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Have you ever wondered how things are shared fairly or how different amounts compare? That's exactly what Ratio & Proportion helps us understand! It's like knowing how many pieces of chocolate you get compared to your friend, or how much milk to add for a perfect cup of tea. Mastering this topic will make you a pro at solving many real-life problems and exam questions quickly.
When you have ratios like A:B and B:C, and you need to find A:B:C, make the 'B' term the same in both ratios by multiplying. Find the LCM of the 'B' values to do this fast.
If you see something like 2A = 3B = 4C, and need A:B:C, think of it as each part being equal to a simple number 'k'. Then express A, B, C in terms of 'k' (e.g., A=k/2). Then take the LCM of the denominators to remove fractions.
When you have a proportion with one missing value (e.g., 2:3 :: 4:x), just cross-multiply the terms. Multiply the outside numbers and the inside numbers, and set them equal. This quickly finds 'x'.
For simple direct proportion problems (more items, more cost), find the value of 'one unit' first. If 5 pens cost ₹50, then 1 pen costs ₹10 (50/5). Then multiply by the new quantity to find the total cost. This is super fast for mental math.
For inverse proportion problems (like workers and time), remember that if one quantity becomes 'n' times, the other becomes '1/n' times. So, if workers double, time halves. If workers become 3 times, time becomes 1/3. Just flip the ratio!
Imagine you have a basket with 5 red apples and 3 green apples. A ratio is simply a way to compare these two amounts. We can say the ratio of red apples to green apples is 5 to 3, or we write it as 5:3. It just tells us how much of one thing there is compared to another.
When two ratios are equal, we say they are in proportion. It's like saying: if the ratio of red to green apples in one basket is 2:1, and in another basket, the ratio of red to green apples is 4:2, then these two ratios are actually the same (because 4:2 simplifies to 2:1). So, we can write it as 2:1 :: 4:2 (read as '2 is to 1 as 4 is to 2').
There are mainly two types of proportion:
Learning to identify whether quantities are directly or inversely proportional is key to solving many problems related to time, work, speed, and distance. Always remember to simplify ratios to their smallest whole numbers to make comparisons easier. These fundamental concepts are the building blocks for more complex math problems in competitive exams.
Ratio (अनुपात)
a : b or a/bProportion (समानुपात)
a : b :: c : d or a/b = c/dProduct Rule of Proportion (समानुपात का गुणनफल नियम)
Product of Extremes = Product of Means (a × d = b × c)Compound Ratio (मिश्रित अनुपात)
(a/b) × (c/d) × (e/f) = (a×c×e) / (b×d×f)Inverse Ratio (विलोम अनुपात)
If ratio is a:b, its inverse ratio is b:a| Concept | Description | Example |
|---|---|---|
| Ratio | Comparison of two quantities | 3 apples : 2 bananas (3:2) |
| Proportion | Equality of two ratios | 1:2 :: 3:6 (1/2 = 3/6) |
| Direct Proportion | Both quantities increase/decrease together | More hours worked = more salary |
| Inverse Proportion | One quantity increases, other decreases | More workers = less time to finish work |
| Compound Ratio | Product of multiple ratios | (2:3) and (4:5) becomes (2x4):(3x5) = 8:15 |
Q: The ratio of boys to girls in a class is 3:2. If there are 30 boys, how many girls are there?
Q: Divide ₹600 among A, B, and C in the ratio 2:3:5.
Q: If 4 workers can build a wall in 12 days, how many days will 6 workers take to build the same wall?
Q: If A:B = 3:4 and B:C = 8:9, find A:B:C.
You and your friend are playing a game. For every 3 points you score, your friend scores 5 points. If your friend scored 45 points in total, how many points did you score?
In a cricket match, the runs scored by Rohit and Virat are in the ratio 4:5. If both together scored 189 runs, how many runs did Virat score?
A recipe for fruit punch says to mix apple juice and orange juice in a ratio of 7:3. If you use 210 ml of apple juice, how much orange juice do you need?
Your dad divides pocket money between you and your sibling in the ratio 5:4. If you got ₹100 more than your sibling, what was the total pocket money?
A sum of money is divided among P, Q, and R such that for every ₹2 P gets, Q gets ₹3, and for every ₹4 Q gets, R gets ₹5. Find the ratio P:Q:R.
If 2A = 3B = 4C, then what is A:B:C?
A mixture contains milk and water in the ratio 5:3. If 8 liters of water are added, the ratio becomes 1:1. What was the original quantity of milk?
Two numbers are in the ratio 7:11. If 7 is added to each number, the ratio becomes 2:3. Find the smaller number.
1The ratio of two numbers is 3:5. If their sum is 80, what is the larger number?
2If 12, 18, x, 30 are in proportion, what is the value of x?
3A bag contains coins of ₹1, 50 paise, and 25 paise in the ratio 2:3:4. If the total value is ₹180, what is the number of 50 paise coins?
4If A:B = 1:2, B:C = 3:4, and C:D = 5:6, then A:B:C:D is:
5The mean proportional between 4 and 16 is:
6If 15 men can finish a piece of work in 20 days, how many men are required to finish it in 10 days?
7The ages of two persons are in the ratio 5:7. Ten years ago, their ages were in the ratio 3:5. Find their present ages.
8The ratio of milk and water in a 40-liter mixture is 7:3. How much water should be added to make the ratio 3:7?
9The ratio of two numbers is 9:7. If 12 is subtracted from each, the new ratio becomes 3:2. Find the sum of the numbers.
10A sum of ₹3100 is distributed among 100 students (boys and girls) such that each boy gets ₹30 and each girl gets ₹25. Find the number of boys.
11What number must be added to each term of the ratio 7:13 to make the ratio 2:3?
When you have ratios like A:B and B:C, and you need to find A:B:C, make the 'B' term the same in both ratios by multiplying. Find the LCM of the 'B' values to do this fast.
If you see something like 2A = 3B = 4C, and need A:B:C, think of it as each part being equal to a simple number 'k'. Then express A, B, C in terms of 'k' (e.g., A=k/2). Then take the LCM of the denominators to remove fractions.
When you have a proportion with one missing value (e.g., 2:3 :: 4:x), just cross-multiply the terms. Multiply the outside numbers and the inside numbers, and set them equal. This quickly finds 'x'.
For simple direct proportion problems (more items, more cost), find the value of 'one unit' first. If 5 pens cost ₹50, then 1 pen costs ₹10 (50/5). Then multiply by the new quantity to find the total cost. This is super fast for mental math.
For inverse proportion problems (like workers and time), remember that if one quantity becomes 'n' times, the other becomes '1/n' times. So, if workers double, time halves. If workers become 3 times, time becomes 1/3. Just flip the ratio!
a : b or a/ba : b :: c : d or a/b = c/dProduct of Extremes = Product of Means (a × d = b × c)+2 more formulas below