In the previous section we completed the discussion on ratios. In this chapter, we will discuss about percentage. We have seen fractions and decimals. Decimals are the next stage of development from fractions. The stage after decimals is percent. Let us see the details:
We know that, comparison of fractions become easy if they have a common denominator. We have also seen that decimals are fractions with denominator 10, 100, 1000 etc.,. Now, Percent is a special decimal in which the denominator is always 100. In fact, percent is two words combined: 'per' and 'cent'. Per means 'for every'. And cent means 'hundred'. So percent means: 'for every hundred'. It is indeed true because, the denominator is always 100.
Consider the fraction 35⁄100. It means 35 parts taken from 100 equal parts. That is 35 'per 100'. In decimal form it is 0.35. But when the denominator is 100, we can write it as 35%. It is read as 'Thirty five Percent'.
Similarly, 22⁄100 is 22%, 74⁄100 is 74%. To write a fraction or decimal as a percent, the denominator must always be 100. Let us see some examples. Fig.8.1(a) below shows a square area divided into 25 equal parts.
In (b), 4 out of 25 is shaded. In fraction form, it is 4⁄25. To write it in decimal form, we must convert it into an equivalent fraction with denominator 10, or 100, or 1000 and so on. We can choose any one out of these 10, 100, 1000 etc., which ever is convenient. But to write it in percent form, there is no choice. The denominator must be exactly 100. So we will select 100.
4⁄25 = x⁄100. By cross products method, we get x = 16.
Thus 4⁄25 = 16⁄100. But 16⁄100 = 16%. So we can write:
In fig.(b), 16% of the square area is shaded.
Similarly in fig.(c), 6⁄25 is shaded. 6⁄25 = 24⁄100 = 24%. So we can write:
In fig.(c), 24% of the square area is shaded.
Now we will see some solved examples
Solved example 8.1
In a class of 20 students, 12 are boys. What percent of the whole class is boys?
Solution:
• Total number of students = 20
• Number of boys = 12
• Number of boys expressed as a fraction of the total number of students = 12⁄20
• Equivalent fraction of 12⁄20 with denominator 100 = 60⁄100
Thus, 60% of the whole class are boys.
Solved example 8.2
In a library, there are 1200 books. Out of them, 240 are on the subject Maths. What percent of the whole books are Maths books?
Solution:
• Total number of books = 1200
• Number of Maths books = 240
• Number of maths books expressed as a fraction of the total number of books = 240⁄1200
• Equivalent fraction of 240⁄1200 with denominator 100 = 20⁄100
Thus, 20% of the total of books in the library are Maths books.
Solved example 8.3
A cake is divided into equal pieces. Student A took 4 pieces. Student B took 5 pieces. Student C took the remaining 3 pieces. What percentage of the whole cake did each student take?
Solution:
Number of pieces that A took = 4
Number of pieces that B took = 5
Number of pieces that C took = 3
∴ Total number of pieces = 12
• Fraction taken by A = 4⁄12 = 33.33⁄100
Proof:
Let 4⁄12 = x⁄100
Cross multiplying we get 400 = 12 x
∴ x = 400/12 = 100/3 = 33.33
• So A took 33.33% of the whole cake
Fraction taken by B = 5⁄12 = 41.67⁄100
• So B took 41.67% of the whole cake
Fraction taken by C = 3⁄12 = 25⁄100
• So C took 25% of the whole cake
This completes the solution to the problem. But we will do an additional calculation:
■ Let us add the number of pieces:
Number of pieces taken by A = 4
Number of pieces taken by B = 5
Number of pieces taken by C = 3
Sum = 4 +5 + 3 = 12 pieces = One full cake
■ Let us add the fractions:
Fraction taken by A = 4⁄12
Fraction taken by B = 5⁄12
Fraction taken by C = 3⁄12
Sum = 4⁄12 + 5⁄12 + 3⁄12 = 12⁄12 = 1 One full cake
■ Let us add all the 3 percentages:
Percentage taken by A = 33.33
Percentage taken by B = 41.67
Percentage taken by C = 25
Sum = 33.33 + 41.67 + 25 = 100% = 1 full cake
This is an interesting result. We add the 'percentage of the whole' taken by each student, and the sum that we get is 100 %, which is 1 full cake.
Solved example 8.4
Some bricks were unloaded from a truck. Three workers A, B and C did the unloading. A unloaded 37 bricks. B unloaded 42 bricks. C unloaded the remaining 21 bricks. What percentage of the whole did each worker unload?
Solution:
Number of bricks that A unloaded = 37
Number of bricks that B unloaded = 42
Number of bricks that C unloaded = 21
• Total number of bricks = 37 + 42 + 21 = 100
Fraction unloaded by A = 37⁄100
• Here the denominator is already 100. This is because, 'the whole' (the total number of bricks) which gives the denominator is 100. So we do not need to find an equivalent fraction.
So A unloaded 37% of the whole
• Fraction unloaded by B = 42⁄100
So B unloaded 42% of the whole
• Fraction unloaded by C = 21⁄100
So C unloaded 21 % of the whole
Solved example 8.5
Convert each of the following fractions into percentage:
(i) 5⁄16 (ii) 1⁄3 (iii) 4⁄9 (iv) 5⁄8 (v) 7⁄7
Solution:
(i) 5⁄16 = x⁄100 ⇒ x = 5⁄16 × 100 = 500⁄16 = 31.25
∴ 5⁄16 = 31.25⁄100 = 31.25%
• The above steps can be written in just one line:
Multiply the given fraction by 100. Put a '%' sign at the end of the product. That is the required percentage.
(ii) 1⁄3 = x⁄100 ⇒ x = 1⁄3 × 100 = 100⁄3 = 33.33
∴ 1⁄3 = 33.33⁄100 = 33.33%
• The above steps can be written in just one line:
Multiply the given fraction by 100. Put a '%' sign at the end of the product. That is the required percentage.
(iii) 4⁄9 = x⁄100 ⇒ x = 4⁄9 × 100 = 400⁄9 = 44.44
∴ 4⁄9 = 44.44⁄100 = 44.44%
• The above steps can be written in just one line:
Multiply the given fraction by 100. Put a '%' sign at the end of the product. That is the required percentage.
■ From the above examples, we can conclude that, to convert a fraction into a percentage, all we need to do is to multiply the fraction by 100. And then, put a '%' sign at the end of the product.
(iv) 5⁄8 × 100 = 500⁄8 = 62.5
∴ 5⁄8 = 62.5%
(v) 7⁄7 × 100 = 700⁄7 = 100
∴ 7⁄7 = 100%
• It may be noted that this problem does not require any steps to find the solution. This is because 7⁄7 (which is equal to 1) is actually not a fraction. It is a 'whole 1'. And a 'whole 1' is 100%
Solved example 8.6
Convert each of the following decimals into percentage:
(i) 0.25 (ii) 0.74 (iii) 0.3 (iv) 0.03 (v) 0.003 (vi) 0.0003 (vii) 2.32
Solution:
(i) 0.25 = 25⁄100 = 25%
(ii) 0.74 = 74⁄100 = 74%
(iii) 0.3 = 3⁄10 = 30⁄100 = 30%
(iv) 0.03 = 3⁄100 = 3%
(v) 0.003 = 0.3⁄100 = 0.3%
(vi) 0.0003 = 0.03⁄100 = 0.03%
(vii) 2.32 = 232⁄100 = 232%
■ From the above examples we can conclude that, to convert a decimal into a percentage, all that we need to do is write the given decimal as an appropriate fraction with 100 as denominator. Then take out the numerator and give a '%' sign.
Solved example 8.7
Convert each of the following percentages into fraction and decimal forms:
(i) 12% (ii) 20% (iii) 84% (iv) 7%
Solution:
■ Whenever we get a fraction with a denominator 100, we can straight away write it into percentage form, just by using the numerator. In the present problem, we are doing the reverse: We are given the percentage. So this given percentage will be the numerator and the denominator will be none other than 100. Using this fact, we can straight away write the fraction form. Also, once we derive the fraction form, we can straight away write the decimal form because the denominator is 100. [The fraction form with denominator 100 may be reduced to the simplest form if possible]
(i) 12%. In the fraction form, this 12 will be the numerator and 100 will be the denominator. So we can write 12% = 12⁄100 = 3⁄25 (dividing both numerator and denominator by 4)
When the fraction form with the denominator 10, or 100, or 1000 etc., is given, we can straight away write the decimal form. Thus 12⁄100 = 0.12
So we can write: 12% = 3⁄25 = 0.12
(ii) 20% = 20⁄100 = 0.20
(iii) 84% = 84⁄100 = 21⁄25 = 0.84
(iv) 7% = 7⁄100 = 0.07
In the next section we will see more details about percentage.
We know that, comparison of fractions become easy if they have a common denominator. We have also seen that decimals are fractions with denominator 10, 100, 1000 etc.,. Now, Percent is a special decimal in which the denominator is always 100. In fact, percent is two words combined: 'per' and 'cent'. Per means 'for every'. And cent means 'hundred'. So percent means: 'for every hundred'. It is indeed true because, the denominator is always 100.
Consider the fraction 35⁄100. It means 35 parts taken from 100 equal parts. That is 35 'per 100'. In decimal form it is 0.35. But when the denominator is 100, we can write it as 35%. It is read as 'Thirty five Percent'.
Similarly, 22⁄100 is 22%, 74⁄100 is 74%. To write a fraction or decimal as a percent, the denominator must always be 100. Let us see some examples. Fig.8.1(a) below shows a square area divided into 25 equal parts.
 |
| Fig.8.1 |
4⁄25 = x⁄100. By cross products method, we get x = 16.
Thus 4⁄25 = 16⁄100. But 16⁄100 = 16%. So we can write:
In fig.(b), 16% of the square area is shaded.
Similarly in fig.(c), 6⁄25 is shaded. 6⁄25 = 24⁄100 = 24%. So we can write:
In fig.(c), 24% of the square area is shaded.
Now we will see some solved examples
Solved example 8.1
In a class of 20 students, 12 are boys. What percent of the whole class is boys?
Solution:
• Total number of students = 20
• Number of boys = 12
• Number of boys expressed as a fraction of the total number of students = 12⁄20
• Equivalent fraction of 12⁄20 with denominator 100 = 60⁄100
Thus, 60% of the whole class are boys.
Solved example 8.2
In a library, there are 1200 books. Out of them, 240 are on the subject Maths. What percent of the whole books are Maths books?
Solution:
• Total number of books = 1200
• Number of Maths books = 240
• Number of maths books expressed as a fraction of the total number of books = 240⁄1200
• Equivalent fraction of 240⁄1200 with denominator 100 = 20⁄100
Thus, 20% of the total of books in the library are Maths books.
Solved example 8.3
A cake is divided into equal pieces. Student A took 4 pieces. Student B took 5 pieces. Student C took the remaining 3 pieces. What percentage of the whole cake did each student take?
Solution:
Number of pieces that A took = 4
Number of pieces that B took = 5
Number of pieces that C took = 3
∴ Total number of pieces = 12
• Fraction taken by A = 4⁄12 = 33.33⁄100
Proof:
Let 4⁄12 = x⁄100
Cross multiplying we get 400 = 12 x
∴ x = 400/12 = 100/3 = 33.33
• So A took 33.33% of the whole cake
Fraction taken by B = 5⁄12 = 41.67⁄100
• So B took 41.67% of the whole cake
Fraction taken by C = 3⁄12 = 25⁄100
• So C took 25% of the whole cake
This completes the solution to the problem. But we will do an additional calculation:
■ Let us add the number of pieces:
Number of pieces taken by A = 4
Number of pieces taken by B = 5
Number of pieces taken by C = 3
Sum = 4 +5 + 3 = 12 pieces = One full cake
■ Let us add the fractions:
Fraction taken by A = 4⁄12
Fraction taken by B = 5⁄12
Fraction taken by C = 3⁄12
Sum = 4⁄12 + 5⁄12 + 3⁄12 = 12⁄12 = 1 One full cake
■ Let us add all the 3 percentages:
Percentage taken by A = 33.33
Percentage taken by B = 41.67
Percentage taken by C = 25
Sum = 33.33 + 41.67 + 25 = 100% = 1 full cake
This is an interesting result. We add the 'percentage of the whole' taken by each student, and the sum that we get is 100 %, which is 1 full cake.
Solved example 8.4
Some bricks were unloaded from a truck. Three workers A, B and C did the unloading. A unloaded 37 bricks. B unloaded 42 bricks. C unloaded the remaining 21 bricks. What percentage of the whole did each worker unload?
Solution:
Number of bricks that A unloaded = 37
Number of bricks that B unloaded = 42
Number of bricks that C unloaded = 21
• Total number of bricks = 37 + 42 + 21 = 100
Fraction unloaded by A = 37⁄100
• Here the denominator is already 100. This is because, 'the whole' (the total number of bricks) which gives the denominator is 100. So we do not need to find an equivalent fraction.
So A unloaded 37% of the whole
• Fraction unloaded by B = 42⁄100
So B unloaded 42% of the whole
• Fraction unloaded by C = 21⁄100
So C unloaded 21 % of the whole
Solved example 8.5
Convert each of the following fractions into percentage:
(i) 5⁄16 (ii) 1⁄3 (iii) 4⁄9 (iv) 5⁄8 (v) 7⁄7
Solution:
(i) 5⁄16 = x⁄100 ⇒ x = 5⁄16 × 100 = 500⁄16 = 31.25
∴ 5⁄16 = 31.25⁄100 = 31.25%
• The above steps can be written in just one line:
Multiply the given fraction by 100. Put a '%' sign at the end of the product. That is the required percentage.
(ii) 1⁄3 = x⁄100 ⇒ x = 1⁄3 × 100 = 100⁄3 = 33.33
∴ 1⁄3 = 33.33⁄100 = 33.33%
• The above steps can be written in just one line:
Multiply the given fraction by 100. Put a '%' sign at the end of the product. That is the required percentage.
(iii) 4⁄9 = x⁄100 ⇒ x = 4⁄9 × 100 = 400⁄9 = 44.44
∴ 4⁄9 = 44.44⁄100 = 44.44%
• The above steps can be written in just one line:
Multiply the given fraction by 100. Put a '%' sign at the end of the product. That is the required percentage.
■ From the above examples, we can conclude that, to convert a fraction into a percentage, all we need to do is to multiply the fraction by 100. And then, put a '%' sign at the end of the product.
(iv) 5⁄8 × 100 = 500⁄8 = 62.5
∴ 5⁄8 = 62.5%
(v) 7⁄7 × 100 = 700⁄7 = 100
∴ 7⁄7 = 100%
• It may be noted that this problem does not require any steps to find the solution. This is because 7⁄7 (which is equal to 1) is actually not a fraction. It is a 'whole 1'. And a 'whole 1' is 100%
Solved example 8.6
Convert each of the following decimals into percentage:
(i) 0.25 (ii) 0.74 (iii) 0.3 (iv) 0.03 (v) 0.003 (vi) 0.0003 (vii) 2.32
Solution:
(i) 0.25 = 25⁄100 = 25%
(ii) 0.74 = 74⁄100 = 74%
(iii) 0.3 = 3⁄10 = 30⁄100 = 30%
(iv) 0.03 = 3⁄100 = 3%
(v) 0.003 = 0.3⁄100 = 0.3%
(vi) 0.0003 = 0.03⁄100 = 0.03%
(vii) 2.32 = 232⁄100 = 232%
■ From the above examples we can conclude that, to convert a decimal into a percentage, all that we need to do is write the given decimal as an appropriate fraction with 100 as denominator. Then take out the numerator and give a '%' sign.
Solved example 8.7
Convert each of the following percentages into fraction and decimal forms:
(i) 12% (ii) 20% (iii) 84% (iv) 7%
Solution:
■ Whenever we get a fraction with a denominator 100, we can straight away write it into percentage form, just by using the numerator. In the present problem, we are doing the reverse: We are given the percentage. So this given percentage will be the numerator and the denominator will be none other than 100. Using this fact, we can straight away write the fraction form. Also, once we derive the fraction form, we can straight away write the decimal form because the denominator is 100. [The fraction form with denominator 100 may be reduced to the simplest form if possible]
(i) 12%. In the fraction form, this 12 will be the numerator and 100 will be the denominator. So we can write 12% = 12⁄100 = 3⁄25 (dividing both numerator and denominator by 4)
When the fraction form with the denominator 10, or 100, or 1000 etc., is given, we can straight away write the decimal form. Thus 12⁄100 = 0.12
So we can write: 12% = 3⁄25 = 0.12
(ii) 20% = 20⁄100 = 0.20
(iii) 84% = 84⁄100 = 21⁄25 = 0.84
(iv) 7% = 7⁄100 = 0.07
In the next section we will see more details about percentage.
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