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Variation
When two quantities related in such a way that change in one quantity causes change in other the quantity, we say that quantities vary with each other.
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 proportion/variation and last two are the examples of indirect proportion/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 the two quantities be x and y. And change in x leads to corresponding change in y in same proportion.
x
|
x1
|
x2
|
x3
|
y
|
y1
|
y2
|
y3
|
| x | = | k |
| y |
always same for direct proportion)
| x1 | = | x2 | = | x3 | = | ... | = | k |
| y1 | y2 | y3 |
| x | = | ky |
| x1 | = | ky1 | , | x2 | = | ky2 | , | x3 | = | ky3 | , | ... | , | xn | = | kyn |
Ex - A machine in a soft drink factory fills 840 bottles in six hours. How many bottles will it fill in five hours?
Sol -
It is the case of direct variation,
∴ 840/6 = x2/5
⇒ 140 = y2/5
⇒ 140 × 5 = y2
⇒ y2 = 700
∴Number of bottles would be filled in 5 hours = 700.
Sol -
x (no. of bottles)
|
840
|
x2
|
y (time in hours)
|
6
|
5
|
It is the case of direct variation,
∴ 840/6 = x2/5
⇒ 140 = y2/5
⇒ 140 × 5 = y2
⇒ y2 = 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 the two quantities be x and y. And change in x leads to corresponding change in y in same proportion.
x
|
x1
|
x2
|
x3
|
y
|
y1
|
y2
|
y3
|
| xy | = | k |
(K is a constant i.e., product of x and y is
always same for indirect proportion)
| x1y1 | = | x2y2 | = | x3y3 | = | ... | = | xnyn | = | k |
| x1 | = | y2 | , | x2 | = | y3 | , | ... | , | xn-1 | = | yn | ||
| x2 | y1 | x3 | y2 | xn | yn-1 |
Sol-
x (No. of children)
|
24
|
24-4
|
y (No. of sweets)
|
5
|
y2
|
It is the case of inverse variation,
∴ x1/x2 = y2/y1
⇒ 24/20 = y2/5
⇒ 6/5 × 5 = y2
⇒ y2 = 6
Now, each children will get 6 sweets.
| Ratio & Proportions |