# Ratios and Proportion

## What is Ratio?

When 3 litres of yellow paint is mixed with 2 litres of red paint, the result is perfectly orange paint.

Will mixing 3 litres of yellow paint with 3 litres of red paint also result in perfectly orange (PO) paint? The answer is no.

If you want to get more orange paint, how much yellow paint and how much red paint must you mix? You will get orange paint when you have 2 portions of red paint and 3 portions of yellow paint.

This means orange paint is created when for every 3 litres of yellow paint, 2 litres of red paint are added. We say that yellow paint and red paint are in the ratio **3 is to 2**, written as **3:2**.

A ratio tells us how much of the proportion of one thing to the other.

The **sequence (or order)** in which the numbers in a ratio are written matters. For example, the ratio of the yellow paint to the red paint paint is 3:2 and not 2:3. Why? Because yellow paint will always be 1 portion more than the red paint.

Ratio of 3 quantities can be expressed as,

If \(a:b = 2:3\) and \(b:c = 3:5,\) then \(a : b : c = 2 : 3 : 5\)

### Equivalent ratios and the simplest form of a ratio

Nina's fruit-basket has 20 apples and 15 oranges in it.

\(\begin{align} \text{The number of apples : the number of oranges} = 20 : 15 = 4 : 3\end{align}\)

Because of this equality, we say that the ratio \(20 :15\) is **equivalent to** the ratio \(4 : 3.\) The ratio \(4 : 3\) is a ratio in its **simplest** or **lowest form**. A ratio is in its lowest form when the two numbers in the ratio **have no factors in common**.

## The basics of proportion

When two ratios are equivalent, they are said to be in **proportion**. For example, \(2 : 5\) and \(10 : 25\) are equivalent ratios. This means they are in proportion. We represent this by writing **\(2 : 5 : : 10 : 25\)** and read this as **‘2 is to 5 as 10 is to 2.**

### Comparison of ratios

### Division of a quantity in the given ratio

Let’s try and understand this using an example.

Example - Sita has 200. She splits the amount between two of her friends, Mia and Isha, in the ratio \(3 : 7.\)

How much money does Mia receive?

\(\begin{align} \text{Mia’s share} = {3\over 3 + 7} \text{ of } 200 = {3\over 10} \times 200 = 60\end{align}\)

How much money does Isha receive?

\(\begin{align} \text{Isha’s share} = {7\over 3 + 7} \text{ of } 200 = {7\over 10} \times 200 = 140\end{align}\)

In the example given we calculated Mia’a and Sita’s share from the total.

### Procedure - Direct variation

When the ratio of the two quantities is constant, the quantities are in **direct variation**. Increase in one quantity causes an increase in the other and decrease in one quantity causes decrease in the other. Let’s try and understand this using an example,

Example: The price of a pen is ₹12. The table below shows the number of pens bought by Dev and their price.

Number of pens (N) |
1 | 2 | 3 | 4 | 5 |

Total price (P) |
12 | 24 | 36 | 48 | 60 |

\(\begin{align}N\over P \end{align}\) | \(\begin{align}1\over 12 \end{align}\) | \(\begin{align}{1\over 12}={2\over 24} \end{align}\) | \(\begin{align}{1\over 12}={3\over 36} \end{align}\) | \(\begin{align}{1\over 12}={4\over 48} \end{align}\) | \(\begin{align}{1\over 12}={5\over 60} \end{align}\) |

As you would notice that there is an increase or decrease in price as there is an increase and decrease in the number of pens. The ratio of the number of pens to the total price is always the same. We can see that \(1 : 12 = 2 : 24 = 3 : 36\) and so on. \(N\) and \(P\) in direct variation.

This direct variation is written as **\(P∝ N,\)** read as \(“P\) is directly proportional to \(N”\) or \(“P\) varies directly as \(N”.\) When two quantities, \(a\) and \(b,\) are directly proportional, we can say that **\(a ∝ b\)** and **\(b ∝ a****.\)**

### Procedure - Inverse variation

When the ratio of one quantity to the reciprocal of the other quantity is constant, the two quantities are in inverse variation.

Number of men (N) | 1 | 2 | 3 | 6 | 8 |

Number of days (D) | 48 | 24 | 16 | 8 | 6 |

\(\begin{align}N\over D \end{align}\) | \(\begin{align}{1\over{1\over48}}=48\end{align}\) | \(\begin{align}{2\over{1\over24}}=48\end{align}\) | \(\begin{align}{3\over{1\over16}}=48\end{align}\) | \(\begin{align}{6\over{1\over8}}=48\end{align}\) | \(\begin{align}{8\over{1\over6}}=48\end{align}\) |

As you can observe with the increase in the number of men, the number of days required to complete the work decreases. The ratio of the number of men and the **reciprocal** of the number of days is constant.

We can see that \(\begin{align}1:{1\over48}=48, \;2:{1\over24}=48, \;3:{1\over16}\end{align}\)

This is written as \(\begin{align}N ∝ {1\over D},\end{align}\) read as “*N* is inversely proportional to *D*” or “*N* varies inversely as *D*”

When two quantities, \(a\) and \(b,\) are inversely proportional, we can say that \(\begin{align}a ∝ {1\over b} \text { and } b ∝ {1\over a}∝ 1 \end{align}\)

More the number of men, the lesser the number of days required to finish the work.

### Constant of proportionality

Time of travel (T) (Hr) |
1 | 2 | 3 | 4 | 5 |

Distance covered (D) (Km) |
50 | 100 | 150 | 200 | 250 |

\(\begin{align}{D\over T}{(Km/hr)} \end{align}\) | 50 | 50 | 50 | 50 | 50 |

As you can see \(D ∝ T,\) \(DT\) is a constant and its value is called the ‘constant of proportionality’. In general, when \(a\) varies directly as \(b,\) we write \(\begin{align}{a\over b} = k\;\;(\text{or}) \;\;a= kb \end{align}\), where ‘*k*’ is called the **constant of proportionality**.

## Tips and Tricks

- Ratios are best thought of as equivalent fractions. Using the trick of finding equivalent fractions is the best way to find equivalent ratios.

E.g. To get a particular shade of paint the two colours need to be mixed in the ratio \(3:4.\) If I have 5 litres of the first paint, how much of the second will I need?

Here we have to find a fraction equivalent to \(\begin{align} 3\over4 \end{align}\) but with numerator 5.

One way to do this… multiply by 10 and then divide by 6.

\(\begin{align} {3\over 4} = {30 \over 40} = {5\over 6.67} \end{align}\)

So I’ll need 6.67 litres of the second paint.

- Proportions can be direct or indirect. A trick is to always think of the situation from real life and determine whether we are dealing with direct or inverse proportions.

E.g. 10 people finish a task in a certain amount of time. How much time will it take 20 people?

At first glance, seeing 10 becoming double raises the temptation of doubling the time. But if you think of the situation, you’ll recognise that more people will get the same work done is a shorter amount of time. So this is a case of inverse variation.

Build your intuition this way so that you follow the correct procedure.

## Test your knowledge

Q1. Answer the following questions

- In a class, the ratio of the number of boys to the number of girls is \(3 : 4.\) If the class has 24 boys, what is the number of girls?
- The ages of Moni and Nivi are in the ratio \(2 : 3.\) Ten years from now, their ages will be in the ratio \(4 : 5.\) What are their present ages?
- In a bus, the ratio of seats allotted to men and women are in the ratio \(7 : 5.\) If the bus has 60 seats, how many women can sit on the bus?