Illustrating an inverse variation relation

Math 9 variation

1. Inverse Variation

2. Activity Consider the table of values A and B. What have you observed about the values in both tables? What do you observe about the values of y when x increases/decreases? 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

3. 3. What happens to the values of y when x is doubled? How do you compare the two relations? 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. 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. 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 INVER If it suggests an inverse variation.

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. Express each of the following statements into mathematical equation. 1. The number of hours h required to complete a certain job varies inversely as the number of machines m used to do the work. 2. The number of rice cake slices r varies inversely as the number of persons n sharing the whole rice cake. 3. The temperature t at which water boils varies inversely as the number of feet h above the sea level. 4. The time it takes a block of ice t to melt varies indirectly with temperature f.  5. The gravitational force (in newtons) n between two objects is inversely proportional to square of the distance (in meters) d between the centers of the objects.

22. 6. In the study of electricity, Ohm's Law says the electrical current a (measured in amps) across a conductor is inversely proportional to electrical resistance r (measured in ohms). 7. In economics, the basic Law of Demand d tells us that as the price for a particular good (or service) is inverse proportional to the demand for that good (or service). 8.When working with electrical circuits, it turns out that the electrical resistance r is inversely proportional to the square of the current c . 9.  To balance a lever (seesaw), the weight w varies inversely with the distance d of the object from the fulcrum.  10. The volume v of gas varies inversely to the pressure p .

23. WRITING INVERSE VARIATION EQUATION

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

25. A very seesaw problem… u sit? fulcrum 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 yo Melissa Jericho

26. 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 S olve for the distance of Melissa to the fulcrum.

27. 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?

28. 1.  If 32 men can reap a field in 15 days, in how many days can 20 men reap the same field?  2.  12 men can dig a pond in 8 days. How many men can dig it in 6 days?  3.  A hostel has enough food for 125 students for 16 days. How long will the food last if75 more students join them?  4.  A fort had enough food for 80 soldiers for 60 days. How long would the food last if 20 more soldiers join after 15 days?  5.  If 12 men or 15 women can finish a piece of work in 66 days, how long will 24 men and 3 women take to finish the work?

29. 6. 70 patients in a hospital consume 1350 litres of milk in 30 days. At the same rate, how many patients will consume 1710 litres in 28 days? 7 .  If 30 labourers working 7 hours a day can finish a piece of work in 18 days, how many labourers working 6 hours a day can finish it in 30 days? 8 .  If 5 men working 6 hours a day can reap a field in 20 days, in how many days will 15 men reap the field if they work for 8 hours a day? 9 .  If 18 binders can bind 900 books in 10 days, how many binders will be required to bind 660 books in 12 days? 10.  If 20 men can build a 112-m-long wall in 6 days, what will be the length of a similar wall that can be built by 25 men in 3 days?