The document discusses inverse variation and provides examples to illustrate the concept. It begins by showing two tables with values that demonstrate an inverse relationship between variables x and y. It then provides the definition of inverse variation as a situation where an increase in one variable causes a decrease in the other, such that their product is constant. Examples are given of relationships that demonstrate inverse variation, such as the number of people sharing a pizza relating inversely to the number of slices. The document also contains a word problem demonstrating how to set up and solve an equation using the inverse variation relationship.

1. Inverse Variation

2. Activity
Consider the table of values A and B.
x - 2 -1 1 2 3
y - 4 -2 2 4 6
x 16 8 4 2 1
y 2 4 8 16 32
Table A Table B
1. What have you observe about the values in
both tables?
2. What do you observe about the values of y
when x increases/decreases?

3. 3. What happens to the values of y when x is
doubled?
4. How do you compare the two relations?
5. How can you determine if the relationship is
a direct variation or an inverse variation?
x - 2 -1 1 2 3
y - 4 -2 2 4 6
x 16 8 4 2 1
y 2 4 8 16 32
Table A Table B

4. Inverse Variation
This occurs whenever a situation
produces pairs of numbers whose product is
constant.
For any quantities x and y, an increase
in x causes a decrease in y or vice versa. We
can say that y varies inversely as x or y = k/x.
The statement, “y varies inversely to x,”
translates to y = k/x, where k is the constant
of variation.

5. The number of persons sharing a pizza to the number
of slices of the pizza.
The number of persons needed to do a job to the
number of days finishing the job.

6. On a ¼ sheet of paper, determine if the following phrases represent direct variation
or inverse variation. Write DIRECT if the phrase suggest a direct variation and INVERSE
If it suggests an inverse variation.
PHRASE
1. The amount of water to the space that water did not occupy in
a particular container.
2. The area of the wall to the amount of paint used to cover it.
3. The time spent walking to the rate at which the person walks.
4. The time a teacher spends checking papers to the number of
students.
5. The age of a used car to its resale value.
6. The amount of money raised in a concert to the number of
tickets sold.
7. The distance an airplane flies to the time travelling.

7. PHRASE
1. The amount of water to the space that water did not
occupy in a particular container.
2. The area of the wall to the amount of paint used to cover it.
3. The time spent walking to the rate at which the person walks.
4. The time a teacher spends checking papers to the number of
students.
5. The age of a used car to its resale value.
6. The amount of money raised in a concert to the number of
tickets sold.
7. The distance an airplane flies to the time travelling.

8. PHRASE
Inverse
1. The amount of water to the space that water did not occupy in
a particular container.
2. The area of the wall to the amount of paint used to cover it.
3. The time spent walking to the rate at which the person walks.
4. The time a teacher spends checking papers to the number of
students.
5. The age of a used car to its resale value.
6. The amount of money raised in a concert to the number of
tickets sold.
7. The distance an airplane flies to the time travelling.

9. PHRASE
Inverse
1. The amount of water to the space that water did not occupy in
a particular container.
2. The area of the wall to the amount of paint used to cover
it.
3. The time spent walking to the rate at which the person walks.
4. The time a teacher spends checking papers to the number of
students.
5. The age of a used car to its resale value.
6. The amount of money raised in a concert to the number of
tickets sold.
7. The distance an airplane flies to the time travelling.

10. PHRASE
Inverse
1. The amount of water to the space that water did not occupy in
a particular container.
Direct 2. The area of the wall to the amount of paint used to cover it.
3. The time spent walking to the rate at which the person walks.
4. The time a teacher spends checking papers to the number of
students.
5. The age of a used car to its resale value.
6. The amount of money raised in a concert to the number of
tickets sold.
7. The distance an airplane flies to the time travelling.

11. PHRASE
Inverse
1. The amount of water to the space that water did not occupy in
a particular container.
Direct 2. The area of the wall to the amount of paint used to cover it.
3. The time spent walking to the rate at which the person
walks.
4. The time a teacher spends checking papers to the number of
students.
5. The age of a used car to its resale value.
6. The amount of money raised in a concert to the number of
tickets sold.
7. The distance an airplane flies to the time travelling.

12. PHRASE
Inverse
1. The amount of water to the space that water did not occupy in
a particular container.
Direct 2. The area of the wall to the amount of paint used to cover it.
Inverse 3. The time spent walking to the rate at which the person walks.
4. The time a teacher spends checking papers to the number of
students.
5. The age of a used car to its resale value.
6. The amount of money raised in a concert to the number of
tickets sold.
7. The distance an airplane flies to the time travelling.

13. PHRASE
Inverse
1. The amount of water to the space that water did not occupy in
a particular container.
Direct 2. The area of the wall to the amount of paint used to cover it.
Inverse 3. The time spent walking to the rate at which the person walks.
4. The time a teacher spends checking papers to the number
of students.
5. The age of a used car to its resale value.
6. The amount of money raised in a concert to the number of
tickets sold.
7. The distance an airplane flies to the time travelling.

14. PHRASE
Inverse
1. The amount of water to the space that water did not occupy in
a particular container.
Direct 2. The area of the wall to the amount of paint used to cover it.
Inverse 3. The time spent walking to the rate at which the person walks.
Direct
4. The time a teacher spends checking papers to the number of
students.
5. The age of a used car to its resale value.
6. The amount of money raised in a concert to the number of
tickets sold.
7. The distance an airplane flies to the time travelling.

15. PHRASE
Inverse
1. The amount of water to the space that water did not occupy in
a particular container.
Direct 2. The area of the wall to the amount of paint used to cover it.
Inverse 3. The time spent walking to the rate at which the person walks.
Direct
4. The time a teacher spends checking papers to the number of
students.
5. The age of a used car to its resale value.
6. The amount of money raised in a concert to the number of
tickets sold.
7. The distance an airplane flies to the time travelling.

16. PHRASE
Inverse
1. The amount of water to the space that water did not occupy in
a particular container.
Direct 2. The area of the wall to the amount of paint used to cover it.
Inverse 3. The time spent walking to the rate at which the person walks.
Direct
4. The time a teacher spends checking papers to the number of
students.
Inverse 5. The age of a used car to its resale value.
6. The amount of money raised in a concert to the number of
tickets sold.
7. The distance an airplane flies to the time travelling.

17. PHRASE
Inverse
1. The amount of water to the space that water did not occupy in
a particular container.
Direct 2. The area of the wall to the amount of paint used to cover it.
Inverse 3. The time spent walking to the rate at which the person walks.
Direct
4. The time a teacher spends checking papers to the number of
students.
Inverse 5. The age of a used car to its resale value.
6. The amount of money raised in a concert to the number of
tickets sold.
7. The distance an airplane flies to the time travelling.

18. PHRASE
Inverse
1. The amount of water to the space that water did not occupy in
a particular container.
Direct 2. The area of the wall to the amount of paint used to cover it.
Inverse 3. The time spent walking to the rate at which the person walks.
Direct
4. The time a teacher spends checking papers to the number of
students.
Inverse 5. The age of a used car to its resale value.
Direct
6. The amount of money raised in a concert to the number of
tickets sold.
7. The distance an airplane flies to the time travelling.

19. PHRASE
Inverse
1. The amount of water to the space that water did not occupy in
a particular container.
Direct 2. The area of the wall to the amount of paint used to cover it.
Inverse 3. The time spent walking to the rate at which the person walks.
Direct
4. The time a teacher spends checking papers to the number of
students.
Inverse 5. The age of a used car to its resale value.
Direct
6. The amount of money raised in a concert to the number of
tickets sold.
7. The distance an airplane flies to the time travelling.

20. PHRASE
Inverse
1. The amount of water to the space that water did not occupy in
a particular container.
Direct 2. The area of the wall to the amount of paint used to cover it.
Inverse 3. The time spent walking to the rate at which the person walks.
Direct
4. The time a teacher spends checking papers to the number of
students.
Inverse 5. The age of a used car to its resale value.
Direct
6. The amount of money raised in a concert to the number of
tickets sold.
Direct 7. The distance an airplane flies to the time travelling.

21. Now let us try solving a
situation that involves an
inverse variation.

22. A very seesaw
problem…
Jericho and Melissa are figuring out a way to balance
themselves on a seesaw. Jericho weighs 15
kilograms sits 2 meters from the fulcrum. Melissa who
weighs 20 kilograms tried sitting at different distances
from the fulcrum in order to balance the weight of
Jericho. If you were Melissa, how far from the fulcrum
should you sit?
fulcrum
Melissa
Jericho

23. To balance the weight of Jericho, Melissa has to sit at
a distance closer to the fulcrum.
The relation shows that the distance d varies inversely
as the weight w and can be transformed into a
mathematical equation as d = k/w.
Solution : Let us solve for k.
Use Jericho’s: w = 15 k and d = 2
k = dw
= 15 (2) k = 30
Solve for the distance of Melissa to the fulcrum.

24. d = k/w,
= 30 / 20 d = 1. 5 meters
Therefore, Melissa has to sit 1.5 meters from the fulcrum.
Try this.
• The number of days needed in repairing a house
varies inversely as the number of men working. It
takes 15 days for 2 men to repair the house. How
many men are needed to complete the job in 6 days?