Brief Summary
This YouTube video by Viral Maths provides a comprehensive guide to ratios and proportions, covering everything from basic definitions to advanced problem-solving techniques. The instructor explains the fundamental concepts, including the relationship between ratios and fractions, and then moves on to more complex topics such as comparing quantities, solving number-based problems, and applying ratios to income, expenditure, and age-related scenarios. The video emphasizes the importance of understanding the underlying principles and provides numerous examples to illustrate each concept.
- Ratios are comparisons between quantities.
- Proportions are equalities between two ratios.
- Understanding the units and internal gaps is crucial for solving ratio problems.
- Various techniques like the "Baharupia Approach," "Reverse N Approach," "Lटकन Approach," "Hide and Seek Approach," "Mahabharat Approach," "Land Grabbing Approach," and "Sandwich Approach" are introduced for efficient problem-solving.
Intro
The video starts with a welcome message and an introduction to the topic of ratios and proportions. The instructor emphasizes that this session will be a one-stop solution, covering everything from the basics to advanced concepts. The session aims to eliminate any doubts about ratios and proportions, making it unnecessary to seek additional resources or paid courses. The topics to be covered include the basics of ratios and proportions, comparison between two, three, and four quantities, number-based problems, coin-related questions, addition and subtraction of ratios, and problems based on income, expenditure, and ages.
What is Ratio?
The instructor explains that a ratio is essentially a comparison between two or more quantities. To illustrate this, he uses the example of Ram and Shyam, who have ₹100 and ₹200, respectively. Instead of stating their individual amounts, they can express their wealth in terms of a ratio. Shyam has twice the amount of Ram, or Ram has half the amount of Shyam. This comparison can be written as a ratio of 1:2. The instructor also points out that a ratio is essentially a fraction, both serving the purpose of comparison. For example, if asked what fraction of Shyam's money Ram has, the answer would be 1/2, which is the same as the ratio 1:2.
Simplifying Ratios
The instructor explains how to simplify ratios by dividing both sides by a common factor. For example, the ratio 45:95 can be simplified by dividing both numbers by 5, resulting in the ratio 9:19. Similarly, the ratio 125:135 can be simplified to 25:27 by dividing both numbers by 5. The instructor provides another example, 64:124, which simplifies to 16:31 by dividing both numbers by 2 twice. He also explains the different ways to express a simplified ratio, such as "16 is to 31" or "16 ratio 31."
Important Rule About Units
The instructor presents a scenario with ₹300 and 600 km, asking for the ratio. He then explains that this comparison is not valid because ratios must be between the same units. You cannot compare rupees and kilometers directly. The units must be the same for a meaningful comparison. This is a crucial rule to remember when dealing with ratios.
What is Proportion?
The instructor introduces the concept of proportion, explaining that it is the equality between two ratios. For example, if one person has ₹2 and another has ₹3, the ratio is 2:3. If another person has ₹4 and another has ₹6, the ratio is 4:6. If both ratios simplify to the same value (in this case, 2:3), then the two pairs are in proportion. The instructor provides another example with the ratios 40:50 and 80:100, both of which simplify to 4:5, indicating that they are in proportion.
Simplifying Ratios and Real Values
The instructor explains that multiple sets of numbers can simplify to the same ratio. For example, the ratios 20:30, 40:60, 100:150, and 30:45 all simplify to 2:3. He emphasizes that while the simplified ratio is the same, the real values are different. To get from the ratio back to the real values, you need to multiply by a unique number. For example, to get from 2:3 to 20:30, you multiply by 10. To get to 40:60, you multiply by 20, and so on. This unique multiplier is often represented by the variable 'x' in mathematical problems.
Subtraction of Ratios
The instructor addresses the concept of subtracting ratios, emphasizing that it's not always straightforward. He explains that you can only subtract ratios if the internal gap between the numbers in each ratio is the same. For example, in the ratios 2:3 and 4:5, the internal gap is 1 in both cases, so you can subtract them. However, in the ratios 3:4 and 7:9, the internal gaps are different (1 and 2, respectively), so you cannot directly subtract them. In such cases, you need to make the internal gaps the same by multiplying one or both ratios by a suitable number before subtracting.
Making Internal Gaps Same
The instructor continues explaining subtraction of ratios, emphasizing the importance of equalizing internal gaps before subtracting. Using the example of ratios 3:5 and 7:11, he notes the gaps are 2 and 4, respectively. To equalize, the first ratio is multiplied by 2, resulting in 6:10, which has a gap of 4, matching the second ratio. Now, subtraction is possible, yielding a new ratio of 1:1. Another example, 5:11 and 4:7, is presented. The gaps are 6 and 3, respectively. Multiplying the second ratio by 2 gives 8:14, with a gap of 6. The ratios can now be subtracted, resulting in 3:3.
Distribution of Money in Ratio
The instructor explains how to distribute a sum of money according to a given ratio. For example, if ₹140 needs to be distributed between two people, Rahul and Owl, in the ratio 2:3, the first step is to find the total ratio, which is 2+3=5. Then, divide the total amount by the total ratio to find the value of one ratio unit (₹140 / 5 = ₹28). Finally, multiply each person's ratio by the value of one ratio unit to find their share (Rahul: 2 * ₹28 = ₹56, Owl: 3 * ₹28 = ₹84). The instructor also presents an alternative method: multiply the total amount by the individual's ratio and divide by the total ratio.
Finding Combined Ratios
The instructor explains how to find the combined ratio of three quantities when given the ratios of two pairs. For example, if A:B = 2:3 and B:C = 6:7, the goal is to find A:B:C. The instructor introduces the "Baharupia Approach," which involves finding the common element (in this case, B) and making its values the same in both ratios. Since B is 3 in the first ratio and 6 in the second, the first ratio is multiplied by 2 to make B equal to 6. The new ratios are A:B = 4:6 and B:C = 6:7, so A:B:C = 4:6:7.
Reverse N Approach
The instructor introduces another method for finding combined ratios, called the "Reverse N Approach." Using the same example as before (A:B = 5:14 and B:C = 7:9), the Reverse N Approach involves multiplying the numbers in a reverse N pattern: 5 * 7 = 35, 14 * 7 = 98, and 14 * 9 = 126. This gives the ratio A:B:C = 35:98:126, which can be simplified to 5:14:18 by dividing each number by 7.
When Reverse N Approach Fails
The instructor explains that the Reverse N Approach doesn't always work. It only works when the ratios are connected in a chain, like A to B and then B to C. If the ratios are not connected in this way, the Reverse N Approach will give the wrong answer. For example, if A:B = 10:3 and A:C = 5:6, the Reverse N Approach would not work. In this case, you need to use the Baharupia Approach, making the common element (A) the same in both ratios.
Finding Ratio from Equations
The instructor explains how to find the ratio between two variables when given an equation relating them. For example, if 2A = 3B, then A:B = 3:2. The instructor provides a simple rule: the value in front of one variable is the value of the other variable in the ratio. He then extends this concept to a more complex equation: 2A = 3B = 4C. In this case, the first step is to find the least common multiple (LCM) of the numbers in front of the variables (2, 3, and 4), which is 12. Then, divide each term by the LCM: 2A/12 = 3B/12 = 4C/12. This simplifies to A/6 = B/4 = C/3, so A:B:C = 6:4:3.
Lटकन Approach and Hide and Seek Approach
The instructor presents two approaches to solve the equation 2A = 3B = 4C. The first is the "Lटकन Approach," where you find the LCM of the coefficients (2, 3, 4), which is 12. Divide each term by the LCM to get A/6 = B/4 = C/3. The ratio A:B:C is then simply the denominators: 6:4:3. The second approach is the "Hide and Seek Approach." To find the value of A, hide A and multiply the other coefficients (3 * 4 = 12). To find the value of B, hide B and multiply the other coefficients (2 * 4 = 8). To find the value of C, hide C and multiply the other coefficients (2 * 3 = 6). The ratio A:B:C is then 12:8:6, which simplifies to 6:4:3.
Fractional Ratios
The instructor explains how to simplify ratios that involve fractions. For example, if the ratio is 1/2 : 16 2/3 : 1/4, the first step is to convert any mixed fractions to improper fractions. In this case, 16 2/3 becomes 50/3. The ratio is now 1/2 : 50/3 : 1/4. To get rid of the fractions, find the least common multiple (LCM) of the denominators (2, 3, and 4), which is 12. Then, multiply each term in the ratio by the LCM: (1/2) * 12 : (50/3) * 12 : (1/4) * 12. This simplifies to 6 : 200 : 3.
Land Grabbing Approach
The instructor introduces the "Land Grabbing Approach" for finding the combined ratio of four quantities when given the ratios of three pairs. For example, if A:B = 2:3, B:C = 4:5, and C:D = 1:2, the goal is to find A:B:C:D. The Land Grabbing Approach involves writing the ratios in a table, filling in the missing values by "grabbing" the values from the adjacent cells. Then, multiply the numbers in each column to get the combined ratio.
Sandwich Approach
The instructor presents the "Sandwich Approach" as an alternative method for solving the same problem. The Sandwich Approach involves multiplying the numbers in a specific pattern to find the values of A, B, C, and D. The instructor also introduces the "Mahabharat Approach" for finding the ratio of the first and last quantities (A:D) without finding the intermediate values.
Mahabharat Approach
The instructor introduces the "Mahabharat Approach" to find the ratio of the first and last terms in a series of ratios, such as finding A:D when given A:B, B:C, and C:D. This method involves multiplying all the numerators together and all the denominators together, simplifying the result to find the ratio of A to D directly.
Problem 1
The instructor presents the first problem: Three numbers are in the ratio 1/2 : 2/3 : 3/4. The difference between the greatest and smallest numbers is 27. Find the smallest number. The first step is to convert the fractional ratio to a whole number ratio by finding the LCM of the denominators (2, 3, and 4), which is 12. Multiplying each term by 12 gives the ratio 6:8:9. The difference between the greatest and smallest numbers (9 and 6) is 3, which is equal to 27. Therefore, 1 ratio unit is equal to 9. The smallest number is 6 ratio units, so the smallest number is 6 * 9 = 54.
Problem 2
A, B, and C divide ₹10,500 among themselves in the ratio 5:7:9. If each one gets ₹500 more, then what will be the ratio of the amounts with A, B, and C? Since each person receives the same additional amount (₹500), the ratio of the additional amounts is 1:1:1. Therefore, you can simply add 1 to each term in the original ratio to get the new ratio: 6:8:10, which simplifies to 3:4:5.
Problem 3
By mistake, instead of dividing ₹702 among Ram, Ramesh, and Naresh in the ratio 1/3 : 1/4 : 1/6, it was divided in the ratio 3:4:6. Who gained the most and by how much? First, convert the correct ratio (1/3 : 1/4 : 1/6) to a whole number ratio by finding the LCM of the denominators (3, 4, and 6), which is 12. Multiplying each term by 12 gives the ratio 4:3:2. Naresh gained the most.
Coin Based Problems
The instructor transitions to coin-based problems, establishing key concepts. Rupees are larger than paise, so converting from paise to rupees involves division, while converting from rupees to paise involves multiplication. He emphasizes the importance of knowing the fractional values of common denominations of paise (25, 20, 10, 5, 50, and 75) for efficient problem-solving. The core principle is " जिसकी लाठी उसकी भैस" (the one with the stick owns the buffalo), meaning if the total value is given in rupees, the ratio should also be in rupees, and if the total number of coins is given, the ratio should be in terms of the number of coins.
Coin Based Problem
In a bag, 50-paise, 25-paise, and ₹1 coins are in the ratio 5:8:1. The total cost of all the coins is ₹55. Find how many coins of 25 paise are there. The total value is given in rupees, so the ratio needs to be in rupees as well. Convert the coin ratio to a rupee ratio by dividing each term by the fractional value of the coin (2 for 50 paise, 4 for 25 paise, and 1 for ₹1), resulting in the ratio 2.5:2:1. The total ratio is 5.5, which is equal to ₹55. Therefore, 1 ratio unit is equal to ₹10. The number of 25-paise coins is 8 ratio units, so there are 8 * 10 = 80 coins.
Problem 5
In a bag, there are coins of ₹5, ₹10, and ₹20. The total number of coins in the bag is 240. The number of coins of ₹5, ₹10, and ₹20 are in the ratio 5:3:2. What is the total amount of money in the bag? The total number of coins is given, so the ratio is already in the correct form. The total ratio is 5+3+2=10, which is equal to 240 coins. Therefore, 1 ratio unit is equal to 24 coins. The number of ₹5 coins is 5 * 24 = 120, the number of ₹10 coins is 3 * 24 = 72, and the number of ₹20 coins is 2 * 24 = 48. The total amount of money is (120 * 5) + (72 * 10) + (48 * 20) = ₹3360.
Problem 6
Two numbers are in the ratio 3:4. On increasing each of them by 30, the numbers become in the ratio 9:10. The numbers are? The difference between the ratios is 6, which corresponds to the increase of 30. Therefore, 6 ratio units equal 30, and 1 ratio unit equals 5. The original numbers are 3 * 5 = 15 and 4 * 5 = 20.
Problem 7
The ratio in the number of students in three classes is 3:4:5. If 20 students are increased in every class, then this ratio becomes 4:5:6. What was the original number of students in classes taken together? The increase in each class is the same, so the difference in the ratios is constant. The difference between the new and old ratios is 1, which corresponds to 20 students. The original number of students is (3+4+5) * 20 = 240.
Problem 8
The number of students in the three sections of grade 10 in a school are in the proportion 3:5:8. If 15, 30, and 15 more students are admitted in these three sections respectively, the new proportion becomes 4:7:9. The total number of students before the new admission is? The increase in each section is proportional to the original ratio. The increase in the first section is 1, in the second section is 2, and in the third section is 1. Since 1 ratio unit corresponds to 15 students, the total number of students before the new admission is (3+5+8) * 15 = 240.
Problem 9
The ratio of the monthly income of A and B is 11:13 and the ratio of their expenditures is 9:11. If both of them manage to save ₹4000 per month, then find the difference in their income. The savings are the same for both A and B. The difference in the income ratio is 2, which corresponds to the savings of ₹4000. Therefore, 1 ratio unit equals ₹2000. The difference in their incomes is 2 ratio units, so the difference in their incomes is 2 * 2000 = ₹4000.
Panda Approach
The instructor introduces the "Panda Approach" for solving income and expenditure problems. This method involves cross-multiplying the income and expenditure ratios, as well as the savings. The differences between the cross-products are then used to find the value of one ratio unit. This approach works even when the savings are different for the two individuals.
Problem 10
The incomes of A and B are in the ratio 5:3. The expenses of A, B, and C are in the ratio 8:5:2. If C spends ₹2000 and B saves ₹700, then A saves? The instructor solves this problem by first finding the value of one ratio unit for the expenses, using the information about C's expenses. Then, he finds B's income by adding B's expenses and savings. Finally, he finds the value of one ratio unit for the incomes and uses that to find A's income and savings.
Problem 11
The ratio of last year's income of A, B, and C is 3:4:5, while the ratio of their last year's income to current year's income is 4:5, 2:3, and 3:4. If their total current year income is ₹98,500, then find out the present income of B+C. The instructor solves this problem by first finding the current year's income ratio for each person. Then, he finds the total current year's income ratio and uses that to find the value of one ratio unit. Finally, he finds the combined income of B and C.
Problem 12
The ratio of the present age of a mother to that of her daughter is 7:1. After 5 years, the ratio will become 4:1. What is the difference in their present ages? The instructor solves this problem by first equalizing the difference between the ratios. Then, he finds the value of one ratio unit and uses that to find the present ages of the mother and daughter. Finally, he finds the difference between their ages.
