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# Proportionality

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## Define Proportional

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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.

### 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?

 X Y 25 5 5 1 35 7 16 3

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.

Q1. What is the Constant of Proportionality?

Answer: Now that you know what the word proportional means in math, let's understand what does constant of proportionality means. Constant of proportionality implies towards the constant value of the ratio between the two proportional quantities. Suppose we get mangoes from the fruit vendor at Rs. 100 for 2. Then, the ratio will be 100/2, which equals to 50. Here, 50 is the constant of proportionality. The two quantities are directly proportional to each other when they increase or decrease at the same rate.

The constant of proportionality is represented by the letter 'k'. It is also called the unit rate. The value of the constant of proportionality is based on the type of proportion between the two assigned quantities.

Q2. Why do we Use Proportionality and Constant of Proportionality?

Answer: In mathematics, we use the Constant of Proportionality for the purpose of calculating the rate of change and at the same time identify if it is direct variation or inverse variation that we have to manage with.

Suppose that it is direct proportion, then the constant of proportionality will be the same throughout in the form of: y = kx.

Simultaneously, when y is inversely proportional to x, the unit rate is in the form of:

y = kx

Q3. What are the Uses of Proportionality in Real-Life?

Answer: Working with proportional association enables one to solve many real-life issues such as:

• Balancing a recipe's ratio of ingredients.

• Identifying percent increase or percent decrease for price mark-ups.

• Computing chance like determining odds and probability of events.

• Discounts marked on products based on unit rate.

• Scaling a diagram for drawing up and architectural uses.

For instance, we can use the concept of proportionality in order to find answers to real-life problems.