# Scale Factors, Ratio and Proportional Reasoning

🏆Practice ratio, proportion and scale

Summary

Ratio, Proportion, and Scale

The ratio between terms describes how many times greater or smaller a certain magnitude is than the other.

Proportion is a constant relationship or ratio between different magnitudes.

Scale is the proportionality between the real dimensions of something and those of the scheme that represents it.

## Test yourself on ratio, proportion and scale!

How many times longer is the radius of the red circle than the radius of the blue circle?

## Ratio

### What is a ratio?

A ratio is a relationship between things or objects and describes how many times greater or smaller a certain magnitude is than another.

### How is a ratio read?

Just as we read in English, mathematics is also read from left to right. So,

we combine the written words according to their order of appearance and convert them into numbers from left to right.

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## Ratio of a certain part in relation to a whole

We can find a ratio of an object to the entire set.

For example, the ratio between apples and all other fruits in the fridge is $3:5$

This means that out of the $5$ fruits in the fridge, $3$ of them are apples.

## Equivalent Ratios

Equivalent ratios are those that we can find written differently but represent the same ratio or relationship. When simplifying or amplifying fractions, the same quotient will be obtained.

Do you remember we said that a ratio can be shown in the form of a fraction?

Then we can apply the same rule to ratios, we can reduce both terms of the ratio or amplify them and arrive at equivalent ratios.

To solve this type of problem easily, we will always try to arrive at the smallest ratio.

To simplify them, we will ask ourselves by what number can we divide both terms of the ratio, in this way we will arrive at the most reduced equivalent ratio possible.

Let's see the following example in this section:

Do you know what the answer is?

## How can you tell if they are equivalent ratios?

We will ask ourselves: Will we get to the same ratio by reduction or by amplification?

To determine if two or more ratios are equivalent, one must look for a number that, by multiplying or dividing both terms of one of the ratios, we arrive at the other given ratio

Let's see some examples:

## Distribution according to a given ratio

In a distribution according to a given ratio, we will have a defined amount that we must divide according to that ratio. That is, this happens when we have to distribute a certain quantity or objects according to a determined ratio.

Let's see how we can define this situation with some examples, like the ones shown below:

## Proportion

Proportionality is synonymous with equivalence relation. In everyday life, we often use expressions like "taking things relatively" and that means comparing and taking things in their due importance... That is, in the precise relation of what is actually happening, without exaggerating.

How to know if there is proportionality between ratios?

In the same way that we have done in the chapter on equivalent ratios, to find out if there is an equivalence relation (proportionality between ratios),

we will simplify the ratios.

We will apply the greatest reduction possible (with the highest number by which we can divide without remainder) and see if we arrive at the same ratio.

Let's see an example in the following article:

## Finding a missing value in a proportion

Sometimes we will be given only one whole ratio with its two corresponding terms and a third piece of data that is part of another ratio.

Usually, it will be stated that there is proportionality between the ratios and that we must find the missing data in the ratio.

How are problems solved where a number from the proportion is missing?

1. We will identify the ratio given in the question between both terms and write it down in fraction form.
2. We will compare it with the other ratio (also in fraction form) that includes the third piece of data from the question and also the unknown $X$.
3. We will solve for $X$.

Let's see an example in the following article:

Do you think you will be able to solve it?

## Direct Proportionality

What is direct proportionality?

Direct proportionality indicates a situation in which, when one term is multiplied by a certain amount, the second term undergoes exactly the same.

Similarly, when one term is divided by a certain amount, the second term undergoes exactly the same.

The ratio between both magnitudes remains constant.

Let's see an example from everyday life in the following:

## Inverse Proportionality

Inverse proportionality indicates a situation in which, when one term is multiplied by a certain amount of times, the second term is decreased by the same amount of times and vice versa.

The ratio between both magnitudes remains constant.

Let's see an example from everyday life in the following:

## Scale

Scale is a synonymous expression to the word ratio.

Questions about scale deal with the relationship between the actual dimensions of an object and those of the drawing that represents it.

On the left appears the dimension of the graphic representation or map

On the right appears the actual dimension.

Suggestion:

How can you remember that the scale of the scheme or drawing is always seen on the left?

In the word left and in the word scheme the letter $e$ appears.

Note: When writing scales we must use the same units of measure in the scheme and in the real world.

If you have, for example, a dimension given in centimeters in the scheme and in reality it is in meters, the units must be converted so they are identical and only then noted in the scale.

Let's see an example:

## How do we determine if two ratios are proportional?

How do we determine if two ratios are proportional?

First, we must arrange the data of the two terms in the two ratios, that is, in the form of a quotient or fraction, then we have two ways to check it:

First way

We have to look for a number that, when multiplied or divided in some of the ratios or in both, gives us the same ratio. That is, we can simplify or amplify the fractions to observe that it is the same ratio, since two ratios are proportional if they are equivalent ratios.

Example:

Let's take the following two ratios, $3\colon8$ and $12\colon32$

Are they proportional?

We arrange them in the form of a quotient or fraction

$\frac{3}{8}$ and $\frac{12}{32}$

Now let's see if we amplify or simplify one or both ratios, in this case, we are going to amplify the first ratio by multiplying both terms by 4, as follows:

$\frac{3\cdot4}{8\cdot4}=\frac{12}{32}$

And we see that by amplifying the ratio we got the second ratio, therefore they are equivalent ratios.

Solution

Being equivalent ratios, then they are proportional.

Second way.

Once the elements of the ratios are arranged, we can multiply crosswise, and if it gives us the same result, then we say that the ratios are proportional.

For example:

Let's take the following two ratios, $5\colon3$ and $15\colon9$

Are these ratios proportional?

We arrange them in the form of a quotient, as follows

$\frac{5}{3}$ and $\frac{15}{9}$

Now let's multiply crosswise, as shown below:

$\left(5\right)\left(9\right)=45$

$\left(3\right)\left(15\right)=45$

We can observe that the cross multiplication gave us $45$ for both cases, so they are equivalent ratios.

Yes, they are proportional.

Do you know what the answer is?

## Review Questions

What is a ratio in mathematics?

A ratio is a relationship or comparison between magnitudes, people, or objects, written in the form of a fraction.

What is proportionality?

It is a relationship between two ratios, where the ratios are equivalent.

What is direct proportionality?

When we relate two ratios or compare two magnitudes, we say they are in direct proportion if one of them increases, the other magnitude also increases in the same way, or if one of them decreases, the other does so in the same proportionality.

What is a scale used for?

A scale is used to represent objects, or parts of reality on a map, plan, or drawing, in such a way that it does not distort the relationships between the elements that compose them.

## Examples and exercises with solutions on ratio, proportionality, and scale

### Exercise #1

Given the rectangle ABCD

AB=X

The ratio between AB and BC is $\sqrt{\frac{x}{2}}$

We mark the length of the diagonal A the rectangle in m

Check the correct argument:

### Step-by-Step Solution

Given that:

$\frac{AB}{BC}=\sqrt{\frac{x}{2}}$

Given that AB equals X

We will substitute accordingly in the formula:

$\frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}$

$x\sqrt{2}=BC\sqrt{x}$

$\frac{x\sqrt{2}}{\sqrt{x}}=BC$

$\frac{\sqrt{x}\times\sqrt{x}\times\sqrt{2}}{\sqrt{x}}=BC$

$\sqrt{x}\times\sqrt{2}=BC$

Now let's focus on triangle ABC and use the Pythagorean theorem:

$AB^2+BC^2=AC^2$

Let's substitute the known values:

$x^2+(\sqrt{x}\times\sqrt{2})^2=m^2$

$x^2+x\times2=m^2$

We'll add 1 to both sides:

$x^2+2x+1=m^2+1$

$(x+1)^2=m^2+1$

$m^2+1=(x+1)^2$

### Exercise #2

Given the rectangle ABCD

AB=X the ratio between AB and BC is equal to$\sqrt{\frac{x}{2}}$

We mark the length of the diagonal $A$ with $m$

Check the correct argument:

### Step-by-Step Solution

Let's find side BC

Based on what we're given:

$\frac{AB}{BC}=\frac{x}{BC}=\sqrt{\frac{x}{2}}$

$\frac{x}{BC}=\frac{\sqrt{x}}{\sqrt{2}}$

$\sqrt{2}x=\sqrt{x}BC$

Let's divide by square root x:

$\frac{\sqrt{2}\times x}{\sqrt{x}}=BC$

$\frac{\sqrt{2}\times\sqrt{x}\times\sqrt{x}}{\sqrt{x}}=BC$

Let's reduce the numerator and denominator by square root x:

$\sqrt{2}\sqrt{x}=BC$

We'll use the Pythagorean theorem to calculate the area of triangle ABC:

$AB^2+BC^2=AC^2$

Let's substitute what we're given:

$x^2+(\sqrt{2}\sqrt{x})^2=m^2$

$x^2+2x=m^2$

$x^2+2x=m^2$

### Exercise #3

How many times longer is the radius of the red circle than the radius of the blue circle?

5

### Exercise #4

How many times longer is the radius of the red circle, which has a diameter of 24, than the radius of the blue circle, which has a diameter of 12?

2

### Exercise #5

How many times longer is the radius of the red circle than the radius of the blue circle?

### Video Solution

$2$