Proportionality

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:

1. Do the variables display any kind of proportion?

2. 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.