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In mathematics, inverse variation is defined as the relationship between two variables expressed as y = k/x, where x and y are two variables and k is a constant number.
- It states that if the value of one item increases, the value of the other quantity falls.
- In our daily lives, we observe the values of one quantity are influenced by the variation in the values of another quantity.
- Inverse variation refers to a variable that varies inversely with respect to another variable.
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Key Terms: Inverse variation, Inversely proportional, Proportionality constant, Variables, Inverse relationship, Inverse variation equation, Inverse variation formula
Inverse Variation Definition
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Unlike direct variation where one quantity increases or decreases along with another, inverse variation involves quantities that change in opposite directions. As a result, the first and second quantities are inversely proportional.
There are many real-life examples of inverse variation, that can be seen in our daily life. For example:
- If a train travels at a constant speed, the time it takes to complete a journey increases as the distance traveled increases (and vice versa).
- Similarly, the time required to finish a job decreases as the number of people working on it increases.
Inverse Variation
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Inverse Variation Formula
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If one variable x is inversely proportional to another variable y, the variables x and y are represented by the inverse variation formula as
| y = k/x or xy = k |
where k is a constant.
Inverse Variation Equation
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In some cases, quantities change in opposite ways. When one quantity increases, the other decreases proportionally, and vice versa. This relationship is called inverse variation, and the quantities are said to be inversely proportional. The inverse variation of two quantities can be given as
| x ∝ 1/y or xy = k |
Where
- x and y are two of two quantities
- k is a constant known as the proportionality constant.
If x1 and y1 are the initial values and x2 and y2 are the final values of inversely varying quantities. They can be written as follows:
x1 / x2 = y2 / y1
Inverse Variation Relationship Between Two Quantities
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The relationship between two quantities or variables is characterized by inverse proportion in an inverse variation. If x and y are two different quantities, then their inverse relation is:
x α 1/y
⇒ x = ky
⇒ xy = k
Some examples below show the relationship between two quantities in inverse variation.
Inverse Variation Table
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An inverse variation table is used to look at how one quantity varies when other changes in an inversely proportional relationship. Assume that two variables x and y are in inverse variation, with y = 20 / x. The inverse variation table may therefore be written like this:
| x | y = 20/x |
|---|---|
| 1 | 20 |
| 2 | 10 |
| 3 | 6.67 |
| 4 | 5 |
| 5 | 4 |
Solved Examples
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| Ques. If x varies inversely with y, and x = 6 when y = 2, what is the value of the constant of variation? Ans. We have
Given that x varies inversely with y, then x ∝ 1/y ⇒ x = k/y ⇒ xy = k Where k is the constant of variation. On substituting the values of x and y, we get 6 x 2 = k ⇒ k = 12 Ques. If x varies inversely with y, and x = 12 when y = 6, what is the value of the constant of variation? Ans. We have
Given that x varies inversely with y, then x ∝ 1/y ⇒ x = k/y ⇒ xy = k Where k is the constant of variation. On substituting the values of x and y, we get 12 x 6 = k ⇒ k = 72 |
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Things to Remember
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- Inverse variation signifies that two variables are inversely proportional.
- Two values are said to exhibit an inverse variation if one decreases while the other increases and vice versa.
- Both numbers in an inverse variation should not be equal to 0.
- The product should be a constant between two inversely proportional quantities.
- In the cartesian coordinate plane, two values following an inverse variation will result in a rectangular hyperbola.
- The inverse variation is given as x α 1/y ⇒ xy = k.
Sample Questions
Ques. If x varies inversely with y, and x = 8 and y = 3, then what is the value of the constant of variation? (3 Marks)
Ans. Given, x varies inversely with y.
Let the constant of variation be k.
x = k/y
or k = xy
But x = 8 and y = 3 (GIVEN)
So,
k = 8 x 3 = 24
Hence, the constant of variation is 24.
Ques. If x and y are in an inverse variation and k/4= 2, and k/x = 4, then find the value of x when y = 2. (2 Marks)
Ans. We are Given k/4= 2
then, k = 4 x 2 = 8
k/x = 4 (GIVEN)
So, x = k/4 = 8/4 = 2
Hence, if y = 2, then;
x = k/y = 8/2 = 4
Ques. The volume V of a gas varies inversely to the pressure P on it. If the volume is 240 cm3 under the pressure of 30 kg/cm2, what pressure must be applied to have a volume of 160 cm3? (4 Marks)
Ans. The volume V varies inversely as the pressure P means when the volume increases, the pressure decreases, and when the volume decreases, the pressure increases.
PV=k
Substitute 240 for V30 for P in the formula and find the constant
(240)(30)=k
7200=k
Now write an equation and solve for the unknown.
We must find the pressure when the volume is 160 cm3.
So,
(160)(P)=7200.
Solve for P.
P=7200/160
P=45
Therefore, a pressure of 45 kg/cm2 is applied to have a volume of 160 cm3.
Ques. The length of a violin string varies inversely to the frequency of its vibrations. A violin string 14 inches long vibrates at a frequency of 450 cycles per second. Find the frequency of a 12-inch violin string. (4 Marks)
Ans. The length (l ) Varies inversely to the frequency (f ), when the length increases, the frequency decreases, and when the length decreases, the frequency increases.
Now write the formula for inverse variation.
lf=k .
Substitute 450 for f14 for l in the formula and find the constant.
(450)(14)=k
6300=k
Now write an equation and solve for the unknown.
We must find the frequency of12 -12-inch violin string.
So,
(12)(f)=6300 .
Solve for f.
f=6300/12
f=525
Therefore, a 12 -12-inch violin string vibrates at a frequency of 525 cycles per second.
Ques. If 48 men can do a piece of work in 24 days, in how many days will 36 men complete the same work? (3 Marks)
Ans. Fewer men will necessitate more days to accomplish the work
The work may be completed in 24 days by 48 Men.
1 man can complete the same amount of work in 48 × 24 hours.
36 men can complete the same amount of labor in (48 24)/36 = 32 days.
As a result, 36 guys can complete the same task in 32 days.
Ques. When x is 7 and y is 2, find an equation that inversely relates y and x. (3 Marks)
Ans. The inverse variation that relates x and y is
Graphically it looks like this:
Compare the direct variation model and the inverse variation model for when x = 2 and y = 3. Do this both numerically and graphically.
a. Numerically:
-
Direct Variation:
The reason is that since k is positive y will increase as x increases. So as x increases by 1, y increases by 1.5.
-
Inverse Variation: Because k is positive, y decreases as x increases.
We can create a table that will show the comparison:
b. Using the values on the table we can plot the points and then connect them to find the solution by graphing.
Ques. If 52 men can do a piece of work in 35 days, then 28 men will complete the same work in how many days? (3 Marks)
Ans. This is a situation of inverse variation, now we solve using the unitary method.
52 men can do the work in 35 days.
1 man can do the work in (35 × 52) days.
28 men can do the work in days. (35 × 52)/28 days,
Therefore, 28 men can do the work in 65 days.
Ques. In a camp, there is enough food for 500 soldiers for 35 days. If 200 more soldiers join the camp, how many days will the food last? (3 Marks)
Ans. This is a situation of inverse variation, now we solve using the unitary method.
For 500 soldiers, food lasts for 35 days.
For 1 soldier, food lasts for (35 × 500) days.
Since 200 more joined. So, now the number of soldiers is (500 + 200) = 700.
For 700 soldiers, food lasts for (35 × 500)/700 days
Therefore, for 700 soldiers, food lasts for = 25 days.
Ques. Sara starts at 8:00 AM by bicycle to reach school. She cycles at the speed of 18 km/hour and reaches the school at 8:22 AM. By how much should she increase the speed so that she can reach the school at 8:12 AM? (2 Marks)
Ans. This is a situation of inverse variation, now we solve using the unitary method.
In 22 minutes the same distance is covered at the speed of 18 km/hr.
In 1 minute the same distance is covered at the speed of (18 × 22) km/hr.
In 12 minutes the same distance is covered at the speed of (18 × 22)/12 km/hr.
Therefore, in 12 minutes the same distance is covered at the speed of 16 km/hr.
Ques. 32 workers can complete work in 84 days. How many workers will complete the same work in 48 days? (2 Marks)
Ans. This is a situation of inverse variation, now we solve using the unitary method.
To complete the work in 84 days, workers required = 32
To complete the work in 1 day, the worker required = (32 × 84)
To complete the work in 48 days workers required = (32 × 84)/48
Therefore, to complete the work in 48 days, 56 workers are required.
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