**What is Proportion?**

If the values of two quantities depend on each other in such a way that a change in one results in a corresponding change in the other, then the two quantities are said to be in proportion. For instance, let us consider the distance covered by a train in a given interval of time and speed. We can observe that if the speed is more, the train will travel more distance in a given period of time. Hence, we can say that the speed of a train and the distance covered by in a given period of time are in proportion. Let us take another example. Now, let us consider the speed of a moving train and the time taken by it to cover a certain distance. We can observe that the time taken by the train to travel a given distance will be less if the speed is more. In other words, as the speed increases the time taken to cover a given distance decreases. Therefore, we can say that the speed of a train and the time taken by it to cover a given distance are also in proportion.

In the above discussion, we can observe that two quantities may be linked together in such a way that both increase or decrease together. They can also be linked in a manner that if one of the quantities increases, the other quantity decreases and vice – versa. This means that we can have quantities in two different types of proportions – direct and inverse.

**What is direct proportion?**

If two quantities are linked with each other in such a way that an increase in one quantity leads to a corresponding increase in the other and vice – versa, then such a relation is called direct proportion. If two quantities are in direct proportion, we can also say that they are proportional to each other. Let us consider the number of articles bought by a person and the amount paid. It is clear that the larger the number of articles, the greater the amount paid will be. Therefore, the number of articles bought by a person and the amount paid is directly proportional to each other.

**Also, if two quantities a and b are in direct proportion, then the ratio **$\frac{a}{b}$** is always constant. This constant is called the constant of variation. **

**What is the Symbol for a direct proportion? **

The symbol for direct proportion is “∝“ . Therefore, we can say that if two quantities a and b are in direct proportion, they can be written as –

a ∝ b

So, we have,

$\frac{a}{b}$ = k ( constant )

⇒ a = b k

**Graph of direct proportion**

Let us now learn about the graphs of some quantities that are in direct proportion to each other. We have learnt how to represent the direct proportion of two quantities in the form of an equation. Another way of representing the same is through the use of graphs. In other words, the direct proportion can be explained and represented by graphing two sets of related quantities. If the relation is proportional, the graph will form a straight line that passes through the origin.

The general graph of two quantities that are in direct proportion with each other will be given by –

When learning about direct proportion, it is important to understand about inverse proportion as well.

**What is inverse proportion?**

If two quantities are linked with each other in such a way that an increase in one quantity leads to a corresponding decrease in the other and vice – versa, then such a relation is called inverse proportion. If two quantities are in inverse proportion, we can also say that they are inversely proportional to each other.

**What is the Symbol for an inverse proportion? **

The symbol for inverse proportion is the same as that of direct proportion, i.e. “∝“ . The difference lies in the positioning of a and b. Therefore, we can say that if two quantities a and b are in inverse proportion, they can be written as –

a ∝ $\frac{1}{b}$

So, we have,

$\frac{a}{b}$= k ( constant )

**Graph of two quantities in Inverse proportion**

**Direct vs Inverse Proportion**

Let us summarise the differences between direct proportion and inverse proportion.

Direct Proportion | Inverse Proportion |

If two quantities are linked with each other in such a way that an increase in one quantity leads to a corresponding increase in the other and vice – versa, then such a relation is called direct proportion. | If two quantities are linked with each other in such a way that an increase in one quantity leads to a corresponding decrease in the other and vice – versa, then such a relation is called inverse proportion. |

It is represented as a b | It is represented as a 1 / b |

In a direct proportion, the ratio between matching quantities stays the same if they are divided. (They form equivalent fractions). | In an indirect (or inverse) proportion, as one quantity increases, the other decreases. |

**Real Life Examples of quantities that are in direct proportion to each other**

Let us now discuss about some real life examples, where two quantities are in direct proportion with each other.

**Relationship between currencies**

We know that a currency is the system of money used in a country or we can say that a currency is a system of money in common use, especially for people in a nation. The British currency is the pound sterling. The sign for the pound is £

GBP = Great British Pound £

Since decimalisation in 1971, the pound has been divided into 100 pence. This means that the pound ( £ ) is made up of 100 pence (p). The singular of pence is “penny”. The symbol for the penny is “p”; hence an amount such as 50p is often pronounced “fifty p” rather than “fifty pence”.

Hence, **£1 = 100p**

Similarly, the dollar is a currency that is used in many western countries and is represented by the ‘\$’ sign. The dollar is the common currency of countries such as Australia, Belize, Canada, Hong Kong, Namibia, New Zealand, Singapore, Taiwan, Zimbabwe, Brunei and the United States. A cent is also a unit of currency that is usually used along with the dollar. Cent is actually one-hundredth of a dollar and is represented by a small case c with a forward slash or a vertical slash through the c. Therefore, $1 = 100 cents

Now, can we say these currencies such as the dollar and pound are directly proportional to each other? Let us use the relation between U.S. Dollars and U.K. Pounds to illustrate this. The exchange rate used in this example is 0.69 U.S. Dollars per 1 U.K. Pound.

Considering that on a given day, 1 USD = 0.69 UKP, we will have

USD | UKP |

0 | 0 |

100 | 169 |

200 | 138 |

300 | 207 |

400 | 276 |

On plotting the above values on a graph we will get –

We can see from the above graph that both the currencies are directly proportional to each other. Also, the table of values and their graph shows above a straight line that passes through the origin. This again indicates that the relationship between the two currencies is in direct proportion. What does this mean in real terms? This means that if we have ten times more dollars than another person when we both exchange our money, we will still have ten times more money. Another point to be noticed is that the graph passes through the origin; which again makes sense as if we have no dollars we will get no pounds! Let us represent the same in the form of an equation.

We will have –

U.S. Dollars = 0.69 x U.K. Pounds

Let y represent U.S. Dollars and p represent U.K. Pounds. We will then have,

y = 0.69p

Now we have learnt that all quantities that are in direct proportion with each other can be expressed in the form y = mx where m represents the slope (or steepness of the line) when the relationship is graphed. This again shows that both the currencies are in direct proportion with each other.

**Graphs of Linear Equations **

We know that an equation in which the highest power of the variables involved is 1 is called a linear equation. In other words, a linear equation is a mathematical equation that defines a line. While each linear equation corresponds to exactly one line, each line corresponds to infinitely many equations. These equations will have a variable whose highest power is 1.

The sign of equality divides an equation into two sides, namely the left-hand side and the right-hand side, written as L.H.S and R.H.S respectively.

A linear equation in one variable is of the form ax + b = 0, where a and b are constants.

Let us consider the equation y = 5 x.

Below are some points that satisfy the above equation –

x | y |

0 | 0 |

1 | 5 |

2 | 10 |

3 | 15 |

4 | 20 |

On plotting the above values on a graph we will get –

We again see from above the variable of a linear equation are in direct proportion with each other.

**Continued Proportion **

What if we have a direct proportion relation defined between more than two quantities? This is where we get the concept of continued proportion.

Three numbers a b c are said to be in continued proportion if a, b, b, c are in proportion.

Thus, if a, b and c are in proportion, then we have a : b : : b : c

Product of extreme terms = Product of middle terms

⇒ a x c = b x b

⇒ a c = b ^{2}

⇒ b ^{2} = a c

Let us understand it through an example.

**Example**

The first, second and fourth terms of a proportion are 6, 18 and 28 respectively. Find its third term.

**Solution**

We have been given that the first, second and fourth terms of a proportion are 6, 18 and 28 respectively. We are required to find the third term. Let us summarise the information given to us.

The proportion of the first, second and fourth terms = 6, 18 and 28

Let the third term be p. Then,

6, 18 and p and 25 are in proportion.

⇒ Product of extreme terms = Product of means terms

⇒ 6 x 25 = 18 x p

⇒ 150 = 18 p

⇒ p = $\frac{150}{18} = \frac{25}{3}$

**Hence, the third term of the proportion is **$\frac{25}{3}$**.**

**Solved Examples**

**Example 1** A car travels 432 km on 48 litres of petrol. How far would it travel on 20 litres of petrol?

**Solution** We have been given that a car travels 432 km on 48 litres of petrol. We need to find how far it would travel on 20 litres of petrol. Let us suppose that the car travels “ p “ km on 20 litres of petrol. Now, let us summarise the information that has been given to us –

Distance a car travels on 48 litres of petrol = 432 km

Distance a car travels on 20 litres of petrol = p km

Now, we can observe that the lesser the petrol is consumed, the smaller would the distance travelled by the car. Therefore, we can rightly say that this is a case of direct proportion. This means that

Ratio of petrol consumed = Ratio of distance travelled

⇒ $\frac{48}{20} = \frac{432}{p}$

⇒ p = $\frac{432 x 20}{48}# = 180

Hence, the car would travel 180 km in 20 litres of petrol.

**Example 2** The ratio of the length of a school ground to its width is 5 : 2. Find the length of the width of the ground is 50 m.

**Solution** We have been given that the ratio of the length of a school ground to its width is 5 : 2. We need to find the length of the width of the ground is 50 m. Now, we can observe that since there is a fixed ratio between the length and the breadth of the school ground, therefore an increase in length will lead to an increase in breadth as well and vice – versa. Hence, we can say that the length and the breadth are directly proportional to each other. So, we will use the concept of continued proportion to solve this question.

Let the length of the school ground be p metres.

Then, the ratio of the length to the width = p : 50

But, the ratio of the length to its width = 5 : 2

Therefore,

p : 50 = 5 : 2

⇒ $\frac{p}{50} = \frac{5}{2}$

⇒ p = $\frac{5}{2}$ x 50

⇒ p = 125

Hence, the length of the school playground = 125 m

**Example 3** The amount of extension in an elastic string varies directly as the weight hung on it. If a weight of 150 gm produces an extension of 2.9 cm, then what weight would produce an extension of 17.4 cm ?

**Solution** We have been given that the amount of extension In an elastic string varies directly as the weight hung on it. Also it is given that If a weight of 150 gm produces an extension of 2.9 cm. We need to figure out what would be the weight produced through an extension of 17.4 cm.

Now, we have been clearly given that there is a direct proportion between the amount of extension in an elastic string and the weight hung on it. So it is a case of direct proportion. Therefore,

Let the required weight be “ p “ gram. This would mean that

Ratio of the weights = Ratio of the extensions

⇒ 150 : p = 2.9 : 17.4

⇒ $\frac{150}{p} = \frac{2.9}{17.4}$

⇒ p = $\frac{150 x 17.4}{2.9}$ = 900

Hence, a weight of 900 grams would produce an extension of 17.4 cm.

**Key Facts and Summary**

- If the values of two quantities depend on each other in such a way that a change in one results in a corresponding change in the other, then the two quantities are said to be in proportion.
- If two quantities are linked with each other in such a way that an increase in one quantity leads to a corresponding increase in the other and vice – versa, then such a relation is called direct proportion.
- If two quantities a and b are in direct proportion, then the ratio $\frac{a}{b}$ is always constant.
- If two quantities are linked with each other in such a way that an increase in one quantity leads to a corresponding decrease in the other and vice – versa, then such a relation is called inverse proportion.
- The symbol for inverse proportion is the same as that of direct proportion, i.e. “ “ . The difference lies in the positioning of a and b.
- Three numbers a b c are said to be in continued proportion if a, b, b, c are in proportion. Thus, if a, b and c are in proportion, then we have a : b : : b : c

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