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Direct and Inverse proportions

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Jetzt kostenlos anmeldenImagine a factory where work needs to be done by a number of people and a timeline has to be fulfilled. There is a maximum time limit given and not too many workers should work on it otherwise it will not be cost-effective.

How would the manager of the factory choose the number of workers to do the task in a certain time given that if 1 person did it, it would be done in 6 hours, if 3 did it, it would take 2 hours, and so on?

This situation gives a vague idea of how two different quantities are related to each other and using such information, many real-world scenarios can be appropriately addressed.

The **proportions **of two quantities can be explicitly determined by observing the relationship between the quantities. Such quantities can be related to each other in various ways, but here, for the scope of this article, we will focus on the most basic of such proportions: **Direct **and **Inverse proportions.**

Let there be two quantities that depend on each other in such a way that as one of them varies, the other one varies at the same rate.

Two quantities are **directly ****proportiona****l** to each other if and only if they are linearly dependent on each other and the ratio between them is a constant.

Suppose the two quantities are the size of a bed and the other is the cost of making it. As the size of the bed increases, the cost of making it will also increase constantly.

Suppose the cost of 4 sq. mt. bed is $400, 8sq. mt. is $800, 16sq. mt. is $1600, and so on. Such two quantities are said to be **directly proportional.**

Two quantities are inversely proportional if we increase one of them and the other one decreases proportionally. This is the exact reason the word ‘inverse‘ is used, to denote the inverse relationship between them.

Two quantities are** inversely** proportional if and only if as one increases, the other one decreases and vice versa.

A typical real-world example of such two quantities would be the population growth of bacteria with respect to time.

Another example would be workers needed to complete a task, the more the workers, the lesser the time needed.

Two quantities are said to be **directly proportional **if one quantity increases, the other quantity increases, and if one quantity decreases the other quantity decreases. In mathematical notation, for two quantities *x *and *y*, the direct proportionality between x and y is expressed as

$x\propto y$

where the symbol $\propto $ denotes the **proportionality **between the two quantities.

The proportionality denotes that the quantities are **linearly related**. But to solve equations and the properties of these equations, we need an ‘=‘ sign and replace the proportionality symbol.

Hence, we have the following equivalence, $x\propto y$ is the same as $x=ky$, where *k *is a non-zero real-valued constant.

In other words, we have for a non-zero real-valued constant k,

$x\propto y\iff x=ky$.

Two quantities are said to be inversely proportional, if one quantity increases the other decreases and vice versa.

In mathematical notation, for two quantities *x *and *y *, then the inverse proportionality of x and y is given by,

$x\propto \frac{1}{y}$

Converting the proportionality into an equation by introducing a constant of proportionality, we get

$x=\frac{k}{y}$

which can be rearranged to get the expression,

$xy=k$

where *k *is a real-valued constant. We notice that the product of two inverse proportion quantities is always constant.

In other words, we have for a non zero real-valued constant k,

$x\alpha \frac{1}{y}\iff xy=k$.

**Direct proportionality **can be visualized geometrically by plotting the equation $x=ky$ as follows,

We notice that the equation is represented by a straight line that passes through the origin (as both intercepts are 0).

If we rearrange the equation in terms of *y *, we get $y=\frac{1}{k}x$ and the slope can be read as $\frac{1}{k}$.

Hence we can set up different proportions using a given piece of data and convert it into an equation using a constant of proportionality. Remember that the constant of proportionality can be any real number, it need not be a positive number.

We saw that two quantities are directly proportional so if we change one, the other one also changes accordingly and the rate of change is constant. That rate of change is *k *and it is also known as the gradient of the curve x=ky.

**Inverse proportionality **can be visualized geometrically by plotting the equation $xy=k$, that can be reformulated to

$y=\frac{k}{x}=k\frac{1}{x}$.

This is a **rectangular hyperbola.**

** **

One can also plot the graph by calculating the horizontal and vertical asymptotes, which turn out to be none other than the axes themselves.

In the graph, we have assumed the constant *k *to be a positive one. For k negative, the graph will be mirrored along the y axis as follows,

The vertical and horizontal asymptotes still remain the same. The property of both the graphs remain the same: as we increase one variable, the other one decreases and as one variable decreases the other one increases.

To solve a direct proportionality from a given set of data, we keep the following steps in mind,

**Step 1. **Convert the proportionality into an equation through a constant of proportionality,

$x\alpha y\iff x=ky$

**Step 2. **Using the given data, determine the value of the constant of proportionality: $k=\frac{{x}_{1}}{{y}_{1}}$.

**Step 3. **Substitute *k *back into the original equation, and now the proportionality is solved: $x=ky$, where *k *is now known.

To solve an inverse proportionality from a given set of data, we keep the following steps in mind,

**Step 1: **Convert the proportionality into an equation through a constant of proportionality:

$x\alpha \frac{1}{y}\iff x=\frac{k}{y}$

**Step 2: **Using the given data, determine the value of the constant of proportionality: $k={x}_{1}{y}_{1}$.

**Step 3: **Substitute *k *back into the original equation, and now the proportionality is solved: $xy=k$, where *k *is now known.

Let's look at some examples regarding direct and inverse proportions.

A bus is moving at 40 km/hr. We assume that the speed is constant throughout the journey.

(i) Find the distance covered by the bus in the first 30 mins.

(ii) Find the time required for the bus to travel a distance of 240 km.

**Solution**

The bus is moving at a constant speed, which implies that the distance covered is uniform for a given interval of time.

Thus, the distance covered is **directly proportional** to the time taken to cover that distance.

Let *t *denote the time taken and *d *denote the distance traveled, so our proportionality is,

$d\propto t$

Converting this into an equation, we get

$d=kt$

Rearranging the equation, we get

$k=\frac{d}{t}$

But the ratio of distance to time is defined as speed, so the constant *k *is the speed itself, hence

$40=\frac{d}{t}$

**(i)** So, to find the distance covered in 30 min=0.5 hour, we just need to substitute *t=0.5 *hour in the above equation,

$d=40\times 0.5=20km$

Hence the distance covered by the bus in 30 minutes is 20 km.

**(ii) **To find the time taken to cover 240km, substituting for d=240 km, we get

$t=\frac{240}{40}=6hr$

Hence the time taken to travel 240 km is 6 hrs.

This problem could have been simply solved just by using the formula for speed but here we solve it from the basics, using the concept of proportions.

The cost of 2 kilograms of apples is $4. Find the cost of 7 kilograms of apples.

**Solution**

Let *x *be the weight of the apples and *y *be the cost of apples.

As the weight of the apple increase, so does its cost. Hence the weight and the cost of apples are linearly proportional, so we can use the formula,

$x=ky$

where *k *is the constant of proportionality. Plugging in the values we are given, we get

$2=4k$

Isolating k we get,

$k=\frac{2}{4}=\frac{1}{2}$This gives us the equation we need,

$y=2x$

Next, to find the cost of 7 kilograms of apples, we substitute x by 7 to get,

* $y=2\times 7=14$*

Thus, the cost of 7 kilograms of apples is $14.

For an ideal gas under constant temperature, the pressure applied upon it is inversely proportional to the volume occupied by the gas.

In an experiment, it is observed that at a pressure of 10 bar, the volume of the gas is 3 cubic meters. What will be its volume when the volume is 5 cubic meters?

**Solution**

Let *P *and *V *denote the pressure and volume of the gas respectively. It is given that they are inversely proportional, which gives,

$PV=k$

where *k *is a constant.

Now substituting for P=10 and V=3, we get

$k=10\times 3=30$,

which gives us the equation,

$PV=30$

Next, we are asked to find the pressure when the volume is 5 m^{3}, so we have,

$P=\frac{30}{V}=\frac{30}{5}$

$P=6bar$

Therefore, the pressure of the gas occupying 5 m^{3} of volume is 6 bar.

In a smartphone manufacturing factory, 10 workers can assemble a phone in 6 hours.

How many workers will be needed to assemble the phone in 4 hours?

**Solution**

Intuitively, we can deduce that as the number of workers will increase, the time needed to assemble the phone will decrease. Hence it will be an inverse proportionality.

Let *w *denote the number of workers it takes to assemble 1 phone and *t *be the time it takes them to assemble it, then the inverse proportionality is,

$w\propto \frac{1}{t}$.

Converting the proportionality into an equation, we get,

$w=\frac{k}{t}$

where *k *is the constant of proportionality.

We are given that 10 workers took 6 hours to do the task so we have, $10=\frac{k}{6}$

Assume that it took *w *workers to complete it in 4 hours, hence

$w=\frac{k}{4}$

Taking the ratio of the above two equations, *k *is eliminated and the only unknown remains *w, *we have

*$\frac{w}{10}=\frac{6}{4}$*

*$w=15$*

Therefore, it will take 15 workers to assemble the phone in 4 hours.

- For any two quantities, if they are related to each other explicitly then they are said to be proportional to each other.
- Direct and Inverse proportions are two primary types of proportionalities.
- Two quantities are
**directly**proportional to each other if and only if they are linearly dependent on each other and the ratio between them is a constant. The direct proportionality between two quantities ax and y is denoted by $x\propto y$. - Two quantities are
**inversely**proportional if and only if the product of them is always constant and as one increases, the other one decreases,. Their inverse proportionality is denoted by $x\propto \frac{1}{y}$.

Two quantities are **directly **proportional to each other if and only if they are linearly dependent on each other and the ratio between them is a constant.

Two quantities are** inversely** proportional if and only if as one increases, the other one decreases and vice versa.

If two quantities x and y are directly proportional, then x=ky describes their direct proportionality, where k is a real valued constant.

If two quantities are inversely proportional, then xy=k describes their inverse proportionality.

A typical example of direct proportion is the cost of making umbrellas and the number of umbrellas. As the number of umbrellas increases, the cost of making umbrellas increases too.

An example of inverse proportionality is the time taken by a number of workers to complete a task. As the number of workers increases, the time taken to complete a task decreases.

Step 1. Convert the proportionality into an equation through a proportionality constant.

Step 2. By using the given data, determine the value of the constant of proportionality.

Step 3. Substitute the calculated constant of proportionality back into the equation.

More about Direct and Inverse proportions

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