**Introduction**

In our day to day life, we come across situations where we need to compare quantities in terms of their magnitude or measurements. For example, at the time of admission to a college, marks obtained by students in the qualifying examination are compared. Similarly, at the time of recruitment in forces measurements of candidates pertaining to their weight, heights etc. are compared. In general, this comparison can be done in two ways –

- The first one is a comparison by finding the difference of magnitude of two quantities. This is known as comparison by difference.
- The second one is a comparison by division of the magnitude of two quantities. This is known as comparison by division.

When we compare two quantities of the same kind by division, we say that we form a ratio of two quantities. So, how do we define ratio? Let us find out.

**What is Ratio?**

The ratio of two quantities of the same kind and in the same units is a fraction that shows how many times a quantity is of another quantity of the same kind. The ratio of two numbers “ a “ and “ b “ where b ≠ 0, is a ÷ b or ab and is denoted by a : b

In the ratio, a : b, the quantities or numbers a and b are called the terms of the ratio. The former “ a” is called the first term or **antecedent** and the latter term “ b” is called the second term or **consequent**.

Let us understand the ratio with the help of an example.

Suppose we have two brothers, Sam and Peter having their weights as 50 kg and d40 kg respectively. Now, if we compare the weight of Sam with the weight of Peter, we will get

Weight of SamWeight of Peter = 5040 = 54 = 5 : 4

Hence, we can say that the ratio of the weight of Sam to the weight of Peter is 5 : 4.

**Can Ratios be Equivalent?**

We already know that a fraction does not change when its numerator and denominator are multiplied or divided by the same non-zero number. It is important to note here that in ratio as well, there is no change in the ratio if the first and the second term are multiplied or divided by the same non-zero number.

Let us understand this by an example.

Suppose we have the ratio 7 : 3. Now if we multiply both the first and the second term by 5, we will get the ratio 35 : 15. Similarly, if we multiply both the first and the second term by 3, we will get the ratio 21 : 9. So, we have

7 : 3 = 35 : 15 = 21 : 9

Hence, the ratios are equivalent in the same manner as fractions are.

**What is Proportion?**

Proportion is an equality of two ratios. For example, consider two ratios, 6 : 18 and 8 : 24. We can see that

6 : 18 = 1 : 3 and 8 : 24 = 1 : 3

Therefore, 6 : 18 = 8 : 24

Thus the ratios 6 : 18 and 8 : 24 are in proportion.

Therefore, we can say that four numbers a, b, c and d are said to be in proportion if the ratio of the first two is equal to the ratio of the last two. This means, four numbers a, b, c and d are said to be in proportion, if a : b = c : d

If four numbers a, b, c and d are said to be in proportion, then we write

a : b : : c : d

which is read as “ a is to b as c is to d” or “ a to b as c to d”. Here a, b, c and are the first second, third and fourth terms of the proportion. The first and the fourth terms of the proportion are called extreme terms or extremes. The second and the third terms are called the middle terms or means.

Let us understand this by an example.

Consider four terms 40, 70, 200 and 350. We find that 40 : 70 = 200 : 350. So, the given numbers are in proportion. Clearly, 40 and 350 are extreme terms and 70 and 200 are middle terms. We find that,

Product of extreme terms = 40 x 350 = 14000

Similarly, product of middle terms = 70 x 20 = 14000

Therefore,

**Product of extreme terms = Product of middle terms**

Thus, we can say that if four numbers are in proportion then the product of the extreme terms is equal to the product of the middle terms.

**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

**Mean Proportion**

If a, b and c are in continued proportion then b is called the mean proportional between a and c. This means that if b is the mean proportional between a and c then b ^{2} is equal to a c.

**What is a directly proportional relationship?**

**If two quantities are linked in such a way that an increase in one quantity leads to a corresponding increase in the other and vice-versa, then such a relationship is termed as directly proportional.** If two quantities are in direct proportion, then we say that they are proportional to each other. Let us consider an example. 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 variation, then the ratio **ab** is always constant. This constant is called the constant of variation. **

**What is the Symbol for a directly proportional relationship?**

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,

ab = k ( constant )

⇒ a = b k

**Direct vs Inverse Proportional Relationship**

We have now learnt about the directly proportional relationship between two quantities. What would be the opposite of a directly proportional relationship? We call it an inverse proportional relationship. so, how do we define an inverse proportional relationship? Two quantities are said to be inversely proportional when one value increases, and the other decreases. Therefore, two quantities a and b are said to be in inverse proportion if an increase in quantity a, there will be a decrease in quantity b, and vice-versa. Let us summarise the differences between direct proportional relationship and inverse proportional relationship

Direct Proportional Relationship | Inverse Proportional Relationship |

If two quantities are linked in such a way that an increase in one quantity leads to a corresponding increase in the other and vice-versa, then such a relationship is termed as directly proportional. | If two quantities are linked in such a way that an increase in one quantity leads to a corresponding decrease in the other and vice-versa, then such a relationship is termed as inversely proportional. |

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. |

**Graphing of directly proportional relationship**

Let us now learn about the graphs of some directly proportional relationships. We have learnt how to represent the direct proportional relationships of two quantities in the form of an equation. Another way of representing the same is through the use of graphs. In other words, directly proportional relationships 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 a directly proportional relationship will be given by –

On the other hand, the graph of an inversely proportional relationship will be given by –

Let us consider some examples.

**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 have the direct proportional relationship between them? Let us use the relationship 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 share a directly proportional relationship between them. 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 directly proportional relationships 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 share a directly proportional relationship between them.

**Time and distance Graph**

Another common example of directly proportional relationships is that between time and distance when travelling at a constant speed. Let us plot a graph that shows the relationship between distance and time for a vehicle travelling at a constant speed of 30 miles per hour. Below we have some values defining the relation between time and distance when travelling at a constant speed of 30 miles per hour –

Time | Distance |

0 | 0 |

1 | 30 |

2 | 60 |

3 | 90 |

4 | 120 |

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

What if the constant speed of the vehicle would have been 50 miles per hour? The vehicle is travelling at a constant speed of 50 miles per hour. The slope of the graph is steeper. The steepness of the slope for directly proportional relationships increases as the value of the constant m (y = mx) increases.

Below we have some values defining the relation between time and distance when travelling at a constant speed of 50 miles per hour –

Time | Distance |

0 | 0 |

1 | 50 |

2 | 100 |

3 | 150 |

4 | 200 |

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

**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 –

**Key Facts and Summary**

- The ratio of two quantities of the same kind and in the same units is a fraction that shows how many times a quantity is of another quantity of the same kind. The ratio of two numbers “ a “ and “ b “ where b ≠ 0, is a ÷ b or ab and is denoted by a : b.
- In the ratio, a : b, the quantities or numbers a and b are called the terms of the ratio. The former “ a” is called the first term or antecedent and the latter term “ b” is called the second term or consequent.
- Proportion is an equality of two ratios.
- Product of extreme terms = Product of middle terms
- If two quantities are linked in such a way that an increase in one quantity leads to a corresponding increase in the other and vice-versa, then such a relationship is termed as directly proportional.
- If two quantities a and b are in direct variation, then the ratio ab is always constant. This constant is called the constant of variation.
- The symbol for direct proportion is “ “ .
- Two quantities a and b are said to be in inverse proportion if an increase in quantity a, there will be a decrease in quantity b, and vice-versa.
- There is no change in the ratio if the first and the second term are multiplied or divided by the same non-zero number.

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