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What is the Symbol used for Direct and Inverse Proportion?
When two quantities a and b are in direct proportion (or vary directly), they are written as a ∝ b. Symbol “∝” stands for ‘is proportional to’. When two quantities a and b are in inverse proportion (or vary inversely) they are written as a ∝ 1/ b .
Direct and Inverse Proportion Signs:
The direct proportion sign is the letter “k” or the symbol “∝”, which is used to indicate that two variables are directly proportional to each other. For example, if y is directly proportional to x, we can write this relationship as y ∝ x.
The inverse proportion sign is the symbol “∞”, which is used to indicate that two variables are inversely proportional to each other. For example, if y is inversely proportional to x, we can write this relationship as y ∞ x.
Here are some examples of how these signs can be used:
- If the speed of a car is directly proportional to the power of the engine, we can write this relationship as “speed ∝ power”.
- If the time it takes to complete a task is inversely proportional to the number of people working on it, we can write this relationship as “time ∞ number of people”.
- If the resistance of a wire is inversely proportional to its cross-sectional area, we can write this relationship as “resistance ∞ cross-sectional area”.
Properties of Direct and Indirect Proportion
Here are some properties of direct proportion:
- The ratio of the two variables is always constant. For example, if y is directly proportional to x, and the value of y is twice the value of x, this ratio will remain constant no matter what values x and y take.
- The product of the two variables is always constant. For example, if y is directly proportional to x, and the value of y is twice the value of x, the product of x and y will always be the same.
- The graph of a direct proportion relationship is a straight line through the origin.
Here are some properties of inverse proportion:
- The ratio of the two variables is always constant, but the reciprocal of the ratio is constant. For example, if y is inversely proportional to x, and the value of y is half the value of x, this ratio will remain constant no matter what values x and y take.
- The product of the two variables is always constant. For example, if y is inversely proportional to x, and the value of y is half the value of x, the product of x and y will always be the same.
- The graph of an inverse proportion relationship is a hyperbola.
Examples of Direct and Indirect Proportion
Here are some examples of direct proportion:
- The cost of a product is directly proportional to the quantity purchased. For example, if the cost of one apple is $1 and you want to buy two apples, the total cost will be $2.
- The weight of an object is directly proportional to its volume. For example, a bucket of water weighs more than a bucket of sand because the volume of water is greater than the volume of sand.
- The speed of a car is directly proportional to the power of the engine. A car with a more powerful engine can go faster than a car with a less powerful engine.
Here are some examples of inverse proportion:
- The time it takes to complete a task is inversely proportional to the number of people working on it. For example, if one person takes an hour to complete a task, two people will take half an hour to complete the same task.
- The resistance of a wire is inversely proportional to its cross-sectional area. A wire with a larger cross-sectional area will have less resistance than a wire with a smaller cross-sectional area.
- The intensity of a light source is inversely proportional to the square of the distance from the source. This means that if you double the distance from the light source, the intensity of the light will be one quarter of its original intensity.
We hope these examples help!
How to discern Whether it is a Direct or Indirect Proportion?
A direct proportion is when two quantities are related in such a way that when one quantity increases, the other quantity also increases. An indirect proportion is when two quantities are related in such a way that when one quantity increases, the other quantity decreases.
Application of Direct and Inverse Proportion in Daily Life
Direct and inverse proportion are mathematical concepts that describe the relationship between two variables.
In direct proportion, the value of one variable increases or decreases in direct proportion to the value of the other variable. For example, the number of hours worked is directly proportional to the amount of money earned. If you work more hours, you will earn more money, and if you work fewer hours, you will earn less money.
In inverse proportion, the value of one variable increases or decreases inversely proportional to the value of the other variable. For example, the speed of a car is inversely proportional to the time it takes to travel a certain distance. If you want to reduce the time it takes to travel a certain distance, you must increase the speed of the car.
Here are some examples of how direct and inverse proportion can be applied in daily life:
- The amount of water in a tank is inversely proportional to the time it takes to fill the tank. If you want to fill the tank more quickly, you must increase the flow of water into the tank.
- The brightness of a light bulb is directly proportional to the amount of current flowing through it. If you want to make the light brighter, you must increase the current flowing through the bulb.
- The distance traveled by a car is directly proportional to the amount of fuel it consumes. If you want to travel a longer distance, you must use more fuel.
- The volume of a sound is inversely proportional to the distance from the source of the sound. If you want to reduce the volume of a sound, you must increase the distance between the source of the sound and your ears.
