Points to Remember- Direct and Inverse Proportions | Mathematics (Maths) Class 8 PDF Download

Points to Remember 

• When two quantities, $\mathrm{<br\; style="word-break:\; break-word;\; line-height:\; 1.8;\; font-family:\; Lato,\; sans-serif;\; font-size:\; 20px;\; letter-spacing:\; inherit;\; color:\; black;">}$x and y, change in a way that their ratio stays the same, they are said to be in direct proportion.
• If x and $y$y change such that when one increases, the other decreases proportionally (and the reverse is true), they are in 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

• Variable Nature: The value of a variable is not constant and changes over time or circumstances.
• Inter-Related Quantities: Some quantities have a relationship with others, meaning when one changes, the other also changes. This interrelation is known as variation.
• Types of Variation:There are two types of variation: (i) Direct Variation (ii) Inverse Variation

Direct Proportion

If two quantities are related in such a way that an increase in one quantity leads to 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.

Inverse Proportion

If two quantities change in such a manner that, if one quantity increases, the other quantity decreases in the same proportion and vice versa, then it is called Inverse Proportion. 

Two quantities x and y are said to be in inverse proportion,if  x ∝ (1 / y)

x = k / y   ⇒ xy = k

Note: 

I. In an inverse proportion, two quantities x and y vary with each other such that xy remains constant.

II. x × y is always a positive number.

III. x × y or k is called the constant of variation in inverse proportion.

Solved Examples

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?
a. 18
b. 20
c. 23
d. 43
Ans : 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
Ans: c

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