# Ratio

Many times when we are given our share of chocolates or cookies, we want to know how these were shared among us and our siblings or friends.

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The concept of ratio which would be discussed herein would assist you in determining that henceforth.

## Ration definition

Ratio is the comparison of two or more quantities by showing the relationship in their various sizes. It tells us how much of a quantity can be found in another quantity.

Ratios show us the relationship between quantities, and it is essential when things are to be shared or divided amongst a group.

Ratios can be expressed in their simplest forms or simplified when they are divided by the highest common factors.

It is worthy of note that ratio comparison may be between quantities individually as a whole or perhaps between a part of a whole and its whole. This would be explained hereafter.

## Ratio notation

Ration notation tells us the various ways ratios can be represented or expressed. There are three notations for ratio: number notation, word notation, and fraction notation.

### Number Notation

Number notation occurs when ratios are expressed by writing numbers and a colon (:) between the numbers or a slash (/).

For example,

$3:4\phantom{\rule{0ex}{0ex}}5:6:1\phantom{\rule{0ex}{0ex}}2:7\phantom{\rule{0ex}{0ex}}7:2:11:5$

### Word notation

Word notation occurs when the word "is to" is used in expressing rations.

For example,

3 is to 4

5 is to 6 is to 1

2 is to 7

7 is to 2 is to 11 is to 15.

### Fraction notation

Fraction notation occurs when ratios are expressed as fractions. However, this is only applicable when comparing just two quantities.

For example,

$\frac{3}{4}\phantom{\rule{0ex}{0ex}}\frac{2}{7}$

## Ratio formula

The ratio formula is the expression used in calculating ratios. The general principle guiding ratio operation and its formula is division. Earlier we mentioned that ratios can be either in as a relationship between whole quantities or between a part of a whole and its whole. This as a matter of fact determines the kind of formula to be applied.

### Ratio between two whole quantities

In order to find the ratio between two whole quantities, we apply the quotient between the first and the second quantity. This means that the first quantity is divided by the second quantity.

The first quantity is known as the antecedent while the second is called the consequent. So if the first quantity is m and the other quantity is n, then,

$m:n=\frac{m}{n}$

Henderson and Robinson have each been given 5 oranges and 7 oranges respectively, find the ratio of oranges between Henderson and Robison.

Solution

Henderson has 5 oranges while Robinson has 7 oranges.

Therefore, the ratio of oranges between Henderson and Robinson is

$Henderson:Robinson=5:7=\frac{5}{7}$

### Ratio between a part and a whole

In order to find the ratio between a part and a whole, we apply the quotient between a part and a whole. Note that sometimes the total quantities may be given, other times, we would need to calculate it by finding the sum of the parts.

For instance, if m is a part of t, where t is the whole or total of the quantities, the ratio of m to t is,

$m:t=\frac{m}{t}$

Meanwhile, the ratio of m to the sum of the quantities m, n and o,

$m:\left(m+n+o\right)=\frac{m}{m+n+o}$

where m + n + o is the total number of quantities.

Out of 6 packs of sweets in a box, Doyle was given 5. What is the ratio of Doyle's share to the sweets in the box?

Solution

The total packs of sweets in the box is 6, while Doyle's share of sweets is 5.

Therefore, the ratio of Doyle's share to the sweets in the box is

$Doyle\text{'}sshare:totalsweets=5:6=\frac{5}{6}$

A bag contains 3 black balls, 2 red balls and 7 white balls. What is the ratio of white balls to all balls in the bag?

Solution

We first identify what ratio are we calculating. In this case it is white balls ratio to all balls.

Next, we are told that the bag contains 7 white balls.

Next, we find the total number of balls in the bag,

$t=3+2+7=12$

Now having found their values, we express them in ratio,

$Whiteballs:allballs=7:12=\frac{7}{12}$.

## Ratio scale

Ratio scale is obtaining equivalent ratios while multiplying or dividing with constants.

While maintaining the same ratio, we can increase or decrease measurements of geometric shapes.

In the illustration below, the length of the rectangle is 4 units while the width is 2 units, thus,

$length:breadth=4:2$

Now notice that the same rectangle was increased and decreased in measurements with respect to the two other rectangles beside it: here we applied respectively scaling to the initial rectangle.

An illustration of ratio scaling - StudySmarter Originals

There are two types of scaling: scaling up and scaling down.

### Scaling up

We scale up a ratio by multiplying the antecedent and the consequent by the same number c, where c is greater than 1.

When this occurs, we say the ratio has been scaled up. The number c is also known as the scaling factor or multiplier.

An illustration of ratio scaling up - StudySmarter Originals

In the above diagram the measurements of the obtained rectangle are multiplied by 2, the ratio of the original rectangle and the scaled up rectangle are equivalent.

### Scaling down

We scale down a ratio by dividing the antecedent and the consequent with the same number d, where d is greater than 1.

When this occurs, we say the ratio has been scaled down. The number d is also known as the scaling factor or multiplier.

An illustration of ratio scaling down- StudySmarter Originals

In the above diagram the measurements of the obtained rectangle are divided by 2, the ratios of the original rectangle and the scaled down rectangle are equivalent.

The length and breadth of a rectangular block is 9cm and 7cm respectively. What would be its new dimensions if scaled up by 5?

Solution

We first find the ratio of length to breadth. Thus,

$length:breadth=9:7$

The ratio is scaled up 5. So, we multiply the ratio by 5;

$\left(9:7\right)×5=\left(9×5\right):\left(7×:5\right)\phantom{\rule{0ex}{0ex}}\left(9:7\right)×5=45:35$

Therefore the new dimensions of the rectangular block are 45cm (length) and 35cm (breadth).

## Ratio and proportion

Proportion compares and gives the relationship between two ratios. It is expressed with an equal to sign (=) or a double colon (::).

Thus, for two ratios a:b and c:d, their proportion is given by

$a:b=c:d$

or

$a:b::c:d$

### Types of proportion

We distinguish two types of proportions: direct proportion and indirect proportion.

A direct proportion occurs when an increase in a quantity leads to an increase in the other related quantity.

An inverse proportion occurs when an increase in a quantity leads to a decrease in the other related quantity.

### Differences between ratio and proportion

Ratios differ from proportions in the following ways.

1. Ratios are comparisons between quantities meanwhile proportions are comparisons between ratios.

2. Ratios are expressions in the form,

$w:x$

However, proportions are equations in the form,

$w:x=y:z$

3. Ratios are represented with just a single colon (:) or a slash (/) while proportions are represented with a double colon (::) or an equal to sign (=).

4. Ratios are mentioned with the phrase "is to" whereas proportions are identified with the phrase "out of".

Some examples hereafter would elaborate more on the relationship as well as the differences in the application of ratio and proportion.

If 5 pairs of a brand shoe cost £120, how many pair(s) of the same brand shoe would Thomas with £48 buy?

Solution

We first determine what type of proportion we have. In order to do so, we answer this question: if the number of shoes increases would we have to pay more or less?

Your answer would tell you if it is a direct or inverse proportion.

The answer is YES. Surely, more shoes will require more money, thus this is a direct proportion.

The next thing is to write out your values,

5 pairs for £120

Next, assign a letter to the unknown value. Thus, let y represent the number of shoes Thomas would buy. Thus we have y pairs for £48.

Recall that the ratio is expressed only with quantities of the same unit.

Hence, we should pair quantities using the ratio and the order in which quantities are mentioned in the question,

5 pairs to y pairs

£120 to £48

Next, remember that proportion is the equation of ratios, thus we have

$5:y=120:48$

Next, we convert ratios to fractions and solve to get

$\frac{5}{y}=\frac{120}{48}$

Now, we cross multiply to get,

$5×48=120×y\phantom{\rule{0ex}{0ex}}240=120y\phantom{\rule{0ex}{0ex}}120y=240\phantom{\rule{0ex}{0ex}}\frac{120y}{120}=\frac{240}{120}\phantom{\rule{0ex}{0ex}}y=2\phantom{\rule{0ex}{0ex}}$

Thus, Thomas can only afford 2 pairs of shoes with £48.

It takes 12 laborers 3 days to clear a certain plot of land, how many days would it take 4 laborers to clear the same plot?

Solution

We first determine what type of proportion we have. In order to do so, we answer this question,

if the number of laborers decreases would it take less time to clear the same plot?

Your answer would tell you if it is a direct or inverse proportion.

The response is NO. Surely, fewer laborers would mean more time spent on clearing the plot, thus, this is an inverse proportion.

Next, we write out our values:

12 laborers in 3 days

Now, we assign a letter to the unknown value, so, let q represent the time it takes 4 laborers to do the job. Thus we have

4 laborers in q days

Next, we recall that the ratio is expressed only with quantities of the same unit. Thus, we should pair quantities using ratios and in the order they have been mentioned in the question.

However, because it is an inverse proportion we would have to swap positions in one of the quantities. This means that the relationship is in a different direction. Hence we have

12 laborers to 4 laborers

q days to 3 days

Now, remember that proportion is the equation of ratios. Therefore,

$12:4=q:3$

Next, we convert from ratio to fraction to get

$\frac{12}{4}=\frac{q}{3}$

We cross multiply;

$\frac{12}{4}=\frac{q}{3}\phantom{\rule{0ex}{0ex}}4×q=12×3\phantom{\rule{0ex}{0ex}}4q=36\phantom{\rule{0ex}{0ex}}\frac{4q}{4}=\frac{36}{4}\phantom{\rule{0ex}{0ex}}q=9$

Hence, it would take 4 laborers 9 days to clear that plot of land.

Note that if it were to be a direct proportion, it would have been 12 laborers to 4 laborers and 3 days to q days, both maintaining their order or position; but because it is inverse we have chosen to swap the position of the second ratio (days)

## Ratio examples

The use of ratio is very important as it translates into our daily activities. Particularly when it comes to sharing as well as determining the portion or fraction out of a whole quantity. Below are some examples to illustrate further.

A man shares his wealth among three of his sons James, John and Peter in the ratio 4:3:2. If he is worth £90,000, how much goes to John?

Solution

We first find the total of the ratio,

$4:3:2=4+3+2=9$

Next we find what fraction of the man's wealth goes to John. This is the same as finding the ration between Johns share value and the total share value;

$John:total=3:10=\frac{3}{10}$

We then multiply the fraction of the man's wealth that goes to join by the worth of the man,

$\frac{3}{10}×90000=27000$

John's share is £27000.

In a graduating class of 125 students, 50 are boys. What is the ratio of boys to girls?

Solution

Since the number of boys and the total number of students have been given, we should solve for the number of girls which is

$numberofgirls=totalstudents-numberofboys=125-50=75$

Since the number of girls have been calculated, we can now find the ratio of boys to girls as,

$boys:girls=50:75=\frac{50}{75}$

We divide the numerator and denominator by the highest common factor which is 25. We divide through by 25 to get

$boys:girls=\frac{2}{3}=2:3$

## Ratio - Key takeaways

• Ratio is the comparison of two or more quantities by showing the relationship in their various sizes. It tells us how much of a quantity can be found in another quantity.
• There are three notations for a ratio which are number notation, word notation, and fraction notation.
• The ratio formula is the expression or equation used in calculating ratios.
• Ratio scaling is the increase or decrease of ratios when they are multiplied or divided.
• Proportion is an equation that compares and gives the relationship between two ratios.

#### Flashcards in Ratio 47

###### Learn with 47 Ratio flashcards in the free StudySmarter app

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What is a ratio?

Ratio is a concept which enables you make comparison between quantities and tells you how much of a quantity can be found in another quantity.

What is an example of a ratio scale?

An example of ratio scale is if the ratio is 3:4 and has been scaled up by 5 it becomes 15:20.

What is the formula for ratio?

The formula of ratio is just the quotient of the two quantities. Like a:b is a/b.

How do you write ratios in notations?

Ratio notation can be written in three ways such as number notation, word notation and fraction notation.

How do I calculate a ratio?

Ratios are calculated by dividing the quantities. The first quantity is divided by the second quantity.

## Test your knowledge with multiple choice flashcards

Convert 5:7 to fraction.

Is 3:4 the same as $$\dfrac{3}{4}$$?

Which among the following is a major difference between ratio and proportion?

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