Define Proportional
Proportional or proportion meaning in math means equality between two ratios. In the mathematical equation a/b = c/d, a and b are in a similar proportion as c and d. A proportion is essentially established for solving a word problem in which one of its four quantities is unknown. Proportionality is solved by multiplying one numerator by the opposite denominator and equating the product to that of the other denominator and numerator. The term proportionality thus defines any relationship which is always in the same ratio.
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Proportional Examples
The number of mangoes in a crop, for example, is proportional to the number of trees in the vineyard, the ratio of proportionality being the average number of mangoes per tree.
How to Know if Two Quantities are Proportional or Not?
Below are the few factors to find if two quantities are proportional or not:
Firstly, we have to determine the ratio of the two quantities for all the assigned values.
If their ratios are equivalent, then they display a proportional relationship.
If all the ratios are not equivalent, then the link between them is not proportional.
Solved Examples on Proportional Examples
Example:
From the table below, find out if:
Do the variables display any kind of proportion?
If so, what will be the constant of proportionality?
Solution:
In order to check the constant of proportionality, we apply:
y = kx
k = y/x
y/x = 5/25 = 1/5 =7/35 ≠ 3/16
We can notice that all the ratios in the table above are not equal.
Thus, these values are NOT said to be in a proportional relationship.
Therefore, the constant of proportionality is not equivalent.
What is proportional is known, now let’s find if the given quantities are proportional.
When quantities have a similar relative size. Specifically, they have the same ratio.
Example: A wire's length and weight are in proportion. When 30m of rope weighs 1kg, then:
30m of that wire weighs 1kg.
150m of that wire weighs 5kg etc.
Example:
Alex booked an Uber cab at the cost of Rs. 60 for 20 km.
Identify the cost of the ride if his destination is 30 km away.
Solution:
We can observe that this is an instance of direct proportion.
The more the distance, the higher the cost of the ride
Let x be the no. of kilometers and y be the cost.
Seeing that this is direct proportion, we have
y = kx
Substituting, x = 20 and y = 60
60 = 20k
K = 3
3 = k
Thus,
y = 3x
Substitute;
x = 30y
= 3(30)
y = 90
Hence, the cost for 30 km is 90 rupees.
Therefore, Uber ride Alex is Rs. 90.
Fun Facts
The constant ratio in a proportional link is known as the constant of proportionality.
If two quantities are proportional to each other, the link between them can be described by y = kx, where ‘k’ is the constant ratio of y-values to corresponding x-values.
The same link can also be described by the formula x = 1/ky, where 1/k is the constant ratio of x-values to y-values.
The constant of proportionality is also called as unit rate.
FAQs on Proportionality
1. What does proportionality mean in Maths?
In mathematics, proportionality describes a relationship between two quantities where their ratio remains constant. If one quantity changes, the other changes in a predictable way. For example, if the cost of one pen is Rs. 10, then the cost of five pens will be Rs. 50. Here, the total cost is proportional to the number of pens purchased.
2. What are the main types of proportionality?
There are two primary types of proportionality that students learn in the CBSE syllabus:
- Direct Proportionality: When one quantity increases, the other quantity also increases at the same rate, and vice versa. The formula is y = kx.
- Inverse Proportionality: When one quantity increases, the other quantity decreases at the same rate, and vice versa. The formula is y = k/x.
3. What is the constant of proportionality (k)?
The constant of proportionality, represented by the letter 'k', is the fixed ratio that connects two proportional quantities. It is also known as the unit rate. For example, if you travel 120 km in 2 hours at a steady speed, the constant of proportionality is 120/2 = 60. This means your speed (the constant) is 60 km/h.
4. How is the proportionality symbol (∝) used in Maths?
The symbol '∝' stands for "is proportional to." It is used to state a relationship before turning it into an equation. For example, if a variable 'y' is directly proportional to 'x', we write y ∝ x. To solve problems, we replace this symbol with an equals sign and the constant of proportionality 'k', making the equation y = kx.
5. What is the key difference between direct and inverse proportionality?
The key difference lies in how the two quantities relate to each other. In direct proportionality, both quantities move in the same direction (both increase or both decrease) and their ratio is constant (y/x = k). In inverse proportionality, the quantities move in opposite directions (one increases while the other decreases) and their product is constant (x * y = k).
6. How do you find the constant of proportionality from a table of values?
To find the constant 'k' from a table:
- For a direct proportion, divide the value of the second quantity (y) by the corresponding value of the first quantity (x). If the result (y/x) is the same for all pairs, that number is your constant 'k'.
- For an inverse proportion, multiply the value of 'y' by the corresponding value of 'x'. If the product (x * y) is the same for all pairs, that is your constant 'k'.
7. If two quantities both increase, are they always directly proportional?
No, this is a common misconception. For two quantities to be directly proportional, they must not only both increase but must do so at a constant rate. This means their ratio (y/x) must always be the same. For instance, a child's age and weight both increase over time, but they are not directly proportional because the rate of weight gain is not constant per year.
8. What are some real-life examples of direct and inverse proportionality?
Proportionality is used in many real-life situations:
- Examples of Direct Proportionality: The distance you travel is proportional to the time you drive at a constant speed. The total cost is proportional to the number of items you buy.
- Examples of Inverse Proportionality: The time it takes to travel a fixed distance is inversely proportional to your speed. The number of people painting a wall is inversely proportional to the time it takes to finish the job.
9. Why is understanding proportionality important for scaling maps or recipes?
Understanding proportionality is crucial for scaling because it ensures accuracy. When you scale a map, the map scale (e.g., 1 cm = 5 km) is a constant of proportionality. It allows you to convert any distance on the map to the correct real-world distance. Similarly, when adjusting a recipe for more or fewer people, you must scale every ingredient proportionally to maintain the original taste and texture.
10. How does the formula y = kx truly represent a direct proportion?
The formula y = kx is the mathematical definition of a direct proportion. It shows that the variable 'y' is always a constant multiple ('k') of the variable 'x'. The constant 'k' acts as the bridge between the two variables. This equation guarantees that if you double 'x', 'y' will also double, and if you halve 'x', 'y' will also be halved, which is the fundamental behaviour of a direct proportional relationship.
