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#### Exercise 11.1 class 8 solution- Direct proportion

A direct proportion, also known as direct variation, exists when two variables are related in such a way that an increase in one variable leads to a proportional increase in the other. In other words, as one variable goes up, the other also goes up, and as one variable goes down, the other goes down. Mathematically, this relationship is expressed as:

y=kx

Where:

- y is the dependent variable.
- x is the independent variable.
- k is the constant of proportionality, which remains the same for all data points in a direct proportion.

In a direct proportion, if you were to plot the data on a graph with x on the horizontal axis and y on the vertical axis, you would get a straight line that passes through the origin (0, 0). The slope of this line is equal to the constant of proportionality k.

Here are some examples of direct proportion:

**Distance and Time:** The distance traveled by a car is directly proportional to the time it travels at a constant speed. If the car is moving at 60 miles per hour, it will travel 120 miles in 2 hours. The distance (d) is directly proportional to the time (t) at a constant speed.

d=60t

**Cost and Quantity:** In a store, the cost of buying a certain quantity of an item is directly proportional to the quantity. For example, if one item costs $5, then 3 items would cost $15. The cost (C) is directly proportional to the quantity (Q).

C=5Q

**Work and Time:** In a work situation, the amount of work completed is directly proportional to the time spent working. If you work for 5 hours and complete 50 units of work, then you complete 10 units of work per hour. The work (W) is directly proportional to the time (T).

W=10T

Direct proportion is the opposite of inverse proportion, where an increase in one variable leads to a proportional decrease in the other variable. In an inverse proportion, the relationship is expressed as y=k/x

*Examples of direct proportion- some examples of direct proportion is given below*

#### Exercise 11.1 class 8 solution- Inverse proportion

Inverse proportion, also known as inverse variation, is a mathematical relationship between two variables in which an increase in one variable results in a decrease in the other variable and vice versa. In simple terms, when one variable goes up, the other goes down, and vice versa, and this relationship can be described using an equation of the form:

xy=k

Where:

- x and y are the two variables.
- k is a constant of proportionality.

In an inversely proportional relationship, as one variable increases, the other variable decreases in such a way that their product remains constant. This means that the product of x and y will always be the same value, k, no matter the values of x and y.

For example, if you’re driving at a constant speed, the time it takes to reach a certain destination is inversely proportional to your speed. If you double your speed, you’ll reach the destination in half the time because the product of speed and time remains constant. Mathematically, this can be expressed as:

speedÃ—Time=Constant

Inverse proportion is often used in physics, engineering, and various other fields to describe relationships where one quantity depends on the reciprocal (or the inverse) of another quantity.

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