# Direct Proportion andInverse 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.

### Direct Proportion

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

1 : 4 &nbsp &nbsp &nbsp , &nbsp &nbsp &nbsp 2 : 8 &nbsp &nbsp &nbsp , &nbsp &nbsp &nbsp 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 &nbsp1,&nbsp the right value increases by &nbsp4.

Examples &nbsp &nbsp

(1.1)&nbsp

A worker at a company earns &nbsp$36&nbsp for &nbsp4&nbsp hours of work. How many dollars would they earn if they worked &nbsp7&nbsp hours? If they are being paid with the same ratio for dollars to hours. Solution&nbsp &nbsp We can start by writing the ratio as, &nbsp &nbsp &nbsp Hours : Dollars &nbsp &nbsp => &nbsp &nbsp 4 : 36. Division by 4 can be simplify this to &nbsp&nbsp 1 : 9. We can see,&nbsp 9&nbsp dollars for &nbsp1&nbsp hour of work. Now multiplying both ratio sides by &nbsp7&nbsp will give the ratio for &nbsp7&nbsp hours of paid work. 1 × 7 = 7 &nbsp &nbsp , &nbsp &nbsp 9 × 7 = 63 &nbsp &nbsp &nbsp => &nbsp &nbsp &nbsp 7 : 63 So for &nbsp7&nbsp hours of work, the worker would earn &nbsp$56.

(1.2)&nbsp

To make &nbsp6&nbsp sweet drinks, a women uses the following ingredients.

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

How much of each ingredient would be needed to make &nbsp14&nbsp drinks, with the same ingredients ratio?

Solution&nbsp &nbsp

Dividing the ratio through by &nbsp6&nbsp will give the ratio for &nbsp1&nbsp drink.

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

Now that we know the amount of ingredients needed for &nbsp1&nbsp drink.

Multiplying through by &nbsp14&nbsp will result in the amount of ingredients for &nbsp14&nbsp drinks.

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

The ratio of ingredients needed to make &nbsp14&nbsp milkshakes is,

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

### Direct Proportion Formula

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

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

Then it can be written that &nbsp DH.

Meaning that &nbspM&nbsp is Directly Proportional to &nbspH.

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

Money earned &nbsp=&nbsp Dollars × Hours

This connection can be written as &nbsp&nbsp D &nbsp=&nbsp kH.

Where &nbspk&nbsp is a constant, known as the “constant of proportionality”.

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

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

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

M &nbsp=&nbsp 8H

Thus for example &nbsp(1.1),&nbsp any time you would like to know the amount of dollars earned by the worker, you just multiply the hours worked, by &nbsp8.

## 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 &nbsp &nbsp &nbsp , &nbsp &nbsp &nbsp 2 : 8 &nbsp &nbsp &nbsp , &nbsp &nbsp &nbsp 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 &nbsp1000&nbsp metres gives a decrease of &nbsp 6°C.

6°C &nbsp&nbsp for every &nbsp&nbsp +1000m.

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

Example &nbsp &nbsp

(2.1)&nbsp

On a road trip, a coach drives at a steady speed of &nbsp48mph.

The road trip eventually takes &nbsp4&nbsp hours, how much less time would the trip have been had the coach speed been &nbsp60mph?

Solution&nbsp &nbsp

Speed : Time &nbsp &nbsp => &nbsp &nbsp 45mph : 4hrs

Firstly, if the coach was travelling at only &nbsp1mph,&nbsp 45 times slower, then the journey time will be 45 times longer.

As speed decreases &nbsp,&nbsp journey time increases.

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

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

Left part of ratio was divided by &nbsp45,&nbsp right side multiplied by &nbsp45.
As the values are inversely related.

From here, we can now multiply &nbsp1mph&nbsp by &nbsp60,&nbsp while at the same time also dividing &nbsp180hrs&nbsp by &nbsp60.

This will give us the trip time taken for a speed of &nbsp60mph.

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

The road trip would take &nbsp3&nbsp hours if the coach travelled at a speed of &nbsp60mph.

So &nbsp1&nbsp hour shorter than the trip time at &nbsp45mph.

### 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 &nbsp(2.1).

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

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

meaning that &nbspS&nbsp is inversely proportional to &nbspT.

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 &nbsp1&nbsp with &nbspk.

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

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

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

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

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

S &nbsp=&nbsp \bf{\frac{180}{T}}

So if you wanted to know what speed to travel at for the journey to take &nbsp4&nbsp hours.

S &nbsp=&nbsp \bf{\frac{180}3}} &nbsp=&nbsp 60

As we saw earlier, a speed of &nbsp60mph&nbsp would result in a journey time of &nbsp3&nbsp hours.

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