Percentage

Percentage
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“Percentages are numerators of fractions with denominator 100 and have been used in comparing results.”

Percentage: Comparison per 100 units

● In ratio we compare quantities by fraction (division) i.e one is how many times of other. Percentage is an equivalent of the ratio in which other quantity is expressed in 100 units.

● In Percentage we compare one quantity is how many times of hundredth (per cent) of other quantity.

● If x : y = 25 : 100, we would say x is 25 per 100 of y or x is 25 percent of y.

Percent is derived from Latin word ‘per centum’ meaning ‘per hundred’.

● Percent is represented by the symbol \% and means hundredths (1/100).

e.g., 1\% = \frac{1}{100}, 5\% = \frac {5}{100}, 21.5\% =\frac{21.5}{100}.

ℹ We can treat percentage as an equivalent ratio whose denominator is 100.

\frac{x}{y} = \frac{Percent}{100}

If \frac{x}{y} = \frac{50}{100}, we would say x is 50 percent of y.

Ex- In a mixture, ratio of milk to water is 3:5. Milk is what percent of water in the mixture?

A) \frac{Milk}{Water}=\frac{3}{5}=\frac{3\times 20}{5\times 20}=\frac{60}{100}
For 100 unit of water there is 60 unit of milk in the mixture.
Therefore, Milk is 60% of Water in the mixture.

ℹ We can treat percentage as an equivalent fraction whose denominator is 100.

\frac{Part}{Whole} = \frac{Percent}{100}

If \frac{Part}{Whole}=\frac{30}{100}, we would say the part is 30 percent of the whole.

● Parts always add to give 100.

(Q) If 65% of students in a class have a bicycle, what per cent of the student do not have bicycles ?

Ans: Students which do not have bicycles = (100−65)% = 35%.

Converting Fraction to Percentage

➢ Get the equivalent fraction whose denominator is 100 by multiplying both numerator and denominator with same number.
➢ Remove the 100 in denominator and express the number as percent.

Q) Write as a percentage: \frac {1}{2}, \frac{1}{7}, \frac{7}{5}

A) \frac{1}{2} = \frac{1\times 50}{2\times 50} = \frac{50}{100} = 50\%,

\frac{1}{7} = \frac{1\times 100}{7\times 100} = \frac{100}{7\times 100} =\frac{100}{7}\% = 14\frac{2}{7}\% ,

\frac{7}{5} = \frac{7\times 20}{5\times 20} = \frac{140}{100} = 140\%,

● Percentages related to proper fractions are less than 100 whereas percentages related to improper fractions are more than 100.

● To find the percentage we can use the formula –

\boxed{Percent = \frac{Part}{Whole}\times 100}

Converting Decimals to Percentage

➢ Convert the decimal into fraction
➢ Multiply or divide both numerator and denominator with same number to get 100 in denominator.
➢ Remove the 100 in denominator and express the number as percent.

Q) Write as a percentage : 0.65, 0.07, 4.57

A) 0.65 = \frac{65}{100} = 65\%,

0.07 = \frac{7}{100} = 7\%,

4.57 = \frac{457}{100} = 457\%.

Converting Percentage to Fraction or Decimals

➢ Remove the % and multiply the number by 1/100.

Q) Write as a fraction or decimal: 57\%, 3.8\%, 175\%

A) 57\% = \frac{57}{100} = 0.57 ,

3.8 \% = \frac{3.8}{100} = 0.038 ,

175 \% = \frac{175}{100} = \frac{7}{4} = 1.75

Converting Percentage to “How Many” (Part or Whole)

\boxed{\frac{Part}{Whole} = \frac{Percent}{100}}

\boxed{Part = \frac{Percent}{100} \times Whole }

\boxed{Whole = \frac{Percent}{100}\times Part }

Finding the part

Q) Find: (a) 50% of 164 (b) 75% of 12 (c)121/2 % of 64

Ans)
(a) 50\%\, of\, 164 = 50\% \times 164

= \frac{50}{100}\times 64 = \frac {1}{2}\times 64 = 32

(b) 75\% \, of\, 12 = 75\% \times 12

= \frac{75}{100}\times 12 = \frac{3}{4}\times 12 =3\times3 = 9

(c) 12\frac{1}{2}\% \, of\, 64 = \frac{25}{2}\% \times 64

= \frac{25}{2\times 100}\times 64 = \frac{1}{8}\times 64 =8

2). 8% children of a class of 25 like getting wet in the rain. How many children like getting wet in the rain.

Ans- Number of children like getting wet in the rain = 8\% \,of\, 25 = \frac{8}{100} \times 25 = 2

Finding the whole

Use the following formula
Whole = \frac{Percent}{100}\times Part

OR,
Algebraic method :
Assign a variable x to the whole, then solve the equation constructed according to the problem.

Q)(1) 9 is 25% of what number?
(2) 75% of what number is 15?

Ans(1). Let, the number be x, then,
25\% \,of\, x = 9

\implies \frac{25}{100}\times x = 9

\implies \frac{1}{4}\times x = 9

\implies x = 9\times 4 = 36

9 is 25\% of 36.

2). Let, the number be x, then,
75\% \,of\, x = 15

\implies \frac{75}{100}\times x = 15

\implies \frac{3}{4}\times x = 15

\implies x = 15\times \frac{4}{3} = 20

15 is 75\% of 20.

Ratio to Percents

Mixture – When two quantities X and Y are mixed in the ratio x to y.

● Percentage of X and Y w.r.t. each other.

\boxed{\%\,of\,X\,in\,Y=\frac{x}{y}\times 100}

\boxed{\%\, of\, Y \,in\, X = \frac{y}{x}\times 100}

● Percentage of X and Y in the mixture

\boxed{\%\, of \,X\, in\, the\, mixture = \frac{x}{x+y} \times 100}

\boxed{\%\, of\, Y\, in\, the\, mixture = \frac{y}{x+y} \times 100}

Q) In a salad, ratio carrot to radish is 2:1. What is the percentage of both in that mixture?
A) Total of all part (whole) = 2 + 1 = 3
Percentage of carrot = \frac{2}{3}\times 100

= \frac{200}{3}\%

= 66\frac{2}{3}\%

Percentage of radish = \frac{1}{3}\times 100

= \frac{100}{3}\%

= 33\frac{1}{3}\%

Q) If angles of a triangle are in the ratio 2 : 3 : 4. Find the value of each angle and its percentage.
A) Total of all part (whole) = 2 +3 +4 = 9
Sum of angles of triangle = 180°

First angle = \frac{2}{9} of 180\degree = \frac{2}{9}\times 180\degree = 40\degree

First angle percentage = \frac{2}{9}\times 100 = \frac{200}{9}\% = 22\frac{2}{9} \%

Second angle = \frac{3}{9} of 180\degree = \frac{3}{9}\times 180\degree = 60\degree

Second angle percentage = \frac{3}{9}\times 100 = \frac{300}{9}\% = 33\frac{1}{9} \%

Third angle = \frac{4}{9} of 180\degree = \frac{4}{9}\times 180\degree = 80\degree

Third angle percentage = \frac{4}{9}\times 100 = \frac{400}{9}\% = 44\frac{4}{9} \%

Increase or Decrease as Percent

\boxed{\frac{Amount\, of\, Change}{Original\, Amount\,(Base)} = \frac{Percent\, of\, change}{100}}

\boxed{\%\, change = \frac{Amount\, of\, Change}{Original\, Amount\,(Base)}\times 100}

\boxed{\%\, increase = \frac{Increase\, in\, Amount}{Original\,Amount\,(Base)}\times 100}

\boxed{\%\, decrease = \frac{Decrease\, in\, Amount}{Original\, Amount\, (Base)} \times 100}

Q1). Find Percentage of increase or decrease:
a) Price of shirt decreased from Rs 80 to Rs 60.
b) Marks in a test increased from 20 to 30.
Q2). My father says, in his childhood petrol was Re 1 a litre. It is Rs 52 per litre today. By what Percentage has the price gone up?

A1). a) Decrease in price of shirt = Rs 80 − Rs 60 = Rs 20
Percentage decrease in price of shirt = \frac{20}{80}\times 100\% = 25\%.

b) Increase in marks = 30 − 20 = 10
Percentage increase in marks= \frac{10}{30}\times 100\% = \frac{100}{3}% = 33\frac{1}{3}\%

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