Direct and Inverse Variations

​Boom, Snap, Clap

​https://www.youtube.com/watch?v=UkFs5glUMjw

​Direct and Inverse Variations

​Objectives:

​1. Illustrate situations that involve the following variations:

(a) direct;

(b) inverse.

 

2. Translate a direct and an inverse variation statements into relationship between two quantities using:

(a) a table of values;

(b) a mathematical equation;

(c) a graph, and vice versa.

 

3. Solve problems involving direct and inverse variations.

​Which of the following quantities are related?

​Variation

​     - is a relationship with rregard to the change in the value of a variable when the values of the related variables change

​Direct Variation or Direct Proportion

​Direct Variation Equations

Let x and y denote two quantities. If y varies directly with x,   y is directly proportional to x, then ​

​Example 1: Determine whether the two quantities in each item vary directly.

​a) The amount of food intake and the weight of a person.

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​Example 1: Determine whether the two quantities in each item vary directly.

​a) The amount of food intake and the weight of a person.

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Answer : Yes, the two quantities vary directly.

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​Example 1: Determine whether the two quantities in each item vary directly.

​b)  The amount of money raised at a school fundraiser and the number of people who attend

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​Example 1: Determine whether the two quantities in each item vary directly.

​b)  The amount of money raised at a school fundraiser and the number of people who attend

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Answer : Yes, the two quantities vary directly.

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​Example 1: Determine whether the two quantities in each item vary directly.

​c) The number of ballpen you bought and the amount you have to pay

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​Example 1: Determine whether the two quantities in each item vary directly.

​c) The number of ballpen you bought and the amount you have to pay

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Answer : Yes, the two quantities vary directly.

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​Example 1: Determine whether the two quantities in each item vary directly.

​d) The number of workers to finish a task and the number of days they finish the task

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​Example 1: Determine whether the two quantities in each item vary directly.

​d) The number of workers to finish a task and the number of days they finish the task

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Answer : No, the two quantities do not vary directly.

​Example 2: Find the constant of variation and the equation of variation in which y varies directly as x, and y = 45 when x =  15.

​Example 2: Find the constant of variation and the equation of variation in which y varies directly as x, and y = 45 when x =  15.

​Solution:

​Example 2: Find the constant of variation and the equation of variation in which y varies directly as x, and y = 45 when x =  15.

​ii)      To find the equation of variation, we will use the formula y = kx. Substituting the constant of variation to that equation, we get the equation of variation, 

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y = 3x

​Solution:

​Example 3: If y varies directly as x, and y = 36 when x is 4, find the value of y when x = 10.

​Example 3: If y varies directly as x, and y = 36 when x is 4, find the value of y when x = 10.

​Solution 1:

​Example 3: If y varies directly as x, and y = 36 when x is 4, find the value of y when x = 10.

​Solution 2:

​Example 4: The shows the number of hours t traveled and the distance y traveled by a car.

​a) Show that y varies directly as t

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b) Draw the graph of y against t.

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c) Write the equation showing the relationship between y and t.

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d) Find y when t = 1.5 and t = 4.25

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e) Find t when y = 140 and y = 260.​

​Example 4: The shows the number of hours t traveled and the distance y traveled by a car.

​a) Show that y varies directly as t

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​Example 4: The shows the number of hours t traveled and the distance y traveled by a car.

​Example 4: The shows the number of hours t traveled and the distance y traveled by a car.

​b) Draw the graph of y against t.

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https://www.geogebra.org/graphing?lang=en

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​Example 4: The shows the number of hours t traveled and the distance y traveled by a car.

​b) Draw the graph of y against t.

Answer : ​

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​Example 4: The shows the number of hours t traveled and the distance y traveled by a car.

​c) Write the equation showing the relationship between y and t.

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​Example 4: The shows the number of hours t traveled and the distance y traveled by a car.

​c) Write the equation showing the relationship between y and t.

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Answer : y = 40t​

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​Example 4: The shows the number of hours t traveled and the distance y traveled by a car.

​d) Find y when t = 1.5 and t = 4.25

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​Example 4: The shows the number of hours t traveled and the distance y traveled by a car.

​d) Find y when t = 1.5 and t = 4.25

Answer :

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​Example 4: The shows the number of hours t traveled and the distance y traveled by a car.

​e) Find t when y = 140 and y = 260.​

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​Example 4: The shows the number of hours t traveled and the distance y traveled by a car.

​e) Find t when y = 140 and y = 260.​

Answer : ​

​Example 5: A worker's pay check P varies directly as the number of hours h worked. For working 20 hours, the payment is ₱1, 000.00. Find the payment for 45 hours of work.

​Solution 1:

​Example 5: A worker's pay check P varies directly as the number of hours h worked. For working 20 hours, the payment is ₱1, 000.00. Find the payment for 45 hours of work.

​Solution 2:

Multiple Choice

Decide if the second quantity increases or decreases.

As the distance a taxi travels increases, the fare ________________.

• increases

• decreases

Multiple Choice

Decide if the second quantity increases or decreases.

The area of a square decreases as its side ________________.

• increases

• dcreases

Multiple Choice

Determine if the two quantities show direct variation or not.

The distance traveled by car and the amount of gas left in the tank.

• Direct Variation

• Not Direct Variation

Multiple Choice

Find the constant of variation and the equation of variation in which y varies directly as x, and y = 51 when x = 3.

• k = 153

  y = 153x

• k = 17

  y = 17x

• k = 17

  y = 17/x

• k = 153

  y = 153/x

​Direct Variation as a Power or Direct Square Variation

​-the value of y varies directly as the power of x if there exists a nonzero real number k such that

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y = kxⁿ or y = kx².

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The​ constant of variation is k.

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​Example 6: If y varies directly as the square of x, and y = 432 when x = 12, find y when x = 20.

​Solution:

​Brain Break - Never Have I Ever

​https://www.youtube.com/watch?v=04Ey9Gw-Lrk

​Inverse Variation

​Inverse Variation Equations

Take note of the following:​

1. The product of two quantities is constant.

2. The graph of y against x is a hyperbola. 

3. The graph goes down from left to right.

​Example 7: Determine whether  the two quantities in each item vary inversely.

​a)  The speed one travels and the time to reach one's destination

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​Example 7: Determine whether  the two quantities in each item vary inversely.

​a)  The speed one travels and the time to reach one's destination

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Answer : Yes, the two quantities vary inversely because as the speed decreases, the time increases and vice versa.​

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​Example 7: Determine whether  the two quantities in each item vary inversely.

​b) The side of a square is related to its perimeter

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​Example 7: Determine whether  the two quantities in each item vary inversely.

​b) The side of a square is related to its perimeter

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Answer : No, the two quantities do not vary inversely because as the side increases, the perimeter increases.

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​Example 7: Determine whether  the two quantities in each item vary inversely.

​c) The number of people sharing a pizza is related to the size of slice each person gets.

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​Example 7: Determine whether  the two quantities in each item vary inversely.

​c) The number of people sharing a pizza is related to the size of slice each person gets.

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Answer : Yes, the two quantities vary inversely because as the number of people sharing the pizza increases, the size of the slice each gets decreases

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​Example 7: Determine whether  the two quantities in each item vary inversely.

​d) The age of a used car is related to the price the owner can get for it.

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​Example 7: Determine whether  the two quantities in each item vary inversely.

​d) The age of a used car is related to the price the owner can get for it.

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Answer : Yes, the two quantities vary inversely because as the age of used car increases, the price the owner can get for it decreases.

​Example 8: If y is inversely proportional to x and y = 3 when x = 4, find

a)   an equation connecting x and y

b)  the value of y when x = 8

c) the value of x when y = 48

​Solution:

​Example 8: If y is inversely proportional to x and y = 3 when x = 4, find

a)   an equation connecting x and y

b)  the value of y when x = 8

c) the value of x when y = 48

​Solution:

​Example 8: If y is inversely proportional to x and y = 3 when x = 4, find

a)   an equation connecting x and y

b)  the value of y when x = 8

c) the value of x when y = 48

​Solution:

​Example 8: If y is inversely proportional to x and y = 3 when x = 4, find

a)   an equation connecting x and y

b)  the value of y when x = 8

c) the value of x when y = 48

​Solution:

​Example 9: The following table shows the time t in hours taken by a car that travels uniformly from Manila to Ilocos at various speeds v in kph.

​a. Show that v varies inversely as t.

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​Solution:

​Example 9: The following table shows the time t in hours taken by a car that travels uniformly from Manila to Ilocos at various speeds v in kph.

​b. Graph t against v.

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​Solution:

​Example 9: The following table shows the time t in hours taken by a car that travels uniformly from Manila to Ilocos at various speeds v in kph.

​c. Write the equation relating t and v.

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vt = 120​

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​Solution:

​Example 9: The following table shows the time t in hours taken by a car that travels uniformly from Manila to Ilocos at various speeds v in kph.

​d. Find t when v is 100 kph.

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​Solution:

​Example 9: The following table shows the time t in hours taken by a car that travels uniformly from Manila to Ilocos at various speeds v in kph.

​e. Find v when t is 0.6 hr.​

​Solution:

​Example 10: A crew of 12 can build a hut in 8 days. How long would it take a crew of 4 to build the same hut?

​Example 10: A crew of 12 can build a hut in 8 days. How long would it take a crew of 4 to build the same hut?

​Let y = number of days to build a hut

      x = number of crew   

      k = constant of variation

​Solution:

Multiple Choice

Determine whether the equation represents direct variation or inverse variation.

$$r=\frac{d}{t}$$  

• direct variation

• inverse variation

Multiple Choice

Which graph represents inverse variation?

Multiple Choice

A car is travelling a distance of 120 km. How long will it take the car to reach its destination if it travels at a speed of 20 kph?

• 4 hours

• 5 hours

• 6 hours

• 7 hours

​Worktext Reference:

Oronce O.A., Mendoza M.O. (2019). E-Math Worktext in Mathematics 9, Manila, Philippines: REX Bookstore Inc.

 

Reference

Nivera G.C., Lapinid M.R.C. (2018). Grade 9 Mathematics Patterns and Practicalities 9, Makati City, Philippines: Salesiana Books by Don Bosco Press, Inc.

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Yeo J., Yee LC., Meng NC., Seng TK., Chow I., Hong OC. (2017). New Syllabus Mathematics 9 Singapore Math Worktext, Manila, Philippines: REX Bookstore Inc.

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Online Reference​

https://www.cuemath.com/commercial-math/direct-variation/

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