Direct & Inverse Proportion

Image Credit: Pixabay.com

Variation

When two quantities are related in such a way that change in one quantity causes change in the other, then the quantities are said to vary with each other.

ℹ Some situations where variation in one quantity brings the variation in the other quantity :
(i) More the number of coins more the height of the pile.
(ii) More the rate of interest more is the interest charged.
(iii) More the distance more the time taken to cover it for the same speed.
(iv) More the speed of a vehicle less the time taken to cover the same distance.
(v) More the number of workers, less will be the time taken to complete the work.

The variation can be direct or indirect.
First three examples are of direct variation and last two are the examples of indirect variation.

Direct Variation / Proportion

Two quantities are said to be in direct proportion if they increase (decrease) together in such a way that the ratio of their corresponding values remains constant.

➢ Let x and y be two quantities such that change in x leads to corresponding change in y in same proportion.

x

x1

x2

x3

y

y1

y2

y3

then,

\boxed{\frac{x}{y}=k}

i.e.,

\boxed{\frac{x_1}{y_1}= \frac{x_2}{y_2}= \frac{x_3}{y_3}=\cdots=\frac{x_n}{y_n}}

(K is a constant i.e., ratio of x and y is always same for direct proportion)

\boxed{x=ky}

\ x_1=ky_1}
\ x_2=ky_2}
\ x_3=ky_3}
\cdots
\ x_n=ky_n}

Ex – A machine in a soft drink factory fills 840 bottles in six hours. How many bottles will it fill in five hours?
Sol –

x (no. of bottles)

840

x2

y (time in hours)

6

5

It is the case of direct variation,
\therefore \frac{840}{6} = \frac{x_2}{5}

\implies 140 = \frac{x_2}{5}
\implies x_2 = 140 \times 5
\implies x_2 = 700
∴ Number of bottles would be filled in 5 hours = 700.

Indirect Variation / Proportion

Two quantities are said to be in inverse proportion if an increase in one causes a proportional decrease in other (and vice-versa) in such a way that the product of their corresponding values remains constant.

➢ Let x and y be the two quantities such that change in x leads to corresponding change in y in inverse proportion.

x

x1

x2

x3

y

y1

y2

y3

then,
\boxed{xy = k}

i.e.,
\boxed{x_1y_1 = x_2y_2= x_3y_3=\, \cdots \,= x_ny_n}

(K is a constant i.e., product of x and y is always same for indirect proportion)

\boxed{x = \frac{k}{y}}

\frac{x_1}{x_2}= \frac{y_2}{y_1},

\frac{x_2}{x_3}= \frac{y_3}{y_2},

\frac{x_3}{x_4}= \frac{y_4}{y_3},

\cdots ,

\frac{x_{n-1}}{x_n}= \frac{y_n}{y_{n-1}}

Ex-If a box of sweets is divided among 24 children, they will get 5 sweets each. How many would each get, if the number of the children is reduced by 4?
Sol-

x (No. of children)

24

24-4

y (No. of sweets)

5

y2

It is the case of inverse variation, i.e., when number of person (x) decreases then number of sweets for each person (y) increases.

\therefore \frac{x_1}{x_2} = \frac{y_2}{y_1}

\implies \frac{24}{20} = \frac{y_2}{5}

\implies y_2 =  \frac{6}{5}\times 5

\implies y_2 = 6
Each children will get 6 sweets.

Leave a Reply

Close Menu
%d bloggers like this: