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.
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
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.
Q 1: Following are the car parking charges near an Airport up to
a. 2 hours Rs 60
b. 6 hours Rs 100
c. 12 hours Rs 14
d. 24 hours Rs 180
Check 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
Q 2. y is directly proportional to x, and y = 24 when x = 4. What is the value of y when x = 3?
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 cm
b. 11.7 cm
c. 13.2 cm
d. 14 cm
Solution: 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