and gets 1/4 of the profit for his services, remaining is shared in the ratio of investment. If A’s total share is Rs. 8000, find B’s share.
Solution:
Let profit = P
A’s total = (1/4)P + (3/4)P×(3600/10800) = 8000
(1/4)P + (3/4)P×(1/3)=8000
(1/4)P + (1/4)P = 8000 → (1/2)P = 8000 → P = 16,000
B’s share = (3/4)×16,000×(4400/10800) = Rs. 4,888.89 - If 20 men or 24 women or 40 boys can do a task in 12 days, how many men working with 6 women and 2 boys can do a task four times as big in 12 days, working 5 hours a day?
Solution:
This is a work equivalence problem.
Let 1 man = x units/day, 1 woman = y, 1 boy = z
20x=24y=40z
x=1, y=20/24=5/6, z=20/40=1/2
Let number of men be n
Total work = 4×20×12×8 (since original is 20 men × 12 days × 8 hours)
Required: (n + 6×5/6 + 2×1/2)×12×5 = 4×20×12×8
(n+5+1)×60 = 7680
(n+6)×60=7680 → n+6=128 → n=122 - If a sum is divided among A, B, C, D in the ratio 5:2:4:3 and the difference between D and B is Rs. 1000, find B’s share.
Solution:
D-B = 3x-2x=1000 → x=1000
B’s share = 2x = Rs. 2,000 - If the ratio of male to female population in a town is 7:5 and total population is 36,000, how many males and females are there?
Solution:
Males = (7/12)×36,000 = 21,000
Females = (5/12)×36,000 = 15,000
Bottom Line: These questions and solutions cover a wide range of ratio and proportion all type questions to help you practice and master the topic.
Multiple Choice Questions (MCQs) on Ratio and Proportion
Test your understanding of ratio and proportion with these 20 objective questions. These MCQs are suitable for competitive exams like CAT, GMAT, GRE, TANCET, and other quantitative reasoning assessments.
- If the ratio of two numbers is 4:5 and their sum is 72, what is the smaller number?
a. 28
b. 32
c. 36
d. 40
Correct answer: b. 32 - The ratio of A to B is 3:4. If A = 18, what is the value of B?
a. 21
b. 22
c. 24
d. 27
Correct answer: c. 24 - If 3x = 2y, what is the ratio of x to y?
a. 2:3
b. 3:2
c. 1:2
d. 2:1
Correct answer: a. 2:3 - Divide Rs. 1800 between P and Q in the ratio 5:4. How much does Q get?
a. Rs. 800
b. Rs. 900
c. Rs. 1000
d. Rs. 1200
Correct answer: b. Rs. 900 - If a:b = 2:3 and b:c = 4:5, what is a:b:c?
a. 8:12:15
b. 2:4:5
c. 4:6:5
d. 2:3:5
Correct answer: a. 8:12:15 - The mean proportional between 9 and 16 is:
a. 12
b. 13
c. 15
d. 14
Correct answer: a. 12 - If 40% of a number is 24, what is 25% of the same number?
a. 10
b. 12
c. 15
d. 20
Correct answer: b. 15 - The ratio of boys to girls in a class is 5:7. If there are 60 boys, how many girls are there?
a. 72
b. 80
c. 84
d. 90
Correct answer: c. 84 - If a:b = 3:5 and b:c = 2:7, what is the ratio a:b:c?
a. 6:10:35
b. 3:2:7
c. 6:5:7
d. 3:10:14
Correct answer: a. 6:10:35 - If 20 liters of a solution contains milk and water in the ratio 3:2, how much water should be added to make the ratio 1:1?
a. 2 liters
b. 4 liters
c. 6 liters
d. 8 liters
Correct answer: b. 4 liters - If the ratio of the present ages of A and B is 5:7 and after 6 years it will be 7:9, what is A’s present age?
a. 24
b. 30
c. 35
d. 42
Correct answer: b. 30 - The ratio of the incomes of A and B is 7:9 and their expenditures are in the ratio 4:5. If each saves Rs. 2000, what is A’s income?
a. Rs. 7000
b. Rs. 9000
c. Rs. 14,000
d. Rs. 18,000
Correct answer: c. Rs. 14,000 - If a:b = 4:5 and b:c = 6:7, what is a:b:c?
a. 24:30:35
b. 4:6:7
c. 8:15:21
d. 12:15:21
Correct answer: a. 24:30:35 - In what ratio should Rs. 480 be divided between A and B so that A gets Rs. 80 more than B?
a. 5:3
b. 7:5
c. 3:2
d. 2:1
Correct answer: a. 5:3 - If x:y = 7:9, what is (x + y):y?
a. 8:9
b. 9:16
c. 16:9
d. 7:2
Correct answer: c. 16:9 - The ratio of two numbers is 2:3. If 6 is subtracted from each, the new ratio becomes 1:2. What are the numbers?
a. 10, 15
b. 12, 18
c. 14, 21
d. 16, 24
Correct answer: b. 12, 18 - If a:b = 5:8 and b:c = 4:7, find a:b:c.
a. 20:32:56
b. 5:4:7
c. 10:8:14
d. 8:32:56
Correct answer: a. 20:32:56 - If Rs. 960 is divided in the ratio 3:5, what is the smaller share?
a. Rs. 320
b. Rs. 360
c. Rs. 480
d. Rs. 600
Correct answer: a. Rs. 360 - If the ratio of the present ages of A and B is 4:3, and after 5 years it will be 9:8, what is B’s present age?
a. 15
b. 20
c. 25
d. 30
Correct answer: b. 20 - The ratio of the number of men to women in a company is 5:3. If there are 120 women, how many men are there?
a. 100
b. 150
c. 180
d. 200
Correct answer: b. 200
Bottom Line: These MCQs are ideal for self-assessment, mock tests, and quick revision for competitive exams and placement interviews.
Downloadable Resources and PDF Practice Sets for Ratio and Proportion
If you want to strengthen your understanding of ratio and proportion questions, having access to downloadable resources can be incredibly helpful. PDF files and eBooks containing curated aptitude questions and answers allow you to study and practice even when you’re offline. These resources typically include a wide variety of problems, detailed solutions, and sometimes even quick-reference formula sheets.
By downloading these materials, you can work through aptitude quizzes at your own pace, revisit tricky topics, and track your progress over time. Look for resources that cover different question patterns, ratio methods, and step-by-step explanations. Whether you’re preparing for competitive exams, job interviews, or simply want to improve your mathematical skills, having a collection of ratio and proportion PDFs or eBooks makes your preparation more flexible and effective.
Commonly available downloadable resources include:
- Aptitude question and answer booklets in PDF format
- Topic-wise quiz sets for self-assessment
- Formula sheets for quick revision
- eBooks with solved examples and practice problems
Make sure to choose reputable sources for your downloads to ensure quality and accuracy in your study material.
Conclusion
Understanding ratio and proportion is essential for excelling in quantitative aptitude sections of various exams. With regular practice and the right approach, you can solve these questions quickly and accurately. Use the sample problems above to test your skills, and remember to apply the key concepts and tips discussed.
Why It Matters?
With mastery of ratio and proportion, complex numerical calculations can be done with ease and speed, which is important for competitive exams and decision-making.
Practical Advice for Learners
- Practice every day using a combination of MCQs and word problems.
- Prepare a formula sheet.
- Attempt to time yourself as you solve the questions to work on your speed.
- Reviewing mistakes can help identify weak areas.
- Use downloadable PDFs for offline use.
- Discuss challenging problems with peers or mentors.
