Ratios

A ratio is a comparison of two quantities. The ratio \(\textcolor{colordef}{3}\) to \(\textcolor{colorprop}{2}\) can be expressed as \(\textcolor{colordef}{3}:\textcolor{colorprop}{2}\) or \(\dfrac{\textcolor{colordef}{3}}{\textcolor{colorprop}{2}}\).

A part-part ratio compares two distinct parts of a whole.$$\textcolor{colordef}{\text{Part 1}}:\textcolor{colorprop}{\text{Part 2}}$$

For one bowl of fruit juice, there are \(\textcolor{colordef}{3}\) cherries and \(\textcolor{colorprop}{2}\) apples.The ratio of cherries to apples is \(\textcolor{colordef}{3}:\textcolor{colorprop}{2}\).

A Part-whole ratio compares one part of a whole to the whole.$$\textcolor{colordef}{\text{Part 1}}:\textcolor{olive}{\text{Whole}}\text{ or }\textcolor{colorprop}{\text{Part 2}}:\textcolor{olive}{\text{Whole}}$$

If a juice is made with \(\textcolor{yellow}{1}\) lemon and \(\textcolor{orange}{2}\) oranges, find the ratio of oranges to the total number of fruits.

• Let's make some fresh juice! For one glass of juice, we need \(\textcolor{yellow}{1}\) lemon and \(\textcolor{orange}{2}\) oranges. The ratio of lemons to oranges is \(\textcolor{yellow}{1}:\textcolor{orange}{2}\).
• Now, if we want to make two glasses of juice, we need to double the ingredients.
• The ratio remains the same. The ratios are equal: \(\textcolor{yellow}{1}:\textcolor{orange}{2}=\textcolor{yellow}{2}:\textcolor{orange}{4}\).

Two ratios are equal if one can be expressed as a multiple of the other.

To show that two ratios are equal, we can compare their related fractions. If the fractions are equal, then the ratios are equal.

As, \(\textcolor{colordef}{1}:\textcolor{colorprop}{2}=\textcolor{colordef}{2}:\textcolor{colorprop}{4}\)

Imagine you're making a fruit juice mix. The recipe calls for \(\textcolor{orange}{3}\) cups of orange juice and \(\textcolor{olive}{2}\) cups of apple juice. This ratio of \(\textcolor{orange}{3}:\textcolor{olive}{2}\) ensures the juice has the right flavor balance. But what if you want to make a larger batch? If you double the amount of orange juice to \(\textcolor{orange}{6}\) cups, you need to double the amount of apple juice to \(\textcolor{olive}{4}\) cups to keep the same taste. The ratio \(\textcolor{orange}{3}:\textcolor{olive}{2}\) is the same as \(\textcolor{orange}{6}:\textcolor{olive}{4}\), meaning the two mixtures will taste the same. This equality of ratios is called a proportion.

A proportion states that two ratios are equal.

To make \(\textcolor{colorprop}{1}\) chocolate cake, \(\textcolor{colordef}{4}\) eggs are needed. How many eggs are needed to make \(\textcolor{colorprop}{2}\) cakes?

The unitary method is an approach used to solve problems involving proportions. The essence of this method is to determine the value of one unit of a quantity and then use that value to find the unknown quantity.

\(\textcolor{colordef}{5}\) apples cost \(\dollar\textcolor{colorprop}{10}\). To calculate the cost of \(\textcolor{colordef}{8}\) apples, follow these steps:

• To the unit: Find the cost of \(\textcolor{colordef}{1}\) apple by dividing the total cost by the initial number of apples \(\textcolor{colordef}{5}\):So, \(\textcolor{colordef}{1}\) apple costs \(\textcolor{colorprop}{2}\) dollars.
• From the unit: Find the cost of \(\textcolor{colordef}{8}\) apples by multiplying the unit ratio by the final number of apples \(\textcolor{colordef}{8}\):So, \(\textcolor{colordef}{8}\) apples cost \(\textcolor{colorprop}{16}\) dollars.
• 

Cross-multiplication is a method used to solve problems involving proportions. This method involves cross-multiplying across the quantities of a proportion to find the unknown quantity.

\(\textcolor{colordef}{5}\) apples cost \(\dollar\textcolor{colorprop}{10}\). To calculate the cost of \(\textcolor{colordef}{8}\) apples, follow these steps:

• Set up the proportion: Write the proportion where the cost of \(\textcolor{colordef}{5}\) apples is to \(\textcolor{colorprop}{10}\) dollars as the cost of \(\textcolor{colordef}{8}\) apples is to \(\textcolor{colorprop}{x}\) dollars:\(\dfrac{\textcolor{colorprop}{10}}{\textcolor{colordef}{5}} = \dfrac{\textcolor{colorprop}{x}}{\textcolor{colordef}{8}}\)
• Solve for \(x\):
• So, \(\textcolor{colordef}{8}\) apples would cost \(\textcolor{colorprop}{16}\) dollars.