# What is Proportion in Math?

### Ratio and Proportion

When we have two ratios, they are classed as being in proportion if they are both equal to each other in measure.

The ratios illustrated above,&nbsp 2 : 3 &nbsp and &nbsp 4 : 6 &nbsp are ratios that are in proportion.

3&nbsp is to &nbsp4“&nbsp &nbsp as &nbsp &nbsp”4&nbsp is to &nbsp6

&nbsp &nbsp &nbsp &nbsp or

3&nbsp out of &nbsp4” &nbsp &nbsp is the same ratios as &nbsp &nbsp “4&nbsp out of &nbsp6“.

Ratios can also be written as fractions too, so we can represent things this way.

\bf{\frac{2}{3}}  =  \bf{\frac{4}{6}}.

### Notation for Proportion

Say we have two different ratios that are in proportion, &nbsp &nbsp a:b &nbsp=&nbsp c:d.

When this is the case we can write the ratios as, &nbsp &nbsp a:b &nbsp::&nbsp c:d.

This means that &nbspa&nbsp is to &nbspb,&nbsp as &nbspc&nbsp is to &nbspd.

The figures that are in the outer places,&nbsp a&nbsp and &nbspd,&nbsp are referred to as &nbsp”extremes”.

The figures in the inner places,&nbsp b&nbsp and &nbspc,&nbsp are referred to as &nbsp”means”.

## What is Proportion in Math?Looking for Proportion

In order to check if &nbsp2&nbsp ratios are in proportion, we can multiply the &nbspmeans,&nbsp and the &nbspextremes&nbsp together.

Because if &nbsp2&nbsp ratios form a proportion, then these multiplications will be equal in value.

a : b &nbsp=&nbsp c : d &nbsp &nbsp&nbsp if &nbsp&nbsp &nbsp b × c &nbsp=&nbsp a × d

Example &nbsp &nbsp

The ratios demonstrated earlier in the page were &nbsp 2 : 3 &nbsp and &nbsp 4 : 6.

Multiplying the &nbspmeans&nbsp and the &nbspextremes&nbsp together gives:

3 × 4 &nbsp=&nbsp 12 &nbsp &nbsp &nbsp , &nbsp &nbsp &nbsp 2 × 6 &nbsp=&nbsp 12

2 : 3&nbsp and &nbsp4 : 6&nbsp do form a proportion. &nbsp &nbsp 2 : 3 &nbsp::&nbsp 4:6

But what if wee looked at the ratios &nbsp3 : 4&nbsp and &nbsp5 : 7.

Multiplication of the &nbspmeans&nbsp and the &nbspextremes&nbsp together results in:

4 × 5 &nbsp=&nbsp 20 &nbsp &nbsp &nbsp , &nbsp &nbsp &nbsp 3 × 7 &nbsp=&nbsp 21

3 : 4&nbsp and &nbsp5 : 7&nbsp do NOT form a proportion.

We could also write the ratios in fraction form, and this approach is the same as comparing the &nbspcross products.

\bf{\frac{3}{4}} &nbsp,&nbsp \bf{\frac{6}{8}} &nbsp &nbsp &nbsp => &nbsp &nbsp &nbsp 3 × 8 &nbsp=&nbsp 24 &nbsp &nbsp &nbsp 4 × 6 &nbsp=&nbsp 24

\bf{\frac{3}{4}} &nbsp,&nbsp \bf{\frac{5}{7}} &nbsp &nbsp &nbsp => &nbsp &nbsp &nbsp 3 × 7 &nbsp=&nbsp 21 &nbsp &nbsp &nbsp 4 × 5 &nbsp=&nbsp 20

### Proportion in Math Examples

(1.1)&nbsp

If &nbsp&nbsp3 : 7 &nbsp&nbsp and &nbsp&nbsp b : 14 &nbsp form a proportion, what is the value of &nbsp b?

Solution&nbsp &nbsp

For the two ratios to form a proportion, the ‘means’ and ‘extremes’ being multiplied together will produce an equal result.

× 14 &nbsp = &nbsp 7 × b &nbsp &nbsp &nbsp &nbsp => &nbsp &nbsp &nbsp &nbsp 42 &nbsp=&nbsp 7b

\boldsymbol{\frac{42}{7}} &nbsp=&nbsp b &nbsp &nbsp &nbsp &nbsp => &nbsp &nbsp &nbsp &nbsp b &nbsp=&nbsp 6

(1.2)&nbsp

If &nbsp &nbsp \boldsymbol{\frac{6}{a}} &nbsp &nbsp and &nbsp &nbsp \bf{\frac{2}{19}} &nbsp form a proportion, what is the value of &nbsp a?

Solution&nbsp &nbsp

For the two ratios to form a proportion, the ‘means’ and ‘extremes’ again being multiplied together will produce an equal result.

6 × 19 &nbsp = &nbsp a × 2 &nbsp &nbsp &nbsp &nbsp => &nbsp &nbsp &nbsp &nbsp 114 &nbsp=&nbsp 2a

\boldsymbol{\frac{114}{2}} &nbsp=&nbsp a &nbsp &nbsp &nbsp &nbsp => &nbsp &nbsp &nbsp &nbsp a &nbsp=&nbsp 57

(1.3)&nbsp

A lesser scale model is to created of a life size 4×4 car.

Based on the information seen in the image below, what height in &nbspcm&nbsp should the smaller 4×4 model car be?

Solution&nbsp &nbsp

As the smaller size model is to be the same scale as the full size car, the width and height of each car will be in proportion with the other.

Thus we can solve for the height of the model car by drawing up the ratios of width to height.

\boldsymbol{\frac{WIDTH}{HEIGHT}} &nbsp&nbsp = &nbsp&nbsp \bf{\frac{450}{180}} &nbsp&nbsp = &nbsp&nbsp \boldsymbol{\frac{80}{h}}

450 &times h &nbsp = &nbsp 180 × 80

450h &nbsp = &nbsp 14’400

h &nbsp = &nbsp \bf{\frac{14’400}{450}} &nbsp = &nbsp 32

The lesser scale 4×4 car model should be &nbsp32cm&nbsp in height for the cars to be in proportion.

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