Ratio and Proportion Problems
Welcome to the section on Ratio and Proportion, a fundamental topic in quantitative aptitude. Understanding ratios and proportions is crucial not only for solving specific problems but also for grasping concepts in other areas like percentages, averages, and partnerships. We will cover the definitions, types of problems, and step-by-step methods to solve them.
1. Understanding Ratios
A ratio is a comparison of two or more quantities of the same kind by division. It tells us how many times one quantity contains another. For example, if we have 5 apples and 10 oranges, the ratio of apples to oranges is 5:10, which can be simplified to 1:2. This means for every 1 apple, there are 2 oranges.
Key points about ratios:
- Ratios are dimensionless.
- The order of quantities in a ratio matters. 5:10 is different from 10:5.
- Ratios are usually expressed in their simplest form.
Types of Ratios:
1. Ratio of Equality: When both terms of the ratio are equal (e.g., 5:5 or 1:1).
2. Ratio of Greater Inequality: When the first term is greater than the second term (e.g., 5:2).
3. Ratio of Lesser Inequality: When the first term is less than the second term (e.g., 2:5).
Operations on Ratios:
1. Duplicate Ratio: The ratio of the squares of the terms (e.g., the duplicate ratio of a:b is a2:b2).
2. Sub-duplicate Ratio: The ratio of the square roots of the terms (e.g., the sub-duplicate ratio of a:b is √a : √b).
3. Triplicate Ratio: The ratio of the cubes of the terms (e.g., the triplicate ratio of a:b is a3:b3).
4. Sub-triplicate Ratio: The ratio of the cube roots of the terms (e.g., the sub-triplicate ratio of a:b is ³√a : ³√b).
5. Inverse Ratio: The ratio obtained by inverting the terms (e.g., the inverse ratio of a:b is b:a).
2. Understanding Proportions
A proportion is an equation stating that two ratios are equal. If a:b and c:d are two ratios, they are in proportion if a:b = c:d. This can be written as a/b = c/d.
In the proportion a:b = c:d, 'a' and 'd' are called the extremes (or extreme terms), and 'b' and 'c' are called the means (or mean terms).
The fundamental property of proportion is: Product of extremes = Product of means. So, in a:b = c:d, we have a * d = b * c.
Types of Proportions:
1. Direct Proportion: Two quantities are said to be in direct proportion if they increase or decrease together in the same ratio. For example, the cost of apples and the number of apples purchased. If you buy more apples, the cost increases proportionally. If x ∠x, then x = ky, where k is a constant.
2. Inverse Proportion: Two quantities are said to be in inverse proportion if, when one quantity increases, the other quantity decreases in the same ratio, and vice versa. For example, the speed of a vehicle and the time taken to cover a certain distance. If you increase the speed, the time taken decreases proportionally. If x ∠1/y, then xy = k, where k is a constant.
3. Types of Problems and Solutions
Type 1: Finding a missing term in a proportion
If four numbers a, b, c, and d are in proportion, then a:b = c:d, which means a * d = b * c. If any one term is missing, we can find it using this property.
Example: Find the fourth proportional to 4, 6, and 8. Let the fourth proportional be x. Then, 4:6 = 8:x Using the property, product of extremes = product of means: 4 * x = 6 * 8 4x = 48 x = 48 / 4 x = 12 So, the fourth proportional is 12.
Type 2: Problems involving continued proportion
Three numbers a, b, and c are said to be in continued proportion if a:b = b:c. This means b2 = ac. Here, 'b' is called the mean proportional between 'a' and 'c'.
Example: Find the mean proportional between 9 and 16. Let the mean proportional be x. Then, 9:x = x:16 x2 = 9 * 16 x2 = 144 x = √144 x = 12 So, the mean proportional is 12.
Example: Find the third proportional to 6 and 12. Let the third proportional be x. Then, 6:12 = 12:x 6 * x = 12 * 12 6x = 144 x = 144 / 6 x = 24 So, the third proportional is 24.
Type 3: Problems involving the ratio of two or more quantities
These problems often involve distributing amounts or comparing quantities based on given ratios.
Example: The ratio of two numbers is 3:5. If the sum of the numbers is 80, find the numbers. Let the two numbers be 3x and 5x. Their sum is 3x + 5x = 8x. We are given that the sum is 80. So, 8x = 80 x = 80 / 8 x = 10 The numbers are 3x = 3 * 10 = 30 and 5x = 5 * 10 = 50. Check: 30 + 50 = 80. The ratio 30:50 simplifies to 3:5.
Example: The ratio of the ages of A and B is 5:7. If the sum of their ages is 72 years, what is the present age of A? Let the ages of A and B be 5x and 7x years respectively. Sum of ages = 5x + 7x = 12x. Given sum = 72 years. 12x = 72 x = 72 / 12 x = 6 Present age of A = 5x = 5 * 6 = 30 years.
Type 4: Problems involving the ratio of three or more quantities
When you have ratios involving three or more quantities, you often need to combine them or find a common term.
Example: If A:B = 2:3 and B:C = 4:5, find A:B:C. To combine these ratios, we need to make the value of 'B' common in both ratios. The LCM of 3 (from A:B) and 4 (from B:C) is 12. Multiply the first ratio (2:3) by 4: A:B = (2*4):(3*4) = 8:12. Multiply the second ratio (4:5) by 3: B:C = (4*3):(5*3) = 12:15. Now, since B is 12 in both, we can combine them: A:B:C = 8:12:15.
Example: The ratio of boys to girls in a school is 5:4. If there are 250 boys, how many girls are there? Let the number of boys be 5x and the number of girls be 4x. We are given that the number of boys is 250. So, 5x = 250 x = 250 / 5 x = 50 The number of girls = 4x = 4 * 50 = 200.
Type 5: Problems involving mixture and alligation
These problems often deal with mixing two or more ingredients in a certain ratio to obtain a desired final product. The concept of alligation is a powerful tool here.
Example: In what ratio should rice costing Rs. 50 per kg be mixed with rice costing Rs. 60 per kg so that the mixture, when sold at Rs. 62 per kg, yields a profit of 10%? First, find the cost price of the mixture. Selling Price (SP) = Rs. 62 per kg Profit = 10% Cost Price (CP) = SP / (1 + Profit%) CP = 62 / (1 + 10/100) = 62 / (1.1) = 620 / 11 ≈ Rs. 56.36 per kg. Now, we need to mix rice costing Rs. 50/kg and Rs. 60/kg to get a mixture with CP of Rs. 620/11 per kg. Using the rule of alligation: (Price of cheaper ingredient) (Price of dearer ingredient) 50 60 \ / (Mean Price) ----> 620/11 / \ (Quantity of dearer) (Quantity of cheaper) Difference = 60 - 620/11 = (660 - 620)/11 = 40/11 Difference = 620/11 - 50 = (620 - 550)/11 = 70/11 The ratio of cheaper rice to dearer rice is (40/11) : (70/11) = 40 : 70 = 4:7. So, the rice should be mixed in the ratio 4:7.
Type 6: Problems involving share distribution in partnerships
Partnership problems are direct applications of ratios. When partners invest different amounts for different durations, their profit share is proportional to the product of their investment and time period.
Example: A, B, and C start a business. A invests Rs. 10,000 for 6 months, B invests Rs. 8,000 for 8 months, and C invests Rs. 12,000 for 4 months. If the total profit is Rs. 46,000, find the share of A. The ratio of their investments is: A's investment = 10,000 * 6 = 60,000 B's investment = 8,000 * 8 = 64,000 C's investment = 12,000 * 4 = 48,000 The ratio of their profits will be the ratio of these products: Profit Ratio A:B:C = 60,000 : 64,000 : 48,000 Simplifying by dividing by 1,000: 60 : 64 : 48 Simplifying further by dividing by 4: 15 : 16 : 12 Total parts = 15 + 16 + 12 = 43 parts. Total profit = Rs. 46,000. Share of A = (A's part / Total parts) * Total Profit Share of A = (15 / 43) * 46,000 Share of A = 15 * (46,000 / 43) Share of A = 15 * 1000 = Rs. 15,000.
Type 7: Problems involving ages and ratios
These problems relate the ratio of ages of individuals at different points in time (past, present, or future).
Example: The ratio of the present ages of Ram and Shyam is 4:5. Five years ago, the ratio of their ages was 3:4. Find their present ages. Let the present ages of Ram and Shyam be 4x and 5x years respectively. Five years ago, their ages were (4x - 5) and (5x - 5) years. The ratio of their ages five years ago was 3:4. So, (4x - 5) / (5x - 5) = 3 / 4 Cross-multiply: 4 * (4x - 5) = 3 * (5x - 5) 16x - 20 = 15x - 15 16x - 15x = 20 - 15 x = 5 Present age of Ram = 4x = 4 * 5 = 20 years. Present age of Shyam = 5x = 5 * 5 = 25 years. Check: 5 years ago, Ram was 15 and Shyam was 20. Ratio 15:20 = 3:4. This matches the condition.
4. Key Formulas and Concepts Recap
Here's a quick summary of the essential formulas and concepts for ratios and proportions:
| Concept | Formula/Definition |
|---|---|
| Ratio a:b | Comparison of two quantities by division. Simplest form is important. |
| Proportion a:b = c:d | ad = bc (Product of extremes = Product of means) |
| Continued Proportion a:b = b:c | b2 = ac |
| Mean Proportional | x = √(ab) between a and b |
| Third Proportional | x = b2/a to a and b |
| Direct Proportion | x ∠x ⇒ x = ky |
| Inverse Proportion | x ∠1/y ⇒ xy = k |
| Combining Ratios A:B=a:b, B:C=c:d | A:B:C = ac : bc : bd |
| Alligation (for mixing two items) | Quantity of cheaper : Quantity of dearer = (Dearer Price - Mean Price) : (Mean Price - Cheaper Price) |
| Partnership Profit Share | Share ∠Investment * Time |
5. Practice Problems and Application
The best way to master ratio and proportion is through consistent practice. Work through various types of problems, paying attention to the details of each question.
Example: The ratio of two numbers is 7:11. If 8 is added to each number, the ratio becomes 2:3. Find the numbers. Let the numbers be 7x and 11x. According to the problem: (7x + 8) / (11x + 8) = 2 / 3 Cross-multiply: 3 * (7x + 8) = 2 * (11x + 8) 21x + 24 = 22x + 16 24 - 16 = 22x - 21x 8 = x So, x = 8. The numbers are: First number = 7x = 7 * 8 = 56 Second number = 11x = 11 * 8 = 88 Check: (56+8) : (88+8) = 64 : 96. Dividing both by 32 gives 2:3. Correct.
Example: A sum of money is divided among P, Q, and R in the ratio 2:5:7. If R's share is Rs. 2800 more than Q's share, find the total sum of money. Let the shares of P, Q, and R be 2x, 5x, and 7x respectively. The difference between R's share and Q's share is R's share - Q's share = 7x - 5x = 2x. We are given that this difference is Rs. 2800. So, 2x = 2800 x = 2800 / 2 x = 1400 The total sum of money is P's share + Q's share + R's share = 2x + 5x + 7x = 14x. Total sum = 14 * x = 14 * 1400 = Rs. 19600.