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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.

Illustration helping show what is Proportion in Math.


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

3  is to  4”    as    “4  is to  6
        or
3  out of  4”     is the same ratios as     “4  out of  6“.


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,     a:b  =  c:d.

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

This means that  a  is to  b,  as  c  is to  d.


The figures that are in the outer places,  a  and  d,  are referred to as  “extremes”.

The figures in the inner places,  b  and  c,  are referred to as  “means”.

Extremes and means in Proportion.






What is Proportion in Math?
Looking for Proportion


In order to check if  2  ratios are in proportion, we can multiply the  means,  and the  extremes  together.

Because if  2  ratios form a proportion, then these multiplications will be equal in value.

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



Example    

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


Multiplying the  means  and the  extremes  together gives:

3 × 4  =  12       ,       2 × 6  =  12

2 : 3  and  4 : 6  do form a proportion.

2 : 3  ::  4 : 6


But what if we looked at the ratios.
3 : 4  and  5 : 7.

Multiplication of the  means  and the  extremes  together results in:

4 × 5  =  20       ,       3 × 7  =  21

3 : 4  and  5 : 7  do NOT form a proportion.


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

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

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





Proportion in Math Examples


(1.1) 

If   3 : 7    and    b : 14   form a proportion, what is the value of   b?

Solution   

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

3 × 14   =   7 × b         =>         42  =  7b

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




(1.2) 

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

Solution   

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

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

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




(1.3) 

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  cm  should the smaller 4×4 model car be?

Car dimensions for a proportion example.


Solution   

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}}    =    \bf{\frac{450}{180}}    =    \boldsymbol{\frac{80}{h}}


450 × h   =   180 × 80

450h   =   14’400

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


The lesser scale 4×4 car model should be  32cm  in height for the cars to be in proportion.





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