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Aptitude Percentage Theory

Formulas

Percentage

  • Percentage = $\dfrac{Value}{Total\:Value}\times 100$

    • CONCEPT OF PERCENTAGE CHANGE

  • Absolute value change:

    Absolute value change = New Value − Actual Value

    Percentage change:

    Percentage change = $\dfrac{Absolute value change}{Actual Value } \times 100 $


    • Successive Percentage Change

  • (i)Both percentage changes are positive = (x + y + $\dfrac{xy}{100}) %$

    (ii)One percentage change is positive and the other is negative = (x - y - $\dfrac{xy}{100}) %$

    (iii)Both percentage changes are negative = (-x - y + $\dfrac{xy}{100}) %$


    • Multiplying Factor (M.F.)

  • When a quantity is increased by certain percentage.

    Final Value= Initial Value + x% of Initial Value

    When a quantity is decreased by certain percentage.

    Final Value= Initial Value - x% of Initial Value


    • RATIO INTO PERCENTAGE AND PERCENTAGE INTO RATIO

  • Ratio into Percentage

    x : y = $\dfrac{x}{y}$ = $\dfrac{x}{y} \times 100 $

    Percentage into Ratio

    x% = $\dfrac{x}{100}$ = x:100

    • Percent means “for every 100” or "out of 100".

    percent is denoted by the symbol %.

    Express x% as a fraction

    x percent is denoted by = x%

  • x% = $\dfrac{x}{100}$


    • Example

    25% = $\dfrac{25}{100}$ = $\dfrac{1}{4}$


    To express $\dfrac{x}{y}$ as a percent

    $\dfrac{x}{y}$ = $\dfrac{x}{y} \times 100$

  • Percentage = $\dfrac{Value}{Total\:Value}\times 100$


    • Example

    What is the percentage of $\dfrac{4}{5}$?

    = $\dfrac{4}{5} \times 100$

    = 80%


    Exercise

    44321.What is the percentage of $\dfrac{5}{10}$?
    50%
    45%
    80%
    20%
    Explanation:

    = $\dfrac{5}{10} \times 100$

    =50%

    CONCEPT OF PERCENTAGE CHANGE

    Whenever the value of a measured quantity changes, the change can be captured through

    (a)Absolute value change

    (b)Percentage change.


    Absolute value change:

    It is the actual change in the measured quantity.

  • Absolute value change = New Value − Actual Value


    • Percentage change:

    It is the percentage change got by the formula

  • Percentage change = $\dfrac{Absolute value change}{Actual Value } \times 100 $


    • Example

    For instance, if sales in year 1 is ` 2500 crore and the sales in year 2 is ` 2600 crore.

    then,

    The absolute value change = New Value − Actual Value = 2600-2500 = 100

    The absolute value change = 100

    Percentage change = $\dfrac{Absolute value change}{Actual Value } \times 100 $

    = $\dfrac{100}{2500} \times 100 $

    = 4%


    Exercise

    44322.The average attendance at a local high schools football games went down from 2000 people in 2003 to 1500 people in 2004. What was the percent decrease in attendance at the football games?
    50%
    45%
    25%
    20%
    Explanation:

    The absolute value change = New Value − Actual Value
    = 2000-1500 = 500
    Percentage change = $\dfrac{Absolute value change}{Actual Value } \times 100 $
    = $\dfrac{500}{2000} \times 100 $
    = 25%

    Successive Percentage Change

    (I) Both percentage changes are positive


    Formula

  • If there are successive percentage increases of a % and b%.the effective percentage increase is: (x + y + $\dfrac{xy}{100}) %$


    • Example

    A product of two variables say 10 × 10. If the first variable changes to 11 and the second variable changes to 12, what will be the percentage change in the product?

    Percentage change of 1st product (x) = $\dfrac{11-10}{10} \times 100 $ = 10%

    Percentage change of 2nd product (y) = $\dfrac{12-10}{10} \times 100 $ = 20%

    Successive Percentage Change = (x + y + $\dfrac{xy}{100}) %$

    = (10+ 20 + $\dfrac{10 \times 20}{100}) %$

    = 32% increase.


    Exercise

    44323.Two successive price increases of 10% and 10% of an article are equivalent to a single price increase of
    19%
    20%
    21%
    22%
    Explanation:

    here , x=10 and y=10
    Successive Percentage Change = (x + y + $\dfrac{xy}{100}) %$
    Successive Percentage Change = (10 + 10 + $\dfrac{10 \times 10}{100}) %$
    = 21%

    (II)One percentage change is positive and the other is negative


    Formula

  • If there are successive percentage increases of a % and decreases b%.the effective percentage change is: (x - y - $\dfrac{xy}{100}) %$


    • Example

    A's salary increases by 20% and then decreases by 20%. What is the net percentage change in A's

    Successive Percentage Change = (x - y - $\dfrac{xy}{100}) %$

    = (20- 20 - $\dfrac{20 \times 20}{100}) %$

    = 0 - 4

    = 4 % decrease


    Exercise

    44324.When the price of article was reduced by 20% its sale increased by 80%. What was the net effect on the revenue?
    44% increase
    44% decrease
    66% increase
    75% increase
    Explanation:

    here, x=80 and y=20
    Successive Percentage Change = (x - y - $\dfrac{xy}{100}) %$
    = (80 - 20 - $\dfrac{80 \times 20}{100}) %$
    = 44%

    (III)Both percentage changes are negative


    Formula

  • If there are successive percentage decreases of a % and decreases b%.the effective percentage change is: (-x - y + $\dfrac{xy}{100}) %$


    • Example

    A number is first decreased by 20% and then decrease by 10%. Find the Value?

    Successive Percentage Change = (-x - y + $\dfrac{xy}{100}) %$

    = (- 20 - 10 + $\dfrac{20 \times 10}{100}) %$

    = 28 % decrease


    Multiplying Factor (M.F.)

    Case 1:


    When a quantity is increased by certain percentage.

  • Final Value= Initial Value + x% of Initial Value


    • Example

    We have to increase a value 120 by 10%. What will be the final value?

    Final Value= Initial Value + 10% of Initial Value

    = 120 + 10% of 120

    =120 (1+ 10%)

    =120(1+$\dfrac{10}{100}$)

    =120(1.1)

    =120 x 1.1=132


    Exercise

    44325.we have to increase a value 120 by 20%. What will be the final value?
    165
    240
    144
    310
    Explanation:

    Final Value= Initial Value + 20% of Initial Value

    = 120 + 20% of 120

    =120 (1+ 20%)

    =120(1+20/100)

    =120(1+0.2)

    =120 x 1.2=144

    Case 2:


    When a quantity is decreased by certain percentage.

  • Final Value= Initial Value - x% of Initial Value


    • Example

    Suppose we have to decrease a value 120 by 20%. What will be the final value?

    Final Value= Initial Value - 20% of Initial Value

    = 120 - 20% of 120

    =120 (1 - 20%)

    =120(1-$\dfrac{20}{100}$)

    =120(1-0.2)

    =120 x 0.8=96


    Exercise

    44326.we decrease 120 by 10%. What will be the final value?
    98
    108
    245
    198
    Explanation:

    Final Value= Initial Value - 10% of Initial Value

    = 120 - 10% of 120

    =120 (1 - 10%)

    =120(1-10/100)

    =120(1- 0.1)

    =120 x 0.9=108

    RATIO INTO PERCENTAGE AND PERCENTAGE INTO RATIO

    Ratio into Percentage


    Step I: Obtain the ratio. Let the ratio be x : y

    Step II: Convert the given ratio into the fraction $\dfrac{x}{y}$.

    Step III: Multiply the fraction obtained in step II by 100 and put the percentage sign(%).


    Formula

  • x : y = $\dfrac{x}{y}$ = $\dfrac{x}{y} \times 100 $


    • Example
    Convert ratio into percentage of 8 : 25

    8 : 25 = $\dfrac{8}{25}$ = ($\dfrac{8}{25}$ × 100) % = 32 %


    Exercise

    44327.Express the given ratios into percent
    4 : 5
    40%
    45%
    95%
    80%
    Explanation:

    4 : 5 = 4/5 = (4/5 × 100) % = 80 %

    Percentage into Ratio


    Step I: Obtain the percentage.

    Step II: Convert the given percentage into fraction by dividing it by 100 and removing percentage symbol (%).

    Step III: Reduce the fraction obtained in step II in the simplest form.

    Step IV: Write the fraction obtained in step III as a ratio.


    Formula

  • x% = $\dfrac{x}{100}$ = x:100


    • Example

    Express the given percent into ratio- 20 %

    20 % = $\dfrac{20}{100}$ =$\dfrac{1}{5}$ = 1 : 5


    Exercise

    44328.Express the given percent into ratio
    1 %
    1 : 100
    1 :400
    1 :200
    1 :350
    Explanation:

    1 % = 1/100 = 1 : 100

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