Proportion

In this section, you will learn about ratios and proportions, the proportion definition in math, as well as the types of proportions with proportion examples.

Check-out the interactive simulations to know more about the lesson and try your hand at solving a few interesting practice questions at the end of the page.

Now that you know what you're going to learn, maybe it's time to give you an introduction. But where is a proper introduction without an example?

Understanding proportion with the help of an example on buying notebooks

Let's say, you and your friend went to a store to purchase some notebooks. The ratio of purchase is \(3:10\) and the ration of cost is \(15: 50\). Whose notebooks were costlier? 

Well, in order to answer that, let's check out both ratios. 

First, let's point out that both ratio values are equal i.e. if you simplify \(15:50\), you get \(3:10\)

This means, you and your friend paid the same amount for the notebooks. 

These ratios are in proportion and can be written as 
\(3:10::15:50\)

Lesson Plan 

What is a Proportion?

Proportion - Definition

Proportion definition says that when two ratios are equivalent, they are in proportion.

Proportion is a mathematical comparison between two numbers.


Ratios and Proportions

Ratio is a way of comparing two quantities of the same kind by using division.

The ratio formula is used to compare the relationship between two numbers of the same kind.

The ratio formula for two numbers \(a\) and \(b\) is given by \(\begin{align}a:b \end{align}\) or \(\begin{align} \frac{a}{b} \end{align}\)

Ratio and Proportion: Standard form of ratio

 
tips and tricks
Tips and Tricks
  1. \(\frac{a}{b} = \frac{c}{d} \implies \frac{b}{a}=\frac{d}{c}\)
  2. \(\frac{a}{b} = \frac{c}{d} \implies \frac{a}{c}=\frac{b}{d}\)
  3. \(\frac{a}{b} = \frac{c}{d} \implies \frac{a+b}{b}=\frac{c+d}{d}\)
  4. \(\frac{a}{b} = \frac{c}{d} \implies \frac{a-b}{b}=\frac{c-d}{d}\)
  5. \(\frac{a}{b} = \frac{c}{d} \implies \frac{a+b}{a-b}=\frac{c+d}{c-d}\)
  6. If both the numbers \(a\) and \(b\) are multiplied or divided by the same number in the ratio \(a:b\), then the resulting ratio remains the same as the original ratio

Proportion Formula with Examples

Proportion Formula

We write it as \(a : b : : c : d\) or \(a : b = c : d\)

We read it as “\(a\) is to \(b\) as \(c\) is to \(d\).”

Standard formula for Proportions

A proportion formula is an equation that can be solved to get the comparison values.

To solve proportion problems, we use the concept that proportion is two ratios that are equal to each other.

We mean this in the sense of two fractions being equal to each other.

For instance,

\[\begin{align}5 : 10 = \frac{5}{10} = \frac{1}{2} = 1 : 2\end{align}\]


Solved Examples

Let's explore some proportions examples below:

Example 1

 

 

My mother's recipe to make pancakes says, "You have to use 2 eggs to make 20 pancakes."

Chef making pancakes using eggs: 2 eggs to make 20 pancakes

How many eggs will I need to make 100 pancakes?

  Eggs Pancakes
Small amount 2 20
Large amount \(x\) 100

Solution

We know that \(\text{eggs} \propto \text{pancakes}\)

If we write the unknown number in the numerator, then we can solve this as any other equation.

\[\begin{align}\frac{x}{100}=\frac{2}{20}\end{align}\]

Multiplying 100 on both sides

\[\begin{align}100\times\frac{x}{100}&=100\times\frac{2}{20}\\x&=\frac{200}{20}\\&=10\end{align}\]

\(\therefore\) We need 10 eggs to make 100 pancakes
Example 2

 

 

Jessy runs 4 miles in 30 minutes.

At this rate, how far could she run in 45 minutes?

Jessy ran 4 km in 30 mins                              

Solution

Let's assume the unknown quantity here is \(x\)

In this case, \(x\) is the number of miles Jessy can run in 45 minutes at the given rate.

It is given that running 4 miles in 30 minutes is the same as running \(x\) miles in 45 minutes.

\[\begin{align}\frac{4}{30} &= \frac{x}{45}\\30 \times x &= 4\times 45\\x&=180\div 30\\ &= 6\end{align}\]

\(\therefore\) Jessy can run 6 miles in 45 minutes
Example 3

 

 

In an apartment, nine taps can fill a full tank in four hours.

How long do you think it will take if twelve taps of same flow-rate are used to fill the same tank?

Nine taps fill the tank in 4 hours.12 taps fill the same tank in how many hours?

Does it fill the tank faster or does it slow the process?

Solution

This is inverse proportion problem.

Therefore, as the number of taps increases, inversely the time taken to fill the tank decreases.

Let us assume that the time taken to fill the same tank using 12 taps is \(x\) hours.

\[\begin{align}\frac{9}{x} &= \frac{12}{4}\\ 12x &= 9 \times 4 \\ x &= 36 \div 12 \\ x&=3 \end{align}\]

\(\therefore\) 12 taps will take 3 hours and fill the tank faster
Example 4

 

 

If \(\Delta ABC\) has a  perimeter of 12 units and is similar to \(\Delta RST\) with a scale factor of \(\begin{align}\frac{1}{3}\end{align}\)

triangle ABC and triangle RST at scale factor of \(\frac{1}{3}\)

Can you find the perimeter of \(\Delta RST\)?

Solution

Let \(x\) represent the perimeter of \(\Delta RST\)

Scale factor implies that the perimeters are in proportion to this ratio.

The proportion will be as follows.

\[\begin{align} \frac{1}{3} &= \frac{12}{x}\\x&= 3\times12 \\ &=36\end{align}\]

\(\therefore\) The perimeter of \(\Delta RST\) is 36 units
Example 5

 

 

I want to make a nice fluffy cake on my mother's birthday.

To bake a perfect cake, I need to use sugar and flour in the proportion of 1:1

If I have 3 ounces of sugar available to make a 6 ounces cake, how many kilograms of flour is required?

The simulation below will show you the right mixing proportions for the cake.

Solution

Let the quantity of flour required be \(x\) ounces.

We know that 3 ounces of sugar is required for a 6 ounces of cake.

We need to find the quantity of flour required for a 6 ounces cake.

The proportion will be

\[\begin{align} \frac{1}{1} &= \frac{x}{3}\\ x&=3\end{align}\]

\(\therefore\) 3 ounces of flour is needed

Types of Proportions

There are two types of proportions.

  • Direct Proportion
  • Indirect Proportion

Two types of proportions are Direct proportions and Indirect proportions

Direct Proportion

This type describes the direct relationship between two quantities.

In simple words, if one quantity increases, the other quantity also increases and vice-versa.

For example, if the speed of a car is increased, it covers more distance in a fixed amount of time.

In notation, direct proportion is written as

\(y\propto x\)

Inverse Proportion

This type describes the indirect relationship between two quantities.

In simple words, if one quantity increases, the other quantity decreases and vice-versa.

For example, increasing the speed of the car will result in covering a fixed distance in less time.

Types of proportion: indirect and Direct

In notation, inverse proportion is written as

\(y\propto\frac{1}{x}\)
 
important notes to remember
Important Notes
  1. Proportion is a mathematical comparison between two numbers.
  2. Basic proportions are of two types : Direct Proportions and Indirect Proportions.
  3. We can apply the concepts of proportions to geography, comparing quantities in physics, dietetics, cooking, etc.

Interactive Questions

Here are few activities for you to practice. Select/type your answer and click the "Check Answer" button to see the result.

 
 
 
 
 
 

Let's Summarize

We hope you enjoyed learning about Proportion with the simulations and practice questions. Now you will be able to easily solve problems on ratios and proportions, types of proportion, proportion-math examples.

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Frequently Asked Questions (FAQs)

1. What are some ratio and proportion examples?

Grocery shopping is a good example of ratio and proportion in daily life. 

You go to a grocery store and the shopkeeper gives you two options:

  1. 10 ounces packet of cereal costs $200
  2. 20 ounces packet of cereal costs $300

Here, comparing by division helps rather than comparing by finding the difference to find the best buy.

2. How do you know if two ratios form a proportion?

If two ratios are equivalent to each other, then they are proportional.

Illustration to explain how two ratios form a proportion

Here, the ratios 1:3, 2:6, and 3:9 are equivalent ratios.

3. How do you calculate proportion?

Proportion is calculated using the proportion formula which says

a : b : : c : d or a : b = c : d

We read it as “a" is to "b" as "c" is to "d”.

Download Free Grade 7 Worksheets
Ratio, Proportion, and Variation
Grade7 | Worksheet 1
Ratio, Proportion, and Variation
Grade7 | Worksheet 2
Download Free Grade 8 Worksheets
Ratio and Variation
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Ratio and Variation
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