In the previous section we saw solved examples on the calculation of a percentage of a quantity. We also saw their reverse problems. In this section, we will see the relation between percentage and ratios.
In chapter 7.1, we have seen the relation between ratios and fractions. There we saw that:
• Each item represented in a ratio can be expressed as a fraction of the final product.
• If they can be expressed as a fraction, they can be expressed as a percentage also.
Let us see an example:
We will take the same problem involving the two chemicals A and B. They are to be present in the ratio 2:3. We saw that, Chemical A will form 2⁄5 of the final product. And Chemical B will form 3⁄5 of the final product. We know how to convert fractions into percentages.
• 2⁄5 = 2⁄5 × 100 = 40% • 3⁄5 = 3⁄5 × 100 = 60%
Thus we can say: Quantity of Chemical A must form 40% of the final product. And that of Chemical B must form 60% of the final product. So, if we divide the final product into 100 equal parts, 40 parts must be of A and 60 parts must be of B.
Another example:
In a cake, sugar and flour are present in the ratio 1:4. Express this in percentage form.
Solution:
From the ratio, it is clear that 1 out of 5 parts of the cake is sugar. This is same as: sugar forms 1⁄5 of the cake. Similarly, 4 out of 5 parts of the cake is flour. Which is same as: Flour forms 4⁄5 of the cake.
• 1⁄5 = 1⁄5 × 100 = 20% • 4⁄5 = 4⁄5 × 100 = 80%
Thus, 20% of the cake is sugar, and 80% is flour.
Solved example 8.22
120 sweets are to be distributed betwee two students A and B in the ratio 9:11. How many sweets should be given to each. Solve by ratio method as well as percentage method.
Solution by ratio method:
• Total number of sweets = 120.
• Ratio between student A and student B = 9:11
• So the total 120 sweets have to be divided into 9 + 11 = 20 equal parts
• 120⁄20 = 6. So each part will have 6 sweets
• A will get 9 parts. That is., A will get 9 × 6 = 54 sweets
The above step is same as: dividing 120 by 20 first, and then multiplying by 9. That is:
Sweets for A = 120 × 9⁄20 = 54. The same result
• B will get 11 parts. That is., B will get 11 × 6 = 66 sweets
The above step is same as: dividing 120 by 20 first, and then multiplying by 11. That is:
Sweets for B = 120 × 11⁄20 = 66. The same result
• Check: 54 + 66 = 120. Also, 54:66 = 9:11 (dividing both sides by 6)
Solution by percentage method:
• Ratio = 9:11. So A will get 9 parts out of 20 parts. That is 9⁄20
• 9⁄20 = 9⁄20 × 100 = 45%
• 45% of 120 = 120 × 0.45 = 54
• B will get 11 parts out of 20. That is 11⁄20
• 11⁄20 = 11⁄20 × 100 = 55%
• 55% of 120 = 120 × 0.55 = 66
So we have seen how to express a ratio in the form of percentages. The reverse is also possible. That is., if we are given some percentages, we can write them in the form of a ratio.
Let us see an example:
₹500 is to be distributed among two workers A and B. A should get 40% of this 500. Write the distribution in the form of a ratio:
Solution:
• Total amount = ₹500
• A get 40%. So if 500 is divided into 100 equal parts, A will get 40 parts
• 500⁄100 = 5. So, when 500 is divided into 100 equal parts, each part will have ₹5
• A gets 40 such parts. So A gets 40 × 5 = ₹200
• When A gets 40%, naturally, B will get 60%
• So B gets 60 parts out of 100 equal parts.
• We have seen that when 500 is divided into 100 equal parts, each part is ₹5.
• So B gets 60 × 5 = ₹300
• So A and B gets money in the ratio 200:300 ⇒ 2:3 (dividing both sides by 100)
■ Thus we can say: Splitting some thing into 40% and 60% is same as splitting it in the ratio 2:3
In the next section we will see 'Increasing percentage'.
In chapter 7.1, we have seen the relation between ratios and fractions. There we saw that:
• Each item represented in a ratio can be expressed as a fraction of the final product.
• If they can be expressed as a fraction, they can be expressed as a percentage also.
Let us see an example:
We will take the same problem involving the two chemicals A and B. They are to be present in the ratio 2:3. We saw that, Chemical A will form 2⁄5 of the final product. And Chemical B will form 3⁄5 of the final product. We know how to convert fractions into percentages.
• 2⁄5 = 2⁄5 × 100 = 40% • 3⁄5 = 3⁄5 × 100 = 60%
Thus we can say: Quantity of Chemical A must form 40% of the final product. And that of Chemical B must form 60% of the final product. So, if we divide the final product into 100 equal parts, 40 parts must be of A and 60 parts must be of B.
Another example:
In a cake, sugar and flour are present in the ratio 1:4. Express this in percentage form.
Solution:
From the ratio, it is clear that 1 out of 5 parts of the cake is sugar. This is same as: sugar forms 1⁄5 of the cake. Similarly, 4 out of 5 parts of the cake is flour. Which is same as: Flour forms 4⁄5 of the cake.
• 1⁄5 = 1⁄5 × 100 = 20% • 4⁄5 = 4⁄5 × 100 = 80%
Thus, 20% of the cake is sugar, and 80% is flour.
Solved example 8.22
120 sweets are to be distributed betwee two students A and B in the ratio 9:11. How many sweets should be given to each. Solve by ratio method as well as percentage method.
Solution by ratio method:
• Total number of sweets = 120.
• Ratio between student A and student B = 9:11
• So the total 120 sweets have to be divided into 9 + 11 = 20 equal parts
• 120⁄20 = 6. So each part will have 6 sweets
• A will get 9 parts. That is., A will get 9 × 6 = 54 sweets
The above step is same as: dividing 120 by 20 first, and then multiplying by 9. That is:
Sweets for A = 120 × 9⁄20 = 54. The same result
• B will get 11 parts. That is., B will get 11 × 6 = 66 sweets
The above step is same as: dividing 120 by 20 first, and then multiplying by 11. That is:
Sweets for B = 120 × 11⁄20 = 66. The same result
• Check: 54 + 66 = 120. Also, 54:66 = 9:11 (dividing both sides by 6)
Solution by percentage method:
• Ratio = 9:11. So A will get 9 parts out of 20 parts. That is 9⁄20
• 9⁄20 = 9⁄20 × 100 = 45%
• 45% of 120 = 120 × 0.45 = 54
• B will get 11 parts out of 20. That is 11⁄20
• 11⁄20 = 11⁄20 × 100 = 55%
• 55% of 120 = 120 × 0.55 = 66
So we have seen how to express a ratio in the form of percentages. The reverse is also possible. That is., if we are given some percentages, we can write them in the form of a ratio.
Let us see an example:
₹500 is to be distributed among two workers A and B. A should get 40% of this 500. Write the distribution in the form of a ratio:
Solution:
• Total amount = ₹500
• A get 40%. So if 500 is divided into 100 equal parts, A will get 40 parts
• 500⁄100 = 5. So, when 500 is divided into 100 equal parts, each part will have ₹5
• A gets 40 such parts. So A gets 40 × 5 = ₹200
• When A gets 40%, naturally, B will get 60%
• So B gets 60 parts out of 100 equal parts.
• We have seen that when 500 is divided into 100 equal parts, each part is ₹5.
• So B gets 60 × 5 = ₹300
• So A and B gets money in the ratio 200:300 ⇒ 2:3 (dividing both sides by 100)
■ Thus we can say: Splitting some thing into 40% and 60% is same as splitting it in the ratio 2:3
In the next section we will see 'Increasing percentage'.
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