Simple ratios

Simple ratios

A simple ratio shows the relative amount of two quantities. When the value of one quantity increases, the value of the other quantity increases too. When the value of one quantity decreases, the value of the other quantity decreases too.

Steps to find simple ratios:

1. Use the ratio of two quantities, that is, $x$ and $y$ which increase or decrease at the same time, as follows: $x_1$ represents the initial value of the quantity $x$. $x_2$ represents the value of the quantity $x$ after an increase or decrease. $y_1$ represents the initial value of quantity $y$. $y_2$ represents the value of the quantity $y$ after an increase or decrease.

2. Write and compare the ratio between those quantities, that is: $\frac{y_1}{y_2} = \frac{x_1}{x_2}$, $y_1 : y_2 = x_1 : x_2$

If the values of $x_1$, $y_1$ and $x_2$ are known, you may obtain the value of $y_2$. Thus, $y_2 = \frac{y_1 x_2}{x_1}$. Also, if the values of $x_1$, $y_1$ and $y_2$ are known, you may obtain the value of $x_2$. Thus, $x_2 = \frac{y_2 x_1}{y_1}$.

Example 1

If the price of 8 kilograms of meat is sh 56 000, find the price of 6 kilograms of meat.

Solution

Find the price of 1 kilogram of meat as follows:

The price of 1 kg = The price of 8 kg of meat/number of kilograms = sh 56 000/8 = sh 7 000.

The price of 6 kg = 6 kg × sh 7 000/kg = sh 42 000.

Alternative solution

Let $x_1$ and $x_2$ represent the kilograms of meat. $y_1$ and $y_2$ represent the price of meat.

Use the expression $\frac{y_1}{y_2} = \frac{x_1}{x_2}$ to find the price of 6 kg of meat as follows:

$x_1 = 8$ kg, $x_2 = 6$ kg, $y_1 =$ sh 56 000, $y_2 = ?$

Find the value of $y_2$, which is the price of 6 kg of meat:

sh 56 000/$y_2$ = 8 kg/6 kg.

Thus, $y_2 = 6$ kg × sh 56 000/8 kg = sh 42 000.

Therefore, the price of 6 kilograms of meat is 42 000 shillings.

Example 2

The price of 5 books is sh 60 000. Find the price of 3 books of the same kind.

Solution

The price of 1 book = The price of 5 books/Total number of books = sh 60 000/5 = sh 12 000.

Thus, the price of 3 books = 3 × sh 12 000 = sh 36 000.

Alternative solution (Cross multiplication)

Let $x_1$ and $x_2$ represent the number of books. $y_1$ and $y_2$ represent the price of books.

Use the expression $\frac{y_1}{y_2} = \frac{x_1}{x_2}$ to find the price of 3 books as follows:

$x_1 = 5$, $x_2 = 3$, $y_1 =$ sh 60 000, $y_2 = ?$

Find the value of $y_2$, which is the price of 3 books:

sh 60 000/$y_2$ = 5/3

$y_2 = 3$ × sh 60 000/5 = sh 36 000.

Therefore, the price of three books of the same kind is sh 36 000.

Example 3

Twelve kilograms of sugar cost 30 000 shillings.

(a) Find the cost of 5 kilograms of sugar.

(b) How much money will be used to buy 65 kilograms of sugar?

Solution

(a) The cost of 1 kg = The cost of 12 kg of sugar/Number of kilograms = sh 30 000/12 = sh 2 500

Thus, the cost of 5 kg = 5 kg × sh 2 500/1 kg = sh 12 500.

Therefore, the cost of 5 kilograms of sugar is 12 500 shillings.

(b) The amount of money = 65 kg × sh 2 500/1 kg = sh 162 500.

Therefore, 162 500 shillings will be used to buy 65 kilograms of sugar.