In the previous section we saw the basics about ratios and proportions. In this section, we will discuss about the relation between ratios and fractions. Consider an example:
A special material for repairing cracks in walls was bought from a hardware shop. On opening the package, it was found that there are two separate boxes inside. The first box contains a chemical substance A. The second box contains another chemical substance B. Instructions on how to use the chemicals is also given with the package. It is as follows:
• Take two parts of Chemical A
• Take three parts of Chemical B
• Mix them well using a trowel, and it is ready to be applied on the cracks.
What do we understand from the above instructions?
The amount of each chemicals A and B is shown in the fig.7.5 below:
We need a measuring vessel. It will have markings on it's side. Using this measuring vessel, we must take the chemical A twice, and chemical B thrice. So the measuring is done 5 times. [Extreme care should be taken to ensure that, in each of these 5 times, the chemicals are taken upto the same level marking of the measuring vessel. So, the quantity taken will be the same in each of the 5 measurements]
It is a clear case of ratios. A and B are to be taken in the ratio 2:3. By following the above procedure, two parts of A and 3 parts of B will be obtained. When they are mixed together, the final product will be ready. This final product will be equal to 5 times the measuring vessel. Out of these 5, two are A and three are B. So:
• The fraction of A in the final product is 2⁄5
• The fraction of B in the final product is 3⁄5
So we succeeded in writing the ratio in the form of fractions. This has many applications. Let us look at a few:
Suppose it is estimated that a total of 15 litres of the final product will be required to repair the cracks. Then how much litres, each of A and B should be taken and mixed?
Solution:
• Final quantity required = 15 litres
• Divide it into 5 equal parts: 15/5 = 3 litres
• So we can do a grouping: There will be 5 groups. Each group will have 3 litres. This is shown in the fig.7.6 below:
All the groups are equal in size. Because each of them have 3 litres. So we succeeded in dividing the 15 litres into 5 equal parts. Now, to maintain the ratio of 2:3,
• Two groups must be of chemical A, and
• Three groups must be of chemical B.
• Two groups of A means 2 × 3 = 6 litres of A
• Three groups of B means 3 × 3 = 9 litres of B. This is shown in the fig.7.7 below:
• 6 + 9 = 15 litres. Thus we get the 'quantities of A and B' that are to be taken, to obtain 15 litres of the final product.
In the 15 litres of the final product, A and B are in the ratio 6 : 9. Dividing both sides by 3 we get 2 : 3. So 6:9 is an equivalent ratio of 2:3. And so, in the final 15 litres also, A and B are indeed present in the ratio 2:3
What we did in the above analysis was this:
• We divided 15 into 5 equal parts. This is because 2:3 is 2 parts + 3 parts = 5 parts
• 15⁄5 = 3. So, there must be 2 'threes' of A. 2 threes = 2 × 3 = 6 litres of A
• Mathematically, we can write the above two steps as 15⁄5 × 2 = 6 ⇒ 15 × 2⁄5 = 6
• So what we did was to multiply the 'required final quantity' by the 'fraction of A'
• Similarly, multiply the 'required final quantity' by the 'fraction of B':
• 15 × 3⁄5 = 9 litres.
If 25 litres of the final product is to be made, we must take 25 × 2⁄5 = 10 litres of A
And 25 × 3⁄5 = 15 litres of B
Check:
10 : 15 = 2:3 (dividing both sides by 5)
Another example:
In the preparation of certain food items, ratios will have to be strictly maintained. For example, in the making of a cake, sugar and flour may have to be in the ratio 1:4. If we are to prepare 12 kg of the cake, we can calculate the quantities of sugar and flour required as follows:
• Total quantity of the final product = 12 kg
• Ratio of sugar to flour = 1:4
• Quantity of sugar required = 12 × 1⁄5 = 2.4 kg
• Quantity of flour required = 12 × 4⁄5 = 9.6 kg
If the final quantity required is 10 kg, we can calculate the quantities as:
Quantity of sugar required = 10 × 1⁄5 = 2 kg
Quantity of flour required = 10 × 4⁄5 = 8 kg
Solved example 7.2
For making a special colour for a painting, an artist has to take yellow and green in the ratio 3:7. How much , each of yellow and green should be taken if he has to prepare 25 ml of the special colour?
Solution:
• Total quantity of the final product = 25 ml
• Ratio of yellow to green = 2:7
• Quantity of yellow required = 25 × 2⁄9 = 5.56 ml
• Quantity of green required = 25 × 7⁄9 = 19.44 ml
Solved example 7.3
For making a cake, sugar, cream and flour are to be taken in the ratio 1:3:5. How much of each quantity should be taken if the final weight of the cake is to be 15 kg?
Solution:
• Total quantity of the final product = 15 kg
• Ratio sugar : cream : flour = 1:3:5
• Quantity of sugar required = 15 × 1⁄9 = 1.67 kg
• Quantity of cream required = 15 × 3⁄9 = 5.0 kg
• Quantity of flour required = 15 × 5⁄9 = 8.33 kg
So we have seen one application for expressing ratios as fractions: It helps us to determine the quantity of each individual component in the final product. Now let us see another application:
Consider our first example: Combination of Chemical A and Chemical B. Let 12 litres of A be already taken. Then how much of B should be taken?
Solution:
• We know that A and B should always be in the ratio 2:3. Other wise the final product will not have the required properties.
• 12 litres of A is already taken. This must be exactly equal to 2⁄5 of the final product. Let the quantity of the final product be x
• Then we can write: 2⁄5 of x is equal to 12 litres. That is., x × 2⁄5 = 12 ⇒ 2x⁄5 = 12⁄1
• Taking cross products we get 2x × 1 = 12 × 5 ⇒ 2x = 60
∴ x = 60⁄2 = 30 litres. So the quantity of the final product is 30 litres.
• Now, B must be exactly equal to 3/5 of the final product.
• So quantity of B = 30 × 3⁄5 = 18 litres
• check: 12 + 18 = 30. Also 12 : 18 = 2:3 (dividing both sides by 6)
• So we get the quantity of B as 18 litres, when 12 litres of A is already taken.
Solved example 7.4
Mr.A and Mr.B decided to start a business together. They decided to invest money in the ratio 4:5. If A invested ₹20000, How much should B invest?
Solution:
• Ratio of investment by A and B = 4:5
• A invested ₹20000. This must be exactly equal to 4⁄9 of the final total amount. Let the final amount be x
• Then x × 4⁄9 = 20000 ⇒ 4x⁄9 = 20000⁄1 ⇒ 4x = 20000 × 9 ⇒ 4x = 180000
∴ x = 180000⁄4 = 45000. So the final amount is ₹45000
• Now, investment by B must be exactly equal to 5⁄9 of the final amount
• So investment made by B = 45000 × 5⁄9 = ₹25000
• Check: 20000 + 25000 = 45000. Also 20000 : 25000 = 4:5 (dividing both sides by 5000)
• So we get the investment to be made by B as 25000, when 20000 is already invested by A
Solved example 7.5
In the above problem 7.4, some profit was made by the business at the end of one year. If B's share from the profit was calculated as ₹32000, then what was the total profit? And what is the share of A?
Solution:
• Ratio of investment by A and B = 4:5
• The profit will be shared in the same ratio 4:5
• B's share is ₹32000. This must be exactly equal to 5⁄9 of the total profit. Let the total profit be x
• Then x × 5⁄9 = 32000 ⇒ 5x⁄9 = 32000⁄1 ⇒ 5x = 32000 × 9 ⇒ 5x = 288000
∴ x = 288000⁄5 = 57600. So the total profit is ₹57600
• Now, share of A must be exactly equal to 4⁄9 of the total profit
• So share of A = 57600 × 4⁄9 = ₹25600
• Check: 25600 + 32000 = 57600. Also 25600 : 32000 = 4:5 (dividing both sides by 6400)
• So we get the total profit as ₹57600 and the share of A as ₹25600
Solved example 7.6
Three friends A, B and C decided to take up a work. They decided to split the wages in the same ratio as the number of days each person works. A worked for 4 days and got ₹3600. B worked for 5 days and got ₹4500. If the total wages is ₹108000, For how many days did C work?
Solution:
• Total wages = 108000. This amount is split in the ratio of the number of days
• No. of days that A worked = 4; No. of days that B worked = 5
• Let the number of days that C worked = x
• Then the ratio = 4 : 5 : x
• A's share = 4⁄(4 +5 +x) of 10800 ⇒4⁄(9 +x) of 10800 ⇒ 10800 × 4⁄ (9 +x)
• B's share = 5⁄(4 +5 +x) of 10800 ⇒5⁄(9 +x) of 10800 ⇒ 10800 × 5⁄ (9 +x)
• But it is given that A's share is 3600. So we can write:
• 10800 × 4⁄(9 +x) = 3600 ⇒ 43200⁄(9 +x) = 3600⁄ 1
• Taking cross products, we get:
43200 = (9 +x) × 3600 ⇒ 43200 = 32400 + 3600x ⇒ 3600x = 43200 -32400 =10800
∴ x = 10800⁄3600 = 3
• So the number of days that C worked is 3
Check: B's share = 5⁄(4 +5 + 3) of 10800 ⇒5⁄12 of 10800 ⇒ 10800 × 5⁄12 = 4500 (same value given in question)
Another method:
• Total wages = 108000. This amount is split in the ratio of the number of days
• A's share = 3600; B's share = 4500
∴ C's share = 10800 -(3600 +4500) =2700
• This 2700 is x⁄(4 +5 + x) of 10800. So we can write:
• 2700 = 10800 × x⁄(4 +5 + x) ⇒ 2700 × (9 +x) = 10800x same as 24300 + 2700x = 10800x ⇒ 24300 = 10800x -2700x ⇒ 24300 = 8100x
∴ x = 24300⁄8100 = 3 (same as before)
So we have completed the discussion on ratios. In the next section we will discuss about percentage.
A special material for repairing cracks in walls was bought from a hardware shop. On opening the package, it was found that there are two separate boxes inside. The first box contains a chemical substance A. The second box contains another chemical substance B. Instructions on how to use the chemicals is also given with the package. It is as follows:
• Take two parts of Chemical A
• Take three parts of Chemical B
• Mix them well using a trowel, and it is ready to be applied on the cracks.
What do we understand from the above instructions?
The amount of each chemicals A and B is shown in the fig.7.5 below:
 |
| Fig.7.5 |
It is a clear case of ratios. A and B are to be taken in the ratio 2:3. By following the above procedure, two parts of A and 3 parts of B will be obtained. When they are mixed together, the final product will be ready. This final product will be equal to 5 times the measuring vessel. Out of these 5, two are A and three are B. So:
• The fraction of A in the final product is 2⁄5
• The fraction of B in the final product is 3⁄5
So we succeeded in writing the ratio in the form of fractions. This has many applications. Let us look at a few:
Suppose it is estimated that a total of 15 litres of the final product will be required to repair the cracks. Then how much litres, each of A and B should be taken and mixed?
Solution:
• Final quantity required = 15 litres
• Divide it into 5 equal parts: 15/5 = 3 litres
• So we can do a grouping: There will be 5 groups. Each group will have 3 litres. This is shown in the fig.7.6 below:
 |
| Fig.7.6 |
• Two groups must be of chemical A, and
• Three groups must be of chemical B.
• Two groups of A means 2 × 3 = 6 litres of A
• Three groups of B means 3 × 3 = 9 litres of B. This is shown in the fig.7.7 below:
 |
| Fig.7.7 |
In the 15 litres of the final product, A and B are in the ratio 6 : 9. Dividing both sides by 3 we get 2 : 3. So 6:9 is an equivalent ratio of 2:3. And so, in the final 15 litres also, A and B are indeed present in the ratio 2:3
What we did in the above analysis was this:
• We divided 15 into 5 equal parts. This is because 2:3 is 2 parts + 3 parts = 5 parts
• 15⁄5 = 3. So, there must be 2 'threes' of A. 2 threes = 2 × 3 = 6 litres of A
• Mathematically, we can write the above two steps as 15⁄5 × 2 = 6 ⇒ 15 × 2⁄5 = 6
• So what we did was to multiply the 'required final quantity' by the 'fraction of A'
• Similarly, multiply the 'required final quantity' by the 'fraction of B':
• 15 × 3⁄5 = 9 litres.
If 25 litres of the final product is to be made, we must take 25 × 2⁄5 = 10 litres of A
And 25 × 3⁄5 = 15 litres of B
Check:
10 : 15 = 2:3 (dividing both sides by 5)
Another example:
In the preparation of certain food items, ratios will have to be strictly maintained. For example, in the making of a cake, sugar and flour may have to be in the ratio 1:4. If we are to prepare 12 kg of the cake, we can calculate the quantities of sugar and flour required as follows:
• Total quantity of the final product = 12 kg
• Ratio of sugar to flour = 1:4
• Quantity of sugar required = 12 × 1⁄5 = 2.4 kg
• Quantity of flour required = 12 × 4⁄5 = 9.6 kg
If the final quantity required is 10 kg, we can calculate the quantities as:
Quantity of sugar required = 10 × 1⁄5 = 2 kg
Quantity of flour required = 10 × 4⁄5 = 8 kg
Solved example 7.2
For making a special colour for a painting, an artist has to take yellow and green in the ratio 3:7. How much , each of yellow and green should be taken if he has to prepare 25 ml of the special colour?
Solution:
• Total quantity of the final product = 25 ml
• Ratio of yellow to green = 2:7
• Quantity of yellow required = 25 × 2⁄9 = 5.56 ml
• Quantity of green required = 25 × 7⁄9 = 19.44 ml
Solved example 7.3
For making a cake, sugar, cream and flour are to be taken in the ratio 1:3:5. How much of each quantity should be taken if the final weight of the cake is to be 15 kg?
Solution:
• Total quantity of the final product = 15 kg
• Ratio sugar : cream : flour = 1:3:5
• Quantity of sugar required = 15 × 1⁄9 = 1.67 kg
• Quantity of cream required = 15 × 3⁄9 = 5.0 kg
• Quantity of flour required = 15 × 5⁄9 = 8.33 kg
So we have seen one application for expressing ratios as fractions: It helps us to determine the quantity of each individual component in the final product. Now let us see another application:
Consider our first example: Combination of Chemical A and Chemical B. Let 12 litres of A be already taken. Then how much of B should be taken?
Solution:
• We know that A and B should always be in the ratio 2:3. Other wise the final product will not have the required properties.
• 12 litres of A is already taken. This must be exactly equal to 2⁄5 of the final product. Let the quantity of the final product be x
• Then we can write: 2⁄5 of x is equal to 12 litres. That is., x × 2⁄5 = 12 ⇒ 2x⁄5 = 12⁄1
• Taking cross products we get 2x × 1 = 12 × 5 ⇒ 2x = 60
∴ x = 60⁄2 = 30 litres. So the quantity of the final product is 30 litres.
• Now, B must be exactly equal to 3/5 of the final product.
• So quantity of B = 30 × 3⁄5 = 18 litres
• check: 12 + 18 = 30. Also 12 : 18 = 2:3 (dividing both sides by 6)
• So we get the quantity of B as 18 litres, when 12 litres of A is already taken.
Solved example 7.4
Mr.A and Mr.B decided to start a business together. They decided to invest money in the ratio 4:5. If A invested ₹20000, How much should B invest?
Solution:
• Ratio of investment by A and B = 4:5
• A invested ₹20000. This must be exactly equal to 4⁄9 of the final total amount. Let the final amount be x
• Then x × 4⁄9 = 20000 ⇒ 4x⁄9 = 20000⁄1 ⇒ 4x = 20000 × 9 ⇒ 4x = 180000
∴ x = 180000⁄4 = 45000. So the final amount is ₹45000
• Now, investment by B must be exactly equal to 5⁄9 of the final amount
• So investment made by B = 45000 × 5⁄9 = ₹25000
• Check: 20000 + 25000 = 45000. Also 20000 : 25000 = 4:5 (dividing both sides by 5000)
• So we get the investment to be made by B as 25000, when 20000 is already invested by A
Solved example 7.5
In the above problem 7.4, some profit was made by the business at the end of one year. If B's share from the profit was calculated as ₹32000, then what was the total profit? And what is the share of A?
Solution:
• Ratio of investment by A and B = 4:5
• The profit will be shared in the same ratio 4:5
• B's share is ₹32000. This must be exactly equal to 5⁄9 of the total profit. Let the total profit be x
• Then x × 5⁄9 = 32000 ⇒ 5x⁄9 = 32000⁄1 ⇒ 5x = 32000 × 9 ⇒ 5x = 288000
∴ x = 288000⁄5 = 57600. So the total profit is ₹57600
• Now, share of A must be exactly equal to 4⁄9 of the total profit
• So share of A = 57600 × 4⁄9 = ₹25600
• Check: 25600 + 32000 = 57600. Also 25600 : 32000 = 4:5 (dividing both sides by 6400)
• So we get the total profit as ₹57600 and the share of A as ₹25600
Solved example 7.6
Three friends A, B and C decided to take up a work. They decided to split the wages in the same ratio as the number of days each person works. A worked for 4 days and got ₹3600. B worked for 5 days and got ₹4500. If the total wages is ₹108000, For how many days did C work?
Solution:
• Total wages = 108000. This amount is split in the ratio of the number of days
• No. of days that A worked = 4; No. of days that B worked = 5
• Let the number of days that C worked = x
• Then the ratio = 4 : 5 : x
• A's share = 4⁄(4 +5 +x) of 10800 ⇒4⁄(9 +x) of 10800 ⇒ 10800 × 4⁄ (9 +x)
• B's share = 5⁄(4 +5 +x) of 10800 ⇒5⁄(9 +x) of 10800 ⇒ 10800 × 5⁄ (9 +x)
• But it is given that A's share is 3600. So we can write:
• 10800 × 4⁄(9 +x) = 3600 ⇒ 43200⁄(9 +x) = 3600⁄ 1
• Taking cross products, we get:
43200 = (9 +x) × 3600 ⇒ 43200 = 32400 + 3600x ⇒ 3600x = 43200 -32400 =10800
∴ x = 10800⁄3600 = 3
• So the number of days that C worked is 3
Check: B's share = 5⁄(4 +5 + 3) of 10800 ⇒5⁄12 of 10800 ⇒ 10800 × 5⁄12 = 4500 (same value given in question)
Another method:
• Total wages = 108000. This amount is split in the ratio of the number of days
• A's share = 3600; B's share = 4500
∴ C's share = 10800 -(3600 +4500) =2700
• This 2700 is x⁄(4 +5 + x) of 10800. So we can write:
• 2700 = 10800 × x⁄(4 +5 + x) ⇒ 2700 × (9 +x) = 10800x same as 24300 + 2700x = 10800x ⇒ 24300 = 10800x -2700x ⇒ 24300 = 8100x
∴ x = 24300⁄8100 = 3 (same as before)
So we have completed the discussion on ratios. In the next section we will discuss about percentage.
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