# Ratio Fractions

In a class of 50 students, we may have 20 girls and 30 boys. This means that for every 20 girls we may see 30 boys in the class.

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This can be represented in the mathematical concept called 'ratio'.

Did you also know that there is a relation between ratios and fractions? In this article, we are going to discuss ratio fractions.

## Ratio fraction definition

Ratio fraction is a way of expressing ratios as fractions.

In mathematics, a ratio indicates how many times one number contains another. It is used to compare two quantities. Ratios are used to express the relationship between entities or groups using the column sign denoted by " : " between the entities.

Consequently, the sign that indicates the distinction between these two parties also indicates a fraction.

## Ratio and proportion in a fraction

Ratios are basically used to compare quantities. They can be expressed in a couple of ways; using the colon (:), or using the division sign (/).

For example, if we were told that John owned 15 cats and 3 dogs, this could be expressed in a ratio form. It can be written as $15:3$. This literally means that there are 15 cats to 3 dogs available.

Now this can also be written in a fraction form, $\frac{15}{3}$.

Ratios are not to be taken as quantities available. They express how many dogs there are to every cat owned. This makes them extremely valid if we have to simplify them.

We can find the greatest common divisor for both figures and divide them by it. Here, we realize that 3 can divide both 15 and 3 to leave us with $5:1$ or $\frac{5}{1}$.

It is at this point that the concept of ratios becomes vivid. However, now we are talking about how many cats there are to every dog. So though there are 15 cats and 3 dogs, the ratio for that is best written as $5:1$ because for every 5 cats, there is one dog.

We are making a cake, the recipe sometimes says mix flour with water in the ratio 2 part 1. That means that if you use 2 cups of flour, mix it with 1 cup of water. Both numbers should be non-zero in order to make meaning out of the comparison.

### How ratios are expressed?

Ratios between two quantities are expressed in the form $a:b$, where a and b are both different quantities. This reads as " a is to b ".

The ratio formula is thus,

$a:b=\frac{a}{b}$

This means that $a:b$ can also be expressed as $\frac{a}{b}$.

What is the ratio of native French speakers to native English speakers if there were 16 native English speakers in a class of 40 students given that the rest of the class are native French speakers?

Solution

We can first note the figures given in the question.

$Totalnumberofstudents=40$

$NumberofnativeEnglishspeakers=16$

If we want to find the number of English speakers in class, we subtract the number of native French speakers from the total number of students to get,

$NumberofnativeFrenchspeakers=40-16=24$

The ratio for this can therefore be expressed as $24:16$ or $\frac{24}{16}$.

However, this can further be simplified using the greatest common divisor 8. It divides 24 to give us 3, and divides 16 to give us 2. And hence, the appropriate ratio for boys to girls in the class can be expressed as $3:2$ or $\frac{3}{2}$.

A university in London organizes a graduation for students, and the people present include 120 graduates and 80 parents. Express the ratio of the number of graduates to parents.

Solution

We can first write down what is given in the question,$Numberofparents=80$$Numberofgraduates=120$

This can be expressed as a ratio as $120:80$.

However, we can simplify this even more. We now find the greatest common factor for both values; 80 and 120m which is 40. We will now divide each of them by the 40 to simplify them,

$\frac{120}{40}:\frac{80}{40}$ to get $3:2$Therefore, we have$3:2$ as the ratio of the number of graduates to parents.

## Convert a ratio to a fraction

The process of converting ratios to fractions involves writing ratios as a fraction. This process involves finding the total parts of the ratio available and making that the denominator.

If we had 3 apples to 5 berries for example, the first step we can take to convert this ratio to a fraction is to find the total parts. Here, the total parts are 8. Hence, if we are asked to find the fraction of apples for the ratio, we just use the figure of apples as the numerator and use the total parts as a denominator.

Mathematically, we can use a simple formula for that to make it easier. To convert ratios to fractions when we have the ratio $a:b$, we state that the fraction of $a$ for the ratio $a:b$ is$\frac{a}{a+b}$. This means that the fraction for $b$ in the ratio $a:b$ can also be expressed as $\frac{b}{a+b}$.

For example, the Austrian flag possesses a ratio of 2 parts in red and a ratio of 1 part in white. This means that there are 3 parts in all.

The fraction for the red part is written as,

$\frac{2}{2+1}=\frac{2}{3}$

The fraction for the white part is also written as

$\frac{1}{2+1}=\frac{1}{3}$

As mentioned earlier, the number of parts in all with regards to the flag is found by summing $a$ and $b$.

Simplify all fractions if possible. However, it is ideal to maintain working with whole numbers.

## Ratio to fraction calculation

In this section, we are going to take many examples of converting ratios to fractions.

John and Mike share cupcakes in the ratio $3:4$. What is the fraction of the cupcake Mike gets?

Solution

We are asked to give the fraction of Mike from the whole cupcake and so, Mike's portion is in the fraction $\frac{b}{a+b}$, where $a=3$ and$b=4$.

Thus, Mike's portion is

$\frac{b}{a+b}=\frac{4}{3+4}=\frac{4}{7}$

Mike gets $\frac{4}{7}$ of the cake.

In a YouTube video, a recipe for cakes was outlined as 4 cups of flour and a cup of milk. What fraction of the cake is the flour?

Solution

The flour's part is given by$\frac{a}{a+b}$where$a=4$ and $b=1$, hence

$\frac{a}{a+b}=\frac{4}{4+1}=\frac{4}{5}$

The fraction of the flour is $\frac{4}{5}$.

Bernard's dog gives birth to 8 female puppies and 4 male ones. What fraction of the puppies are female?

Solution

The female fraction of the puppies is given as $\frac{a}{a+b}$ where $a=8$and $b=4$, thus we have

$\frac{a}{a+b}=\frac{8}{8+4}=\frac{8}{12}$

We notice here that the resulting fraction can be simplified further. Both the denominator and the numerator can be divided by 4, so we will divide both by 4.

$\frac{8÷4}{12÷4}=\frac{2}{3}$

The fraction of the female puppies is $\frac{2}{3}$.

## Difference between a ratio and a fraction

The fundamental difference between a ratio and a fraction is that ratios compare two quantities whilst fractions depict what is a portion of a larger whole. This means that the denominator for fractions is the whole of which the numerator is a part.

For example, if we were given the fraction of female puppies born to be $\frac{4}{5}$, what we can make from this, is that with the part of the total number of puppies that were born, 4 out of 5 of them were female.

On the other hand, if we are given the ratio of female puppies to male puppies born are $4:1$, then we know that for every 4 female puppies born, one is male.

## Ratio Fractions - Key takeaways

• Ratio fraction is a way of expressing ratios as fractions.
• Ratios are used to compare two quantities.
• Ratios denote how two distinct entities are related using the column sign " : " .
• To convert ratios to fractions when we have the ratio $a:b$, we state that the fraction of $a$ for the ratio $a:b$ is$\frac{a}{a+b}$ .
• The fundamental difference between ratios and fractions is that ratios compare two quantities whilst fractions depict what is a portion of a larger whole.

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How can you express a ratio as a fraction?

Ratios are best written as fractions when the first term, called the antecedent term, is the numerator and when the second term, called the consequent  is the denominator.

How do you find the ratio of a fraction?

First, write down the numerator. Second, write a colon. Thirdly, write down the denominator.

How to solve ratio problems with fractions?

Always simplify the fractions first. Get rid of the fractions if the solution is not required to be in fractions.

How are ratios and fractions similar?

A ratio can be expressed as a fraction. The first number in the ratio is the numerator, and the second number after the colon is the denominator.

What is difference between ratio and fraction?

The fundamental difference between a ratio and a fraction is that ratios compare two quantities whilst fractions depict what is a portion of a larger whole.

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