If the values of two quantities A and B are 4 and 6 respectively, then we say that they are in the ratio 4:6 (read as "four is to six"). Ratio is the relation which one quantity bears another of the same kind, the comparison being made by considering what multiple, part or parts, one quantity is of the other. The ratio of two quantities "a" and "b" is represented as a:b and reads as "a is to b". here, "a" is called antecedent, "b" is the consequent. since the ratio express the number of times one quantity contains the other, it's an abstract quantity.
Ratio of any number of quantities is expressed after removing any common factors that all the terms of the ratio have. For example, if there are two quantities having values of 4 and 6, their ratio is 4:6, i.e., 2:3 after taking the common factor 2 between them out. similarly, if there are three quantities 6,8 and 18, there is a common factor among all three of them. so, dividing each of the term by 2, we get the ratio ass 3:4:9.
If two quantities whose values are A and B respectively are the in the ratio a:b, since we know that some common factor k(>0) would have been removed from A and B to get the ratio a:b, we can write the original values of the two quantities (i.e., A and B) as ak and bk respectively.For example, if the salaries of two persons are in the ratio 7:5, we can write their individual salaries as 7k and 5k respectively.
A ratio a:b can also be expressed as a/b. so if two items are in the ratio 2:3, we can say that their ratio is 2/3. if two terms are in the ratio 2, it means that they are in the ratio of 2, it means that they are in the ratio of 2/1 i.e., 2:1.
"A ratio is said to be ratio of greater or less quantity or of quantity according as antecedent is greater than, less than or equal to consequent". In other words,
- the ratio a:b where a>b is called a ratio of greater inequality (example 3:2)
- the ratio a:b where a<b is called a ratio of less inequality (example 3:2)
- the ratio a:b where a=b is called a ratio of equality (example 1:1)
From this we can find that a ratio of greater inequality is diminished and a ratio of less inequality is increased by adding the same quantity to both terms, i.e., in the ratio a:b, when we add the same quantity x (positive) to both the terms of the ratio, we have the following results
If a<b then (a+x):(b+x) > a:b
If a>b then (a+x):(b+x) < a:b
If a=b then (a+x):(b+x) = a:b
this idea can be helpful in questions on data interpretation when we need to compare fractions to find the larger of two given fractions.
If two quantities are in the ratio a:b , then the first quantity will be a/(a+b) times the total of two quantities and the second quantity will be equal to b/(a+b) times the total of two quantities.
To understand the concept of Ratio, we have to come across the more examples.
Example 1:The sum of two numbers is 84. if the two numbers are in the ratio 4:3, find the two numbers ?
solution: As the two numbers are in the ratio 4:3, let their actual values be 4x and 3x. As the sum of two numbers is 84, we have 4x + 3x = 84,which is equal to 7x = 84. so, x = 12.
Hence, 4x = 48 and 3x = 36.
Alternatively, the two numbers are (4/7)*84 = 48 and (3/7)*84 = 36 i.e., 48 and 36 respectively since the ratio of two numbers is 4:3..
Example 2: If 4a = 3b, find (7a+9b) : (4a:5b) .
Solution: it is given that 4a = 3b, hence, (a/b) = (3/4)
=> a =3k and b=4k, where k is the common factor of a and b. Required expression (7a+9b):(4a +5b)
= [(7*3k)+(9*4k)]:[(4*3k)+(5*4k)] = (21k + 36k) : (12k+20k) = 57k:32k = 57 : 32.
Example 3: The number of red balls and green balls in a bag are in the ratio of 16:7. if there are 45 more red balls than the green balls, find the number of green balls in the bag.
Solution: Since, the ratio of number of red and green balls is 16:7, let the number of red balls and green balls in the bag be 16x and 7x. so, the difference of red and green balls is 9x. 16x-7x = 9x = 45.
That means x =5. Hence, the number of green balls = 7x i.e., 35
Alternatively, 7x = (7/9)*(9x) = (7/9)*(45) = 35. Hence, there are 45 green balls in the bag.
Example 4: What least number must be added to each of a pair of numbers which are in the ratio
7 : 16 . so, that the ratio between the terms becomes 13 : 22 .
Solution : Let the number to be added to each number be a. Let the actual values of the numbers be 7x and 16x, since their ratio is 7 : 16. Given that, (7x+a)/(16x+a) = (13/22)
=> 154x + 22a = 208x + 13a => 9a = 54x.
=> a = 6x. when x =1, a is the least number required and is equal to 6.
Example 5: A number is divided into four parts such that 4 times the first part, 3 times the second part, 6 times the third part and 8 times the fourth part are all equal. in what ratio is number is divided .
Solution: lets the four parts into which the number is divided be a,b,c and d. 4a = 3b =6c = 8d.
Let the value of each of these equal to e. a = e/4, b = e/3, c = e/6, and d = e/8.
Hence, a:b:c:d = e/4 : e/3 : e/6 : e/8 = 6/24 : 8/24 : 4/24 : 3/24.The ratio of the parts into which the number is divided is 6 : 8 : 4 : 3.
Example 6: If x:y = 4:3, y:z = 2:3, find x : y : z ?
Solution: As y is common to both ratios , make y in both ratios equal. this is done by making y have the value equal to the LCM of two parts corresponding to y in the ratios i.e., LCM(3,2) is 6. if y = 6,
x = (4/3) * 6 = 8, z= (3/2)*6 = 9. Hence x : y : z = 8 : 6 : 9.
Example 7: In a mixture of milk and water, ratio of milk to water is 5:1 . if 5 liters of water is added to the mixture, the ratio becomes 5:2. Determine the quantity of milk in mixture initially?
Solution: The ratio of milk and water is 5 : 1. let water and milk are 5k and k. when the k is common factor. (5k +5) / (k + 5) = 5/2. 10k +10 = 5k + 25, k = 3. So the answer is 15 liters of milk is there in the initial mixture of milk and water.
Ratio of any number of quantities is expressed after removing any common factors that all the terms of the ratio have. For example, if there are two quantities having values of 4 and 6, their ratio is 4:6, i.e., 2:3 after taking the common factor 2 between them out. similarly, if there are three quantities 6,8 and 18, there is a common factor among all three of them. so, dividing each of the term by 2, we get the ratio ass 3:4:9.
If two quantities whose values are A and B respectively are the in the ratio a:b, since we know that some common factor k(>0) would have been removed from A and B to get the ratio a:b, we can write the original values of the two quantities (i.e., A and B) as ak and bk respectively.For example, if the salaries of two persons are in the ratio 7:5, we can write their individual salaries as 7k and 5k respectively.
A ratio a:b can also be expressed as a/b. so if two items are in the ratio 2:3, we can say that their ratio is 2/3. if two terms are in the ratio 2, it means that they are in the ratio of 2, it means that they are in the ratio of 2/1 i.e., 2:1.
"A ratio is said to be ratio of greater or less quantity or of quantity according as antecedent is greater than, less than or equal to consequent". In other words,
- the ratio a:b where a>b is called a ratio of greater inequality (example 3:2)
- the ratio a:b where a<b is called a ratio of less inequality (example 3:2)
- the ratio a:b where a=b is called a ratio of equality (example 1:1)
From this we can find that a ratio of greater inequality is diminished and a ratio of less inequality is increased by adding the same quantity to both terms, i.e., in the ratio a:b, when we add the same quantity x (positive) to both the terms of the ratio, we have the following results
If a<b then (a+x):(b+x) > a:b
If a>b then (a+x):(b+x) < a:b
If a=b then (a+x):(b+x) = a:b
this idea can be helpful in questions on data interpretation when we need to compare fractions to find the larger of two given fractions.
If two quantities are in the ratio a:b , then the first quantity will be a/(a+b) times the total of two quantities and the second quantity will be equal to b/(a+b) times the total of two quantities.
To understand the concept of Ratio, we have to come across the more examples.
Example 1:The sum of two numbers is 84. if the two numbers are in the ratio 4:3, find the two numbers ?
solution: As the two numbers are in the ratio 4:3, let their actual values be 4x and 3x. As the sum of two numbers is 84, we have 4x + 3x = 84,which is equal to 7x = 84. so, x = 12.
Hence, 4x = 48 and 3x = 36.
Alternatively, the two numbers are (4/7)*84 = 48 and (3/7)*84 = 36 i.e., 48 and 36 respectively since the ratio of two numbers is 4:3..
Example 2: If 4a = 3b, find (7a+9b) : (4a:5b) .
Solution: it is given that 4a = 3b, hence, (a/b) = (3/4)
=> a =3k and b=4k, where k is the common factor of a and b. Required expression (7a+9b):(4a +5b)
= [(7*3k)+(9*4k)]:[(4*3k)+(5*4k)] = (21k + 36k) : (12k+20k) = 57k:32k = 57 : 32.
Example 3: The number of red balls and green balls in a bag are in the ratio of 16:7. if there are 45 more red balls than the green balls, find the number of green balls in the bag.
Solution: Since, the ratio of number of red and green balls is 16:7, let the number of red balls and green balls in the bag be 16x and 7x. so, the difference of red and green balls is 9x. 16x-7x = 9x = 45.
That means x =5. Hence, the number of green balls = 7x i.e., 35
Alternatively, 7x = (7/9)*(9x) = (7/9)*(45) = 35. Hence, there are 45 green balls in the bag.
Example 4: What least number must be added to each of a pair of numbers which are in the ratio
7 : 16 . so, that the ratio between the terms becomes 13 : 22 .
Solution : Let the number to be added to each number be a. Let the actual values of the numbers be 7x and 16x, since their ratio is 7 : 16. Given that, (7x+a)/(16x+a) = (13/22)
=> 154x + 22a = 208x + 13a => 9a = 54x.
=> a = 6x. when x =1, a is the least number required and is equal to 6.
Example 5: A number is divided into four parts such that 4 times the first part, 3 times the second part, 6 times the third part and 8 times the fourth part are all equal. in what ratio is number is divided .
Solution: lets the four parts into which the number is divided be a,b,c and d. 4a = 3b =6c = 8d.
Let the value of each of these equal to e. a = e/4, b = e/3, c = e/6, and d = e/8.
Hence, a:b:c:d = e/4 : e/3 : e/6 : e/8 = 6/24 : 8/24 : 4/24 : 3/24.The ratio of the parts into which the number is divided is 6 : 8 : 4 : 3.
Example 6: If x:y = 4:3, y:z = 2:3, find x : y : z ?
Solution: As y is common to both ratios , make y in both ratios equal. this is done by making y have the value equal to the LCM of two parts corresponding to y in the ratios i.e., LCM(3,2) is 6. if y = 6,
x = (4/3) * 6 = 8, z= (3/2)*6 = 9. Hence x : y : z = 8 : 6 : 9.
Example 7: In a mixture of milk and water, ratio of milk to water is 5:1 . if 5 liters of water is added to the mixture, the ratio becomes 5:2. Determine the quantity of milk in mixture initially?
Solution: The ratio of milk and water is 5 : 1. let water and milk are 5k and k. when the k is common factor. (5k +5) / (k + 5) = 5/2. 10k +10 = 5k + 25, k = 3. So the answer is 15 liters of milk is there in the initial mixture of milk and water.
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