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Direct Proportion and
Inverse Proportion


Leading on from understanding the basics of ratio and proportion. Direct Proportion and inverse proportion generally involve using ratios to compare and establish values and amounts.

We will look at both concepts on this page.




Direct Proportion

If two or more ratios are equal, then they are classed as being in Direct Proportion.


1 : 4       ,       2 : 8       ,       3 : 12

The three above ratios are in direct proportion, both parts of the ratios increase at the same rate.

Each time the left value increases by  1,  the right value increases by  4.



Examples    


(1.1) 

A worker at a company earns  $36  for  4  hours of work.

How many dollars would they earn if they worked  7  hours?

If they are being paid with the same ratio for dollars to hours.

Solution   

We can start by writing the ratio as,       Hours : Dollars     =>     4 : 36.

Division by 4 can be simplify this to    1 : 9.

We can see,  9  dollars for  1  hour of work.

Now multiplying both ratio sides by  7  will give the ratio for  7  hours of paid work.

1 × 7 = 7     ,     9 × 7 = 63       =>       7 : 63

So for  7  hours of work, the worker would earn  $56.




(1.2) 

To make  6  sweet drinks, a women uses the following ingredients.

960ml Milk   :   36g Strawberries   :   6g Sugar.

How much of each ingredient would be needed to make  14  drinks, with the same ingredients ratio?


Solution   

Dividing the ratio through by  6  will give the ratio for  1  drink.

960 ÷ 6 = 160     ,     36 ÷ 6 = 6     ,     6 ÷ 6 = 1

Now that we know the amount of ingredients needed for  1  drink.

Multiplying through by  14  will result in the amount of ingredients for  14  drinks.


160 × 14 = 2240     ,     6 × 14 = 84     ,     1 × 14 = 14

The ratio of ingredients needed to make  14  milkshakes is,

2240ml Milk   :   84g Strawberries   :   14g Sugar.





Direct Proportion Formula

When we have one value that is Directly Proportional to another, such as is the case in example  (1.1),  where dollars earned was directly proportional to hours worked.


If we label money ( D ),  and hours ( H ).

Then it can be written that   DH.

Meaning that  M  is Directly Proportional to  H.



In example  (1.1),  money earned was directly proportional to hours worked.

Money earned  =  Dollars × Hours


This connection can be written as    D  =  kH.

Where  k  is a constant, known as the “constant of proportionality”.



To establish the value of k. We just need two corresponding values of  M  and  H.
In example  (1.1)  this was given, with  “2”  hours and  “16”  dollars.


36  =  k4     =>     \bf{\frac{36}{4}}  =  k     =>     9  =  k


Which results in a complete direct proportion formula for example  (1.1).

M  =  8H

Thus for example  (1.1),  any time you would like to know the amount of dollars earned by the worker, you just multiply the hours worked, by  8.







Inverse Proportion

With direct proportion and inverse proportion, inverse proportion follows a similar idea to direct proportion, but there is a slight difference.


With direct proportion each part of a ratio increased at the same rate.


1 : 4       ,       2 : 8       ,       3 : 12


With inverse proportion however, what happens is that one part of a ratio will increase, as one part decreases.


A good example is to think of climbing up a mountain.
As the height/altitude increases and when climb higher, the air temperature will go down and decrease.

The height value goes up, and the temperature value goes down.

We can see this illustrated in an image below, where going up every  1000  metres gives a decrease of   6°C.


Direct and Inverse Proportion mountain to help understand inverse proportion.

6°C    for every    +1000m.

Temperature : Height     =>     0 : 1000     ,     –6 : 2000     ,     –12 : 3000



Example    


(2.1) 

On a road trip, a coach drives at a steady speed of  48mph.

The road trip eventually takes  4  hours, how much less time would the trip have been had the coach speed been  60mph?

Solution   

Speed : Time     =>     45mph : 4hrs

Firstly, if the coach was travelling at only  1mph,  45 times slower, then the journey time will be 45 times longer.


As speed decreases  ,  journey time increases.

With an inverse ratio, we can divide one side and do the reverse on the other, multiplication


( 45mph ÷ 45 ) : (4hours × 45)    =    1mph : 180hrs

Left part of ratio was divided by  45,  right side multiplied by  45.
As the values are inversely related.

From here, we can now multiply  1mph  by  60,  while at the same time also dividing  180hrs  by  60.

This will give us the trip time taken for a speed of  60mph.


1mph : 180hrs       =>       1mph × 60  :  \bf{\frac{180}{60}}     =     60mph : 3hrs

The road trip would take  3  hours if the coach travelled at a speed of  60mph.

So  1  hour shorter than the trip time at  45mph.





Inverse Proportion Formula

As is the case with direct proportion, situations where the values are inversely proportional can also be modelled with a standard formula.

Situations such as the situation in example  (2.1).

Where as the coach speed increased, the time taken decreased.


It can be written that  S \bf{\frac{1}{T}},

meaning that  S  is inversely proportional to  T.



To obtain a general inverse proportion formula for this situation,
we first start off by writing the proportion symbol as an equals sign, and replacing  1  with  k.

S  =  \bf{\frac{k}{T}}     ,    multiply through by  T     =>     ST  =  k

Like in the case with direct proportion, here  k  is again a constant, and again called the “constant of proportionality”.
In a given ratio, two corresponding values of  S  and  T  can be used in order to establish the value  k.



In example  (2.1),  when   S = 45,   T = 4.

ST = k      =>      45 × 4  =  180     =>     180  =  k

Which results in the formula for the direct proportion in example  (2.1):

S  =  \bf{\frac{180}{T}}


So if you wanted to know what speed to travel at for the journey to take  4  hours.

S  =  \bf{\frac{180}3}}  =  60

As we saw earlier, a speed of  60mph  would result in a journey time of  3  hours.





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