Explainer: Inverse Variation

In this explainer, we will learn how to create formulas linking two quantities that vary directly and indirectly.

In the physical world, there are many examples of quantities that vary inversely. For example, the frequency of vibration on a stringed instrument varies inversely with the length of the string, and gravitational force is inversely proportional to the square of the distance between objects: magnitudeofgravitationalforcedistance∝1().

Before we talk about inverse variation, let us recap the definition of direct variation.

Direct Variation

Two variables are said to be in direct variation, or direct proportion, if their ratio is constant.

This type of relationship is often written as π‘¦βˆπ‘₯, for variables π‘₯ and 𝑦. It is mathematically described as 𝑦=π‘˜π‘₯, where π‘˜ is the constant of variation.

By dividing both sides of the previous equation by π‘₯, we can see that π‘˜=𝑦π‘₯, and this will be the same for all values of π‘₯ and 𝑦.

With direct variation, if one quantity increases, the other also increases. However, for inverse variation, if one quantity increases, the other decreases. Formally, we define this as follows.

Inverse Variation

Two variables are said to be in inverse variation if, as one variable increases, the other decreases, in which case their product is a constant value.

This type of relationship is often written as π‘¦βˆ1π‘₯, for variables π‘₯ and 𝑦. It is mathematically described as 𝑦=π‘˜π‘₯, where π‘˜ is the constant of variation.

With inverse proportionality, the equation of proportionality looks slightly different to the directly proportional situation. For example, if β€œπ‘¦ is inversely proportional to π‘₯,” then we write this in the form β€œπ‘¦ is directly proportional to the inverse of π‘₯”: π‘¦βˆ1π‘₯.

The equation of proportionality, then, looks like this: 𝑦=π‘˜Γ—1π‘₯, or 𝑦=π‘˜π‘₯.

Alternatively, if β€œπ‘¦ is inversely proportional to the square of π‘₯,” then we write this in the form β€œπ‘¦ is directly proportional to the inverse of the square of π‘₯”: π‘¦βˆ1π‘₯.

The equation of proportionality, then, looks like this: 𝑦=π‘˜Γ—1π‘₯, or 𝑦=π‘˜π‘₯.

Further, if β€œπ‘¦ is inversely proportional to the cube root of π‘₯,” then we write this in the form β€œπ‘¦ is directly proportional to the inverse of the cube root of π‘₯”: π‘¦βˆ1√π‘₯.

The equation of proportionality, then, looks like this: 𝑦=π‘˜Γ—1√π‘₯, or 𝑦=π‘˜βˆšπ‘₯.

Let us now examine two examples of the different ways inversely proportional relationships can be described:

  • β€œπ‘¦ varies inversely as the cube of π‘₯” means π‘¦βˆ1π‘₯.
  • β€œπ‘¦ is inversely proportional to the square root of π‘₯” means π‘¦βˆ1√π‘₯.

An inversely proportional relationship may also be described in a longer form. For example, β€œthe pressure, in atmospheres, in a glider varies inversely as the square root of its height above sea level, in yards.” If we let 𝑝 represent the pressure (in atmospheres) and β„Ž represent the height above sea level (in yards), then we can express the proportionality as π‘βˆ1βˆšβ„Ž or as an equation 𝑝=π‘˜βˆšβ„Ž, where π‘˜ is the constant of proportionality.

If we look at the graph of an inversely proportional relationship we can see that it looks very different to that of a directly proportional relationship.

The graph of 𝑦 = π‘˜/π‘₯

We can see that as the value of π‘₯ increases, the value of π‘˜π‘₯ will get closer to zero, and the curve will approach the π‘₯-axis.

We can also see that, as the value of π‘₯ decreases to zero, the value of π‘˜π‘₯ will get larger, and the curve will approach the 𝑦-axis.

Let us consider a few examples involving inverse proportionality.

Example 1: Finding the Proportional Relationship between Variables

Decide if π‘₯ varies directly or inversely with 𝑦 and use this to find the value of 𝑦 when π‘₯=3.

π‘₯2470
𝑦70352

Answer

The table shows that 𝑦 decreases as π‘₯ increases; this means that we have an inverse relationship. So 𝑦 varies inversely with π‘₯, which we write as π‘¦βˆ1π‘₯, which is equivalent to 𝑦=π‘˜π‘₯ or π‘₯𝑦=π‘˜.

Therefore, when 𝑦 varies inversely with π‘₯, the product of π‘₯ and 𝑦 is constant.

We can therefore check to see whether the products of the pairs of π‘₯ and 𝑦 in the table are constant. Taking the first two, we have 2Γ—70=140.

We can now consider the product of the second pair: 4Γ—35=140.

Similarly, for the last pair, we have 70Γ—2=140.

Therefore, π‘˜=140. Hence, at π‘₯=3, we have 𝑦=1403=4623.

Therefore, 𝑦 varies inversely with π‘₯ and when π‘₯=3, 𝑦=4623.

Example 2: Solving Direct Proportion Equations Involving Inverse Variation of One Variable with Another

Variable 𝑦 is inversely proportional to π‘₯. When π‘₯=3, 𝑦=6.

Find the value of 𝑦 when π‘₯=8.

Answer

Firstly, write down the statement of proportionality: π‘¦βˆ1π‘₯.

Using π‘˜ as the constant of proportionality, we can say 𝑦=π‘˜Γ—1π‘₯𝑦=π‘˜π‘₯.

Now we can substitute the values we were given for π‘₯ and 𝑦 in the question and solve for π‘˜: 6=π‘˜318=π‘˜.

Now that we know the value of π‘˜, we can complete the proportionality equation: 𝑦=18π‘₯.

We can now substitute the value we were given for π‘₯ in the question and calculate the corresponding value of 𝑦: 𝑦=188𝑦=214.

The answer is that when π‘₯=8, 𝑦=214.

Example 3: Solving Direct Proportion Equations Involving Inverse Variation of One Variable with the Square Root of Another

Variable 𝑦 varies inversely with the square root of π‘₯. When π‘₯=25, 𝑦=4.

Find the value of π‘₯ when 𝑦=2.

Answer

Firstly, write down the statement of proportionality: π‘¦βˆ1√π‘₯.

Using π‘˜ as the constant of proportionality, we can say 𝑦=π‘˜Γ—1√π‘₯𝑦=π‘˜βˆšπ‘₯.

Now we can substitute the values we were given for π‘₯ and 𝑦 in the question and solve for π‘˜: 4=π‘˜βˆš254=π‘˜520=π‘˜.

Now that we know the value of π‘˜, we can complete the proportionality equation: 𝑦=20√π‘₯.

We can now substitute the value we were given for 𝑦 in the question and calculate the corresponding value of π‘₯: 2=20√π‘₯2√π‘₯=20√π‘₯=10π‘₯=100.

The answer is that when 𝑦=2, π‘₯=100.

Example 4: Solving Direct Proportion Equations Involving Inverse Variation of One Variable with a Linear Function of Another

Variable 𝐴 varies inversely with (𝐡+5). If 𝐴=1 when 𝐡=5, find the value of 𝐡 when 𝐴=10.

Answer

Firstly, write down the statement of proportionality: 𝐴∝1(𝐡+5).

Using π‘˜ as the constant of proportionality, we can say 𝐴=π‘˜Γ—1(𝐡+5)𝐴=π‘˜(𝐡+5).

Now we can substitute the values we were given for 𝐴 and 𝐡 in the question and solve for π‘˜: 1=π‘˜(5+5)1=π‘˜1010=π‘˜.

Now that we know the value of π‘˜, we can complete the proportionality equation: 𝐴=10(𝐡+5).

We can now substitute the value we were given for 𝐴 in the question and calculate the corresponding value of 𝐡: 10=10(𝐡+5)10(𝐡+5)=10(𝐡+5)=1𝐡+5=1𝐡=βˆ’4.

The answer is that when 𝐴=10, 𝐡=βˆ’4.

Example 5: Word Problem on Inverse Variation

For a rectangle of fixed area, the length 𝑙 varies inversely with the width 𝑀. Given that 𝑙=22cm when 𝑀=16cm, determine the value of 𝑙 when 𝑀=44cm.

Answer

Given that the area is fixed, we have 𝑙𝑀=𝐴, where 𝐴 is the area, which is constant. This statement is equivalent to saying that 𝑙 varies inversely with the width 𝑀. We know a particular width and length: 𝑙=22cm when 𝑀=16cm; therefore, we can find the area by multiplying as follows: 22Γ—16=352.cm

We can now find the length when the width is 44 cm by dividing the area by 44. Hence, 𝑙=35244=8.cm

Key Points

  • Two variables are said to be in inverse variation if their product is a constant value.
  • Inverse variation is written as π‘¦βˆ1π‘₯ and is mathematically described as 𝑦=π‘˜π‘₯, where π‘˜ is the constant of variation.
  • When we solve problems involving an inverse variation, we use two corresponding values for the variables to identify the constant, π‘˜, and then use the equation 𝑦=π‘˜π‘₯ to find any unknown values.
  • The graph of two quantities that have a simple direct variation is a straight line passing through the origin, whereas the graph of an inverse variation is a reciprocal graph.

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