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**Points to Remember**** **

• If two quantities x and y vary (change) together in such a manner that the ratio of their corresponding values remains constant, then x and y are said to be in direct proportion.

• If two quantities x and y vary (change) in such a manner that an increase in x causes a proportional decrease in y (and vice versa), then x and y are said to be inverse proportion.

• If x and y are in a direct proportion, then (x/y) = constant.

• If x and y are in an inverse variation, then xy = constant.**WE KNOW THAT**

The value of a variable is not constant and keeps on changing. There are many quantities whose value varies as per the circumstances. Some quantities have a relation with other quantities such that when one changes the other also changes. Such quantities are inter-related. This is called a variation. Variation is of two types:

(i) Direct variation and (ii) Inverse variation.**DIRECT PROPORTION**

If two quantities are related in such a way that an increase in one quantity leads a corresponding increase in the other and vice versa, then this is a case of direct variation. Also, a decrease in one quantity brings a corresponding decrease in the other.

Two quantities x and y are said to be in direct proportion, if

(x/y) = k or x = ky

Note:I.In a direct proportion two quantities x and y vary with each other such that (x/y) remains constant.

II. (x/y) is always a positive number.

III. (x/y) or k is called the constant of variation.

**Solved ExamplesQ 1: Following are the car parking charges near an Airport up toa. 2 hours Rs 60b. 6 hours Rs 100c. 12 hours Rs 14d. 24 hours Rs 180Check if the parking charges are in direct proportion to the parking time.Solution: **We know that two quantities are in direct proportion if whenever the values of one quantity increase, then the value of another quantity increase in such a way that ratio of the quantities remains same. Here, the charges are not increasing in direct proportion to the parking time because of 2/60 ≠ 6/100 ≠ 12/140 ≠ 24/180

a. 18

b. 20

c. 23

Answer : a

**Solution: **Step 1 Find the constant of proportionality:

y is directly proportional to x ⇒ y ∝ x ⇒ y = kx where k is the constant of proportionality.

But y = 24 when x = 4

⇒ 24 = k × 4

⇒ k = 6

Step 2 Write down the equation connecting y and x:

y = kx ⇒ y = 6x

Step 3 Substitute x = 3 into this equation to find the corresponding value of y:

When x = 3, y = 6 × 3 = 18**Q 3. The circumference (C cm) of a circle is directly proportional to its diameter (d cm). The circumference of a circle of diameter 3.5 cm is 11 cm. What is the circumference of a circle of diameter 4.2 cm?a. 9.17 cmb. 11.7 cmc. 13.2 cmd. 14 cmAnswer: cSolution: **We are told C = 11 when d = 3.5

We need to find the value of C when d = 4.2

Step 1 Find the constant of proportionality:

C is directly proportional to d ⇒ C ∝ d ⇒ C = kd where k is the constant of proportionality.

But C = 11 when d = 3.5

⇒ 11 = k × 3.5

⇒ k = 11/3.5 = 22/7

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