## Chapter 6: Proportions and Modeling Using Variation

Solve problems involving joint variation.

Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation . For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable c , cost, varies jointly with the number of students, n , and the distance, d .

## A General Note: Joint Variation

Joint variation occurs when a variable varies directly or inversely with multiple variables.

For instance, if x  varies directly with both y  and z , we have x  = kyz . If x  varies directly with y  and inversely with z , we have $x=\frac{ky}{z}$. Notice that we only use one constant in a joint variation equation.

## Example 4: Solving Problems Involving Joint Variation

A quantity x  varies directly with the square of y  and inversely with the cube root of z . If x  = 6 when y  = 2 and z  = 8, find x  when y  = 1 and z  = 27.

Begin by writing an equation to show the relationship between the variables.

Substitute x  = 6, y  = 2, and z  = 8 to find the value of the constant k .

Now we can substitute the value of the constant into the equation for the relationship.

To find x  when y  = 1 and z  = 27, we will substitute values for y  and z  into our equation.

x  varies directly with the square of y  and inversely with z . If x  = 40 when y  = 4 and z  = 2, find x  when y  = 10 and z  = 25.

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## Joint Variation – Formula, Examples | How to Solve Problems Involving Joint Variation?

Joint Variation definition, rules, methods and formulae are here. Check the joint variation problems and solutions to prepare for the exam. Refer to problems of direct and inverse variations and the relationship between the variables. Know the different type of variations like inverse, direct, combined and joint variation. Go through the below sections to check definition, various properties, example problems, value tables, concepts etc.

## Joint Variation – Introduction

Joint Variation refers to the scenario where the value of 1 variable depends on 2 or more and other variables that are held constant. For example, if C varies jointly as A and B, then C = ABX for which constant “X”. The joint variation will be useful to represent interactions of multiple variables at one time.

Most of the situations are complicated than the basic inverse or direct variation model. One or the other variables depends on the multiple other variables. Joint Variation is nothing but the variable depending on 2 or more variables quotient or product. To understand clearly with an example, The amount of busing candidates for each of the school trip varies with the no of candidates attending the distance from the school. The variable c (cost) varies jointly with n (number of students) and d (distance).

Joint Variation problems are very easy once you get the perfection of the lingo. These problems involve simple formulae or relationships which involves one variable which is equal to the “one” term which may be linear (with just an “x” axis), a quadratic equation (like “x²) where more than one variable (like “hr²”), and square root (like “\sqrt{4 – r^2\,}4−r2​”) etc.

## Functions of 2 or More Variables

It is very uncommon for the output variable to depend on 2 or more inputs. Most of the familiar formulas describe the several variables functions. For suppose, if the rectangle perimeter depends on the length and width. The cylinder volume depends on its height and radius. The travelled distance depends on the time and speed while travelling. The function notation of the formulas can be written as

P = f(l,w) = 2l + 2w where P is the perimeter and is a function of width and length

V = f(r,h) = Πr²h where V is the volume and is a function of radius and height

d = f(r,t) = rt where d is the distance and is a function of time and rate.

## Tables of Values

Just for the single variable functions, we use the tables to describe two-variable functions. The heading of the table shows row and column and it shows the value if two input variables and the complete table shows the values of the output variable.

You can easily make graphs in three dimensions for two-variable functions. Instead of representing graphs, we represent functions by holding two or one variable constants.

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## How to Solve Joint Variation Problems?

Follow the step by step procedure provided below to solve problems involving Joint Variation and arrive at the solution easily. They are along the lines

Step 1: Write the exact equation. The problems of joint variation can be solved using the equation y =kxz. While dealing with the word problems. you should also consider using variables other than x,y and z. Use the variables which are relevant to the problem being solved. Read the problem carefully and determine the changes in the equation of joint variation such as cubes, squares or square roots.

Step 2: With the help of the information in the problem, you have to find the value of k which is called the constant of proportionality and variation.

Step 3: Rewrite the equation starting with 1 substituting the value of k and found in step 2.

Step 4: Use the equation in step 3 and the information in the problem to answer the question. While solving the word problems, remember including the units in the final answer.

## Joint Variation Problems with Solutions

The area of a triangle varies jointly as the base and the height. Area = 12m² when base = 6m and height = 4m. Find base when Area = 36m² and height = 8m?

The area of the triangle is represented with A

The base is represented with b

Height is represented with h

As given in the question,

A = 12m² when B = 6m and H = 4m

We know the equation,

A = kbh where k is the constant value

12 = k(6)(4)

Divide by 24 on both sides, we get

12/24 = k(24)/24

The value of k = 1/2

As the equation is

To find the base of the triangle of A = 36m² and H = 8m

36 = 1/2(b)(8)

Dividing both sides by 4, we get

36/4 = 4b/4

The value of base = 9m

Hence, the base of the triangle when A = 36m² and H = 8m is 9m

Wind resistance varies jointly as an object’s surface velocity and area. If the object travels at 80 miles per hour and has a surface area of 30 square feet which experiences 540 newtons wind resistance. How much fast will the car move with 40 square feet of the surface area in order to experience a wind resistance of 495 newtons?

Let w be the wind resistance

Let s be the object’s surface area

Let v be the object velocity

The object’s surface area = 80 newtons

The wind resistance = 540 newtons

The object velocity = 30

w = ksv where k is the constant

(540) = k (80) (30)

540 = k (2400)

540/2400 = k

The value of k is 9/40

To find the velocity of the car with s = 40, w = 495 newtons and k = 9/40

Substitute the values in the equation

495 = (9/40) (40) v

The velocity of a car is 55mph for which the object’s surface area is 40 and wind resistance is 495 newtons

Hence, the final solution is 55mph

For the given interest, SI (simple interest) varies jointly as principal and time. If 2,500 Rs left in an account for 5 years, then the interest of 625 Rs. How much interest would be earned, if you deposit 7,000 Rs for 9 years?

Let i be the interest

Let p be the principal

Let t be the time

The interest is 625 Rs

The principal is 2500

The time is 5 hours

i = kpt where k is the constant

Substituting the values in the equation,

(625) = k(2500)(5)

625 = k(12,500)

Dividing 12,500 on both the sides

625/12,500 = k (12,500)/12,500

The value of k = 1/20

To find the interest where the deposit is 7000Rs for 9 years, use the equation

i = (1/20) (7000) (9)

i = (350) (9)

Therefore, the interest is 3,150 Rs, if you deposit 7,000 Rs for 9 years

Thus, the final solution is Rs. 3,150

The volume of a pyramid varies jointly as its height and the area of the base. A pyramid with a height of 21 feet and a base with an area of 24 square feet has a volume of 168 cubic feet. Find the volume of a pyramid with a height of 18 feet and a base with an area of 42 square feet?

Let v be the volume of a pyramid

Let h be the height of a pyramid

Let a be the area of a pyramid

The volume v = 168 cubic feet

The height h = 21 feet

The area a = 24 square feet

V = Kha where K is the constant,

168 = k(21)(24)

168 = k(504)

Divide 504 on both sides

168/504 = k(504)/504

The value of k = 1/3

To find the volume of a pyramid with a height of 18 feet and a base with an area of 42 square feet

h = 18 feet

a = 42 square feet

V = (1/3) (18) (42)

V = (6) (42)

V = 252 ft³

The volume of the pyramid = 252 ft³ which has a height of 18 feet and a base with an area of 42 square feet

Therefore, the final solution is 252 ft³

The amount of oil used by a ship travelling at a uniform speed varies jointly with the distance and the square of the speed. If the ship uses 200 barrels of oil in travelling 200 miles at 36 miles per hour, determine how many barrels of oil are used when the ship travels 360 miles at 18 miles per hour?

No of barrels of oil = 200

The distance at which the oil is travelling = 200 miles

The distance at which the ship is travelling = 36 miles per hour

A = kds² where k is constant

200 = k.200.(36)²

Dividing both sides by 200

200/200 = k.200.(36)²/200

1 = k.(36)²

The value of k is 1/1296

To find the no of barrels when the ship travels 360 miles at 18 miles per hour

A = 1/1296 * 360 * 18²

Therefore, 90 barrels of oil is used when the ship travels 360 miles at 18 miles per hour

Thus, the final solution is 90 barrels

## Study Guides > College Algebra

Solve problems involving joint variation.

Many situations are more complicated than a basic direct variation or inverse variation model. One variable often depends on multiple other variables. When a variable is dependent on the product or quotient of two or more variables, this is called joint variation . For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable c , cost, varies jointly with the number of students, n , and the distance, d .

## A General Note: Joint Variation

Joint variation occurs when a variable varies directly or inversely with multiple variables.

For instance, if x  varies directly with both y  and z , we have x  = kyz . If x  varies directly with y  and inversely with z , we have $x=\frac{ky}{z}$. Notice that we only use one constant in a joint variation equation.

## Example 4: Solving Problems Involving Joint Variation

A quantity x  varies directly with the square of y  and inversely with the cube root of z . If x  = 6 when y  = 2 and z  = 8, find x  when y  = 1 and z  = 27.

Begin by writing an equation to show the relationship between the variables.

Substitute x  = 6, y  = 2, and z  = 8 to find the value of the constant k .

Now we can substitute the value of the constant into the equation for the relationship.

To find x  when y  = 1 and z  = 27, we will substitute values for y  and z  into our equation.

x  varies directly with the square of y  and inversely with z . If x  = 40 when y  = 4 and z  = 2, find x  when y  = 10 and z  = 25.

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## Joint or Combined Variation

These lessons help Algebra students learn about joint or combined variation.

Related Pages: Proportions Joint Variation Word Problems Direct Variation Inverse Variation More Algebra Lessons

The following figure shows Joint Variation. Scroll down the page for more examples and solutions of Joint and Combine Variations.

## What is Joint Variation or Combined Variation?

Joint Variation or Combined Variation is when one quantity varies directly as the product of at least two other quantities.

For example: y = kxz y varies jointly as x and z, when there is some nonzero constant k

Joint Variation Examples

Example: Suppose y varies jointly as x and z. What is y when x = 2 and z = 3, if y = 20 when x = 4 and z = 3?

Example: z varies jointly with x and y. When x = 3, y = 8, z = 6. Find z, when x = 6 and y = 4.

Joint Variation Application

Example: The energy that an item possesses due to its motion is called kinetic energy. The kinetic energy of an object (which is measured in joules) varies jointly with the mass of the object and the square of its velocity. If the kinetic energy of a 3 kg ball traveling 12 m/s is 216 Joules, how is the mass of a ball that generates 250 Joules of energy when traveling at 10 m/s?

Distinguish between Direct, Inverse and Joint Variation

Example: Determine whether the data in the table is an example of direct, inverse or joint variation. Then, identify the equation that represents the relationship.

Combined Variation

In Algebra, sometimes we have functions that vary in more than one element. When this happens, we say that the functions have joint variation or combined variation. Joint variation is direct variation to more than one variable (for example, d = (r)(t)). With combined variation, we have both direct variation and indirect variation.

How to set up and solve combined variation problems?

Example: Suppose y varies jointly with x and z. When y = 20, x = 6 and z = 10. Find y when x = 8 and z =15.

Lesson on combining direct and inverse or joint and inverse variation

Example: y varies directly as x and inversely as the square of z, and when x = 32, y = 6 and z = 4. Find x when y = 10 and z = 3.

How to solve problems involving joint and combined variation?

If t varies jointly with u and the square of v, and t is 1152 when u is 8 and v is 4, find t when v is 5 and u is 5.

The amount of oil used by a ship traveling at a uniform speed varies jointly with the distance and the square of the speed. If the ship uses 200 barrels of oil in traveling 200 miles at 36 miles per hour, determine how many barrels of oil are used when the ship travels 360 miles at 18 miles per hour.

Designer Dolls found that its number of Dress-Up Dolls sold, N, varies directly with their advertising budget, A, and inversely proportional with the price of each doll, P. When $54,00 was spent on advertising and the price of the doll is$90, then 9,600 units are sold. Determine the number of dolls sold if the amount of advertising budget is increased to $144,000. Example: y varies jointly as x and z and inversely as w, and y = 3/2, when x = 2, z =3 and w = 4. Find the equation of variation. We welcome your feedback, comments and questions about this site or page. 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Operations (3) • World's HARDEST Easy Geometry problems (1) • Wronskian (1) • Yield of Chemical Reactions (2) ## Joint Variation and Proportion problems Joint variation and proportion example question. If z varies jointly as x and y , and z = 3 when x = 4 and y = 6, find z when x = 20 and y = 9. Solution to this Joint Variation and Proportion practice problem is provided in the video below! Tags: joint proportion example problems , joint proportion example questions , joint proportion example solutions , joint variation example problems , joint variation example questions , joint variation example solutions , joint variation problems and solutions , joint variation video tutorial Thank you so much, this tutorial is awesome! Enjoyed the clarity of explanation, thank you Very nice tutorial and good drill. My son understands now how to distinguish many different types of variation! ## Leave a comment Cancel reply Your email address will not be published. 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Find the value of z when x = 2 and y = 4. Since z varies directly with the sum of squares of x and y, z ∝ x 2 + x 2 z = k(x 2 + y 2 ) ----(1) Substitute z = 5, x = 3 and y = 4 to find the value k. 5 = k(3 2 + 4 2 ) 5 = k(9 + 16) Divide both sides by 25. Substitute k = 1/5 in (1). z = (1/5)(x 2 + y 2 ) Substitute x = 2, y = 4 and evaluate z. z = (1/5)( (2 2 + 4 2 ) z = (1/5)( (4 + 16) z = (1/5)( (20) Problem 2 : M varies directly with the square of d and inversely with the square root of x. M = 24 when d = 4 and x = 9. Find the value of M when d = 5 and x = 4. Since m varies directly with the square of d and inversely with the square root of x M ∝ d 2 √ x M = kd 2 √ x ----(1) Substitute M = 24, d = 4 and x = 9 to find the value k. 24 = k4 2 √9 24 = k(16)(3) Divide both sides by 48. Substitute k = 1/2 in (1). M = (1/2)(d 2 √ x ) Substitute d = 5, x = 4 and evaluate M. M = (1/2) (5 2 √4 ) M = (1/2)( (25)(2) Problem 3 : Square of T varies directly with the cube of a and inversely with the square of d. T = 2 when a = 2 and d = 4. Find the value of s quare of T when a = 4 and d = 2 Since square of T varies directly with the cube of a and inversely with the square of d T 2 ∝ a 3 d 2 T 2 = ka 3 d 2 ----(1) Substitute T = 2, a = 2 and d = 4 to find the value k. 2 2 = k2 3 4 2 4 = k(4)(16) Divide both sides by 64. Substitute k = 1/16 in (1). T 2 = (1/16)a 3 d 2 Substitute a = 4, d = 2 and evaluate T 2 . T 2 = (1/16)(4 3 )(2 2 ) T 2 = (1/16)(64)(4) T 2 = 16 Problem 4 : The area of a rectangle varies directly with its length and square of its width. When the length is 5 cm and width is 4 cm, the area is 160 cm 2 . Find the area of the rectangle when the length is 7 cm and the width is 3 cm. Let A represent the area of the rectangle, l represent the length and w represent width. Since the area of the rectangle varies directly with its length and square of its width, A ∝ lw 2 A = klw 2 ----(1) Substitute A = 160, l = 5 and d = 4 to find the value k. 160 = k(5)(4 2 ) 160 = k(5)(16 ) 160 = 80k Divide both sides by 80. Substitute k = 2 in (1). Substitute l = 7, w = 3 and evaluate A. A = 2(7)(3 2 ) A = 2(7)(9) Area of the rectangle = 126 cm 2 Problem 5 : The volume of a cylinder varies jointly as the square of radius and two times of its height. A cylinder with radius 4 cm and height 8 cm has a volume 128 π cm 3 . Find the volume of a cylinder with radius 3 cm and height 10 cm. Let V represent volume of the cylinder, r represent radius and h represent height. Since t he volume of a cylinder varies jointly as the radius and the sum of the radius and the height. V ∝ r 2 (2h) V = kr 2 (2h) ----(1) Substitute V = 128 π , r = 4 and h = 8 to find the value of k. 128π = k(4 2 )(2 ⋅ 8) 128π = k(16)(16) 128π = 256k Divide both sides by 256. π/2 = k Substitute k = π/2 in (1). V = ( π/2) r 2 (2h) V = π r 2 h Substitute r = 3, h = 10 and evaluate V. V = π(3 2 )(10) V = π(9) (10) Volume of the cylinder = 90 π cm 3 Kindly mail your feedback to [email protected] We always appreciate your feedback. © All rights reserved. onlinemath4all.com • Sat Math Practice • SAT Math Worksheets • PEMDAS Rule • BODMAS rule • GEMDAS Order of Operations • Math Calculators • Transformations of Functions • Order of rotational symmetry • Lines of symmetry • Compound Angles • Quantitative Aptitude Tricks • Trigonometric ratio table • Word Problems • Times Table Shortcuts • 10th CBSE solution • PSAT Math Preparation • Privacy Policy • Laws of Exponents ## Recent Articles Best way to learn mathematics. Jul 05, 24 02:28 AM ## SAT Math Preparation Videos (Part - 4) Jul 04, 24 11:47 PM ## Integration Problems with Solutions Jul 04, 24 07:11 PM ## {{ article.displayTitle }}  {{ 'ml-lesson-number-slides' | message : article.intro.bblockCount }} {{ 'ml-lesson-number-exercises' | message : article.intro.exerciseCount }} {{ 'ml-lesson-time-estimation' | message }} • {{ item.file.title }} {{ presentation }} ## Catch-Up and Review Here are a few recommended readings before getting started with this lesson. • Reciprocal Functions • Graphing Reciprocal Functions • Direct Variation and Proportional Relationships ## Number of Phone Calls Per Day Between Two Cities The average number of phone calls per day between two cities N varies directly with the populations of the cities P 1 ​ and P 2 ​ , and inversely with the square of distance d between the two cities. ## Joint Variation A joint variation , also known as joint proportionality , occurs when one variable varies directly with two or more variables . In other words, if a variable varies directly with the product of other variables, it is called joint variation. Here, the variable z varies jointly with x and y , and k is the constant of variation . Here are some examples of joint variation. Examples of Joint Variation Example Rule Comment The A=w Here, is the rectangle's length, w its width, and the constant of variation k is 1. The V=31wh Here, and w are the length and the width of the base, respectively, while h is the pyramid's height. The constant of variation k is 31. ## Television Series Vincenzo and Emily are having a lively chat about television series they love. Emily managed to watch 1 6 4 episodes of The Flash in just 5 0 days! Each episode typically lasts 4 0 minutes. Use the fact that if z varies jointly with x and y , the equation of variation is z = k x y , where k is the constant of variation . Substitute values LHS / 6 5 6 0 = RHS / 6 5 6 0 b a ​ = b / 1 0 a / 1 0 ​ Rearrange equation c a ​ ⋅ b = c a ⋅ b ​ Calculate quotient Round to nearest integer ## Recognzing Inverse Variation Width Length 2 4 8 16 32 ## Inverse Variation An inverse variation , or inverse proportionality , occurs when two non-zero variables have a relationship such that their product is constant . This relationship is often written with one of the variables isolated on the left-hand side. x y = k or y = x k ​ Example Rule Comment The gas pressure in a sealed container if the container's volume is changed, given constant temperature and constant amount of gas. P=VnRT The variables are the pressure P and the volume V. The amount of gas n, temperature T, and universal gas constant R are fixed values. Therefore, the constant of variation is nRT. The time it takes to travel a given distance at various speeds. t=sd The constant of variation is the distance d and the variables are the time t and the speed s. ## Identifying the Type of Variation Determine whether the relationship between the variables in the table shows direct or inverse variation, or neither. ## Number of Songs on Emily's Phone Emily, tired of watching shows, wants to update the playlist on her phone before starting a family road trip from Portland to San Francisco. The number of songs that can be stored on her phone varies inversely with the average size of a song. Emily's phone can store 4 1 0 0 songs when the average size of a song is 4 megabytes (MB). x = 4 , y = 4 1 0 0 LHS ⋅ 4 = RHS ⋅ 4 Size, x x16400 Number of Songs, y 3 216400 5466 4 416400 4100 5 516400 3280 6 616400 2733 In the table, as the size gets larger, the number of songs that the phone can store gets smaller. Therefore, the number of songs decreases as the average size increases. ## Emily's Trip to San Francisco Example Graph: r r640 t 10 10640 64 20 20640 32 30 30640 21 40 40640 16 50 50640 12.8 60 60640 10.7 Ordered pairs ( r , t ) are the coordinates of the points on the graph. Plot the points and connect them with a smooth curve. LHS ⋅ r = RHS ⋅ r LHS / 1 2 = RHS / 1 2 ## Combined Variation A combined variation , or combined proportionality , occurs when one variable depends on two or more variables, either directly , inversely , or a combination of both. This means that any joint variation is also a combined variation. z = y k x ​ The variable z varies directly with x and inversely with y , and k is the constant of variation . Therefore, this is a combined variation. Here are some examples. Examples of Combined Variation Example Rule Comment Newton's Law of Gravitational Force F=d2Gm1m2 The gravitational force F varies directly as the masses of the objects m1 and m2, and inversely as the square of the distance d2 between the objects. The gravitational constant G is the constant of variation. The Ideal Gas Law P=VnRT The pressure P varies directly as the number of moles n and the temperature T, and inversely as the volume V. The universal gas constant R is the constant of variation. ## Number of T-Shirts Sold Emily is wandering around a gift shop to buy gifts for some of her friends. Emily overhears a conversation between the shopkeeper and an employee. The shopkeeper says that the number of t-shirts sold is directly proportional to their advertising budget and inversely proportional to the price of each t-shirt. When$ 1 2 0 0 are spent on advertising and the price of each t-shirt is $4 . 8 0 , the number of t-shirts sold is 6 5 0 0 . How many t-shirts are sold when the advertising budget is$ 1 8 0 0 and the price of each t-shirt is \$ 6 ?

Use the equation of the combined variation , z = y k x ​ , where k is the constant of variation .

When one quantity varies with respect to two or more quantities, this variation can be regarded as a combined variation.

Combined Variation Equation Form
a varies with b and c. a=kbc
a varies jointly with b and c, and inversely with d. a=dkbc
a varies directly with b and inversely with the product dc. a=dckb

LHS ⋅ 4 . 8 0 = RHS ⋅ 4 . 8 0

LHS / 1 2 0 0 = RHS / 1 2 0 0

b = 1 8 0 0 , p = 6

## Alternative Solution

Cross multiply

LHS / 7 2 0 0 = RHS / 7 2 0 0

## Finding the Value of z

In the applet, various types of variations are shown randomly. Find the value of z by using the given values. If necessary, round the answer to the two decimal places.

In this lesson, variation types are explained with real-life examples. Considering those examples, the challenge presented at the beginning of the lesson can be solved with confidence. Recall that the average number of phone calls per day between two cities varies directly with the populations of the cities and inversely with the square of the distance between the two cities.

San Francisco Portland
Population 806000 585000
Distance 650
Number of Calls 42000

Calculate power

c a ⋅ b ​ = a ⋅ c b ​

LHS / 1 1 1 6 0 0 0 = RHS / 1 1 1 6 0 0 0

Round to 3 decimal place(s)

San Francisco Los Angeles
Population 806000 3800000
Distance d
Number of Calls 806000

LHS ⋅ d 2 = RHS ⋅ d 2

LHS / 8 0 6 0 0 0 = RHS / 8 0 6 0 0 0

LHS ​ = RHS ​

a 2 ​ = ± a

Calculate root

## Joint Variation – Formula, Examples | How to Solve Problems Involving Joint Variation?

Joint Variation definition, rules, methods and formulae are here. Check the joint variation problems and solutions to prepare for the exam. Refer to problems of direct and inverse variations and the relationship between the variables. Know the different type of variations like inverse, direct, combined and joint variation. Go through the below sections to check definition, various properties, example problems, value tables, concepts etc.

## Joint Variation – Introduction

Joint Variation refers to the scenario where the value of 1 variable depends on 2 or more and other variables that are held constant. For example, if C varies jointly as A and B, then C = ABX for which constant “X”. The joint variation will be useful to represent interactions of multiple variables at one time.

Most of the situations are complicated than the basic inverse or direct variation model. One or the other variables depends on the multiple other variables. Joint Variation is nothing but the variable depending on 2 or more variables quotient or product. To understand clearly with an example, The amount of busing candidates for each of the school trip varies with the no of candidates attending the distance from the school. The variable c (cost) varies jointly with n (number of students) and d (distance).

Joint Variation problems are very easy once you get the perfection of the lingo. These problems involve simple formulae or relationships which involves one variable which is equal to the “one” term which may be linear (with just an “x” axis), a quadratic equation (like “x²) where more than one variable (like “hr²”), and square root (like “\sqrt{4 – r^2\,}4−r2​”) etc.

## Functions of 2 or More Variables

It is very uncommon for the output variable to depend on 2 or more inputs. Most of the familiar formulas describe the several variables functions. For suppose, if the rectangle perimeter depends on the length and width. The cylinder volume depends on its height and radius. The travelled distance depends on the time and speed while travelling. The function notation of the formulas can be written as

P = f(l,w) = 2l + 2w where P is the perimeter and is a function of width and length

V = f(r,h) = Πr²h where V is the volume and is a function of radius and height

d = f(r,t) = rt where d is the distance and is a function of time and rate.

## Tables of Values

Just for the single variable functions, we use the tables to describe two-variable functions. The heading of the table shows row and column and it shows the value if two input variables and the complete table shows the values of the output variable.

You can easily make graphs in three dimensions for two-variable functions. Instead of representing graphs, we represent functions by holding two or one variable constants.

• What is Variation
• Practice Test on Ratio and Proportion

## How to Solve Joint Variation Problems?

Follow the step by step procedure provided below to solve problems involving Joint Variation and arrive at the solution easily. They are along the lines

Step 1: Write the exact equation. The problems of joint variation can be solved using the equation y =kxz. While dealing with the word problems. you should also consider using variables other than x,y and z. Use the variables which are relevant to the problem being solved. Read the problem carefully and determine the changes in the equation of joint variation such as cubes, squares or square roots.

Step 2: With the help of the information in the problem, you have to find the value of k which is called the constant of proportionality and variation.

Step 3: Rewrite the equation starting with 1 substituting the value of k and found in step 2.

Step 4: Use the equation in step 3 and the information in the problem to answer the question. While solving the word problems, remember including the units in the final answer.

## Joint Variation Problems with Solutions

The area of a triangle varies jointly as the base and the height. Area = 12m² when base = 6m and height = 4m. Find base when Area = 36m² and height = 8m?

The area of the triangle is represented with A

The base is represented with b

Height is represented with h

As given in the question,

A = 12m² when B = 6m and H = 4m

We know the equation,

A = kbh where k is the constant value

12 = k(6)(4)

Divide by 24 on both sides, we get

12/24 = k(24)/24

The value of k = 1/2

As the equation is

To find the base of the triangle of A = 36m² and H = 8m

36 = 1/2(b)(8)

Dividing both sides by 4, we get

36/4 = 4b/4

The value of base = 9m

Hence, the base of the triangle when A = 36m² and H = 8m is 9m

Wind resistance varies jointly as an object’s surface velocity and area. If the object travels at 80 miles per hour and has a surface area of 30 square feet which experiences 540 newtons wind resistance. How much fast will the car move with 40 square feet of the surface area in order to experience a wind resistance of 495 newtons?

Let w be the wind resistance

Let s be the object’s surface area

Let v be the object velocity

The object’s surface area = 80 newtons

The wind resistance = 540 newtons

The object velocity = 30

w = ksv where k is the constant

(540) = k (80) (30)

540 = k (2400)

540/2400 = k

The value of k is 9/40

To find the velocity of the car with s = 40, w = 495 newtons and k = 9/40

Substitute the values in the equation

495 = (9/40) (40) v

The velocity of a car is 55mph for which the object’s surface area is 40 and wind resistance is 495 newtons

Hence, the final solution is 55mph

For the given interest, SI (simple interest) varies jointly as principal and time. If 2,500 Rs left in an account for 5 years, then the interest of 625 Rs. How much interest would be earned, if you deposit 7,000 Rs for 9 years?

Let i be the interest

Let p be the principal

Let t be the time

The interest is 625 Rs

The principal is 2500

The time is 5 hours

i = kpt where k is the constant

Substituting the values in the equation,

(625) = k(2500)(5)

625 = k(12,500)

Dividing 12,500 on both the sides

625/12,500 = k (12,500)/12,500

The value of k = 1/20

To find the interest where the deposit is 7000Rs for 9 years, use the equation

i = (1/20) (7000) (9)

i = (350) (9)

Therefore, the interest is 3,150 Rs, if you deposit 7,000 Rs for 9 years

Thus, the final solution is Rs. 3,150

The volume of a pyramid varies jointly as its height and the area of the base. A pyramid with a height of 21 feet and a base with an area of 24 square feet has a volume of 168 cubic feet. Find the volume of a pyramid with a height of 18 feet and a base with an area of 42 square feet?

Let v be the volume of a pyramid

Let h be the height of a pyramid

Let a be the area of a pyramid

The volume v = 168 cubic feet

The height h = 21 feet

The area a = 24 square feet

V = Kha where K is the constant,

168 = k(21)(24)

168 = k(504)

Divide 504 on both sides

168/504 = k(504)/504

The value of k = 1/3

To find the volume of a pyramid with a height of 18 feet and a base with an area of 42 square feet

h = 18 feet

a = 42 square feet

V = (1/3) (18) (42)

V = (6) (42)

V = 252 ft³

The volume of the pyramid = 252 ft³ which has a height of 18 feet and a base with an area of 42 square feet

Therefore, the final solution is 252 ft³

The amount of oil used by a ship travelling at a uniform speed varies jointly with the distance and the square of the speed. If the ship uses 200 barrels of oil in travelling 200 miles at 36 miles per hour, determine how many barrels of oil are used when the ship travels 360 miles at 18 miles per hour?

No of barrels of oil = 200

The distance at which the oil is travelling = 200 miles

The distance at which the ship is travelling = 36 miles per hour

A = kds² where k is constant

200 = k.200.(36)²

Dividing both sides by 200

200/200 = k.200.(36)²/200

1 = k.(36)²

The value of k is 1/1296

To find the no of barrels when the ship travels 360 miles at 18 miles per hour

A = 1/1296 * 360 * 18²

Therefore, 90 barrels of oil is used when the ship travels 360 miles at 18 miles per hour

Thus, the final solution is 90 barrels

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## Joint Variation and Combined Variation - Definitions - Expii

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## Variation Word Problems

Equations Word Problems More Prob's

It's one thing to be able to take the words for a variation equation (such as " y varies directly as the square of x and inversely as the cube root of z ") and turn this into an equation that you can solve or use. It's another thing to extract the words from a word problem. But, because the lingo for variation equations is so specific, it's not really that hard. Just look for the keywords, and you're nearly home and dry.

The only other keywords (or "key-phrases", really) you might need to know are "is proportional to" which, in the strictly-mathematical sense, means "varies directly as"; and "is inversely proportional to" which means "varies inversely with".

Content Continues Below

## Suppose that y is inversely proportional to x , and that y  = 0.4 when x  = 2.5 . Find y when x  = 4 .

Translating the above from the English into algebra, I see the key-phrase "inversely proportional to", which means "varies indirectly as". In practical terms, it means that the variable part that does the varying is going to be in the denominator. So I get the formula:

Plugging in the data point they gave me, I can solve for the value of k :

0.4 = k /(2.5)

(0.4)(2.5) = k = 1

Now that I have found the value of the variation constant, I can plug in the x -value they gave me, and find the value of y when x  = 4 :

Most word problems, of course, are not nearly as simple as the above example (or the ones on the previous page). Instead, you have to figure out which values go where, what the equation is, and how to interpret it. Fortunately, the keywords and key-phrases should generally be fairly clear, telling you exactly what format to use.

## According to Hooke's Law, the force needed to stretch a spring is proportional to the amount the spring is stretched. If fifty pounds of force stretches a spring five inches, how much will the spring be stretched by a force of 120 pounds?

"Is proportional to" means "varies directly with", so the formula for Hooke's Law is:

...where " F " is the force and " d " is the distance that the spring is stretched.

Note: In physics, "weight" is a force. These Hooke's Law word problems, among other types, are often stated in terms of weight, and the weight they list is the force they mean.

First I have to solve for the value of k . They've given me the data point ( d , F) = (5, 50) , so I'll plug this in to the formula:

Now I know that the formula for this particular spring is:

(Hooke's Law doesn't change, but each spring is different, so each spring will have its own " k ".)

Once I know the formula, I can answer their question: "How much will the spring be stretched by a force of 120 pounds?" I'll plug the value they've given me for the force into the equation I've found:

Note that they did not ask "What is the value of ' d '?". They asked me for a distance. I need to be sure to answer the question that they actually asked. That final answer is the distance that the spring is stretched, including the units (which are "inches", in this case):

Note: If you give the above answer as being only " 12 ", the grader will be perfectly correct to count your answer as being at least partly wrong. The answer is not a number, but is a number of units.

## Kepler's third law of planetary motion states that the square of the time required for a planet to make one revolution about the sun varies directly as the cube of the average distance of the planet from the sun. If you assume that Mars is 1.5 times as far from the sun as is the earth, find the approximate length of a Martian year.

This one is a bit different from the previous exercise. Normally, I've given given a relation in terms of variation with a plain old variable, like y . In this case, though, the variation relationship is between the square of the time and the cube of the distance. This means that the left-hand side of my equation will have a squared variable!

My variation formula is:

t 2 = k d 3

If I take " d  = 1 " to mean "the distance is one AU", an AU being an "astronomical unit" (the distance of earth from the sun), then the distance for Mars is 1.5 AU. Also, I will take " t  = 1 " to stand for "one earth year". Then, in terms of the planet Earth, I get:

(1) 2 = k  (1) 3

Then the formula, in terms of Earth, is:

Now I'll plug in the information for Mars (in comparison to earth):

t 2 = (1.5) 3

This is one of those times when a calculator's decimal approximation is probably going to be a little more useful in answering the question. I'll show the exact answer in my working, but I'll use a sensible approximation in my final answer. The decimal expansion starts as:

In other words, the Martian year is approximately the length of:

1.837 earth years

By the way, you can make the above answer more intuitive by finding the number of (Earth) days, approximately, represented by that " 0.837 " part of the answer above. Since the average Earth year, technically, has about 365.25 days, then the 0.837 of an Earth year is:

0.837 × 365.25 = 305.71425

Letting the "average" month have 365.25 ÷ 12 = 30.4375 days, then the above number of (Earth) days is:

305.71425 ÷ 30.4375 = 10.044

In other words, the Martian year is almost exactly one Earth year and ten Earth months long.

If you were writing for an audience (like a fellow student, as you'll be required to do in some class projects or essay questions), this "one year and ten months" form would probably be the best way to go.

## The weight of a body varies inversely as the square of its distance from the center of the earth. If the radius of the earth is 4000 miles, how much would a 200 -pound man weigh 1000 miles above the surface of the earth?

Remembering that "weight" is a force, I'll let the weight be designated by " F ". The distance of a body from the center of the earth is " d   ". Then my variation formula is the following:

(200)(16,000,000) = k

3,200,000,000 = k

(Hey; there's nothing that says that the value of the variation constant k has to be small!)

The distance is always measured from the center of the earth. If the guy is in orbit a thousand miles up (from the surface of the planet), then his distance (from the center of the planet) is the 4000 miles from the center to the surface plus the 1000 miles from the surface to his ship. That is, d = 5000 . I'll plug this in to my equation, and solve for the value of the force (which is here called "weight") F :

Remembering my units, my answer is that, up in his spacecraft, the guy weighs:

URL: https://www.purplemath.com/modules/variatn2.htm

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• Problems on Variation

In math variation we solved numerous types of problems on variation by using different types of variation like direct variation, inverse variation and joint variation. The problems on variation are mainly related to the questions based on word problems of constant variation, word problems of direct variation, word problems of inverse variation and also word problems of joint variation. Each word problems on variation are explained step by step so that students can understand the question and their solution easily.

1. The area of an umbrella varies directly as the square of its radius. If the radius of the umbrella is doubled, how much will be the area of the umbrella?

If the area of the umbrella is C and radius is R then C α R 2 or C= KR 2 where K is the constant of variation.

So the area of the umbrella is KR 2 .

Now if the radius is doubled the area will be

K(2R) 2  = 4KR 2  = 4C.

So the area will be by 4 times of normal the area of the umbrella.

3. If 5 men take 8 days to type 10 books, apply the principle of variation to find out how many days 8 men will take to type 2 books?

If N represents numbers of men, D is number of days and B is number of books the from the principle of variation

N α  $$\frac{1}{D}$$  or, N is in inverse variation with D as when numbers of men increase it will take less time, so the numbers of days will decrease.

N α B or N is in direct variation with B as when numbers of men increase they can type more, so the numbers of books can be typed will increase.

From the theorem of joint theorem

N α $$\frac{B}{D}$$

or, N = K $$\frac{B}{D}$$  where K is constant of variation.

For the given data

5 = K ×   $$\frac{10}{8}$$

or, K = $$\frac{40}{10}$$  = 4.

Substituting the value of K in the variation equation

N = 4 $$\frac{B}{D}$$

For 8 men to type 2 books number

8 = 4 x  $$\frac{2}{D}$$

So it will 1 day.

5. In X is in indirect variation with square of Y and when X is 3, Y is 4. What is the value of X when Y is 4?

From the given problem indirect variation equation can be expressed as

X =  $$\frac{K}{Y^{2}}$$

or, K = XY 2

For the given case

K = 3 x 4 2 =48.

So when Y is 4,

or, X =  $$\frac{48}{Y^{2}}$$

=  $$\frac{K}{4^{2}}$$

= 3

So the value of X is 3.

Note:  Variation is a very important part of algebra in higher grade and college grade. By practicing the problems of variation student get very clear concept on different types of variation.

7. If a car runs at a average speed of 40kmph with some regular intervals and takes 3 hrs to run a distance of 90 km, what time it will take to run at a average speed of 60 kmph with same intervals to run 120 km?

If T is the time taken to cover the distance and S is the distance and V is the average speed of the car,

The from the theory of variation

V α S or V varies directly with S when T is constant as when average speed will increase for a fixed time, distance covered by the car will increase.

V α $$\frac{1}{T}$$ or, V varies inversely with T when S is constant as when average speed will increase to cover a fixed distance, time taken by the car to will decrease.

So V α $$\frac{S}{T}$$

or, V = K $$\frac{S}{T}$$  where K is the constant of variation.

For the case given in the problem

V = K $$\frac{S}{T}$$

40 = K x  $$\frac{90}{3}$$

or, K =  $$\frac{4}{3}$$

V =  $$\frac{4S}{3T}$$

So at a average speed of the car is 60kmph to run 120 km it will take

or, 60 =  $$\frac{4 × 120}{3T}$$

or, T =$$\frac{60}{160}$$

= $$\frac{3}{8}$$  hrs

=  $$\frac{3}{8}$$  ×  60  mins

= 22.5 mins.

●   Variation

• What is Variation?
• Direct Variation
• Inverse Variation
• Joint Variation
• Theorem of Joint Variation
• Worked out Examples on Variation

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#### IMAGES

1. Solve using Joint Variation

2. Pre-Calculus

3. Joint Variation: Solving Joint Variation Problems in Algebra

4. Joint Variation

5. Solving Joint Variation Problems (3 of 3)

6. Joint Variation

#### VIDEO

1. JOINT VARIATIONS

2. JOINT VARIATION

3. Joint Variation

4. Joint Variation

5. SP3-3: Solved Example Problem (Module 3

6. Introduction of the Topic Joint Variation with some Examples|| Episode 3

1. Joint And Combined Variation Word Problems (video lessons, examples and

Example 1: A quantity varies inversely as two or more other quantities. The figure below shows a rectangular solid with a fixed volume. Express its width, w, as a joint variation in terms of its length, l, and height, h. Solution: w ∝ 1/ (lh) In other words, the longer the length l or the height h, the narrower is the width w.

2. Joint Variation: Solving Joint Variation Problems in Algebra

Joint variation is a variation in which the quotient of a variable and the product of two variables is a constant. Joint variation states that if y varies directly as the product of x and z, if there is a constant k such that y = kxz or k = y / xz, y varies jointly as x and z. It occurs when a variable varies directly or inversely with multiple ...

3. Solve problems involving joint variation

Joint variation occurs when a variable varies directly or inversely with multiple variables. ... Notice that we only use one constant in a joint variation equation. Example 4: Solving Problems Involving Joint Variation. A quantity x varies directly with the square of y and inversely with the cube root of z.

4. Joint Variation

Go through the below sections to check definition, various properties, example problems, value tables, concepts etc. Joint Variation - Introduction. Joint Variation refers to the scenario where the value of 1 variable depends on 2 or more and other variables that are held constant. For example, if C varies jointly as A and B, then C = ABX for ...

5. Study Guide

When a variable is dependent on the product or quotient of two or more variables, this is called joint variation. For example, the cost of busing students for each school trip varies with the number of students attending and the distance from the school. The variable c, cost, varies jointly with the number of students, n, and the distance, d.

6. JOINT VARIATION (Definition, Examples, Solving Problems) Made Easy

This video is about the definition and examples of joint variation and translating statements into the equation of variation. It also includes examples of so...

7. Joint Variation

The equation for the given problem of joint variation is. x = Kyz where K is the constant. For the given data. 16 = K × 4 × 6. or, K = 46 4 6. So substituting the value of K the equation becomes. x = 4yz 6 4 y z 6. Now for the required condition. x = 4×8×126 4 × 8 × 12 6.

8. Joint or Combined Variation (video lessons, examples and solutions)

Example: Suppose y varies jointly with x and z. When y = 20, x = 6 and z = 10. Find y when x = 8 and z =15. Show Video Lesson. Lesson on combining direct and inverse or joint and inverse variation. Example: y varies directly as x and inversely as the square of z, and when x = 32, y = 6 and z = 4. Find x when y = 10 and z = 3.

9. 1.8: Variation

Solving Problems involving Direct, Inverse, and Joint variation ... Joint variation is a relationship in which one quantity is proportional to the product of two or more quantities. Combined variation exists when combinations of direct and/or inverse variation occurs . Example $$\PageIndex{3}$$: Joint Variation. The area of an ellipse varies ...

10. Direct Inverse and Joint Variation Word Problems

This algebra video tutorial focuses on solving direct, inverse, and joint variation word problems. It shows you how to write the appropriate equation / form...

11. Joint Variation

This video by Fort Bend Tutoring shows the process of solving joint variation problems. Six (6) examples are shown in this FBT video. Instruction by Larry "M...

12. Joint and Combined Variation

Combined variation is a mix of direct and indirect variation. The joint variation equation is z = k x m y n where k ≠ 0 and m > 0, n > 0. Review. For questions 1-5, write an equation that represents relationship between the variables. w varies inversely with respect to x and y. r varies inversely with the square of q. z varies jointly with x ...

13. PDF Joint and Combined Variation Practice Problems

Joint and Combined Variation Practice Problems. *Make sure to use correct UNITS, when applicable. 1) If f varies jointly as g and the cube of h, and f = 200 when g = 5 and h = 4, find f when g = 3 and h = 6. 2) If y varies jointly as x and z, and y = 33 when x = 9 and z = 12, find y when x = 16 and z = 22. 3) If a varies jointly as b and the ...

14. Joint Variation and Proportion problems

Joint Variation and Proportion example question. If z varies jointly as x and y, and z = 3 when x = 4 and y = 6, find z when x = 20 and y = 9. Solution to this Joint Variation and Proportion practice problem is provided in the video below! Joint Variation problem ! ! ! ! ! Video tutorial on how to solve example questions and word problems ...

15. Joint Variation Worksheet

Write the equation for the following joint variation. "w varies jointly as x and y and inversely as z". Problem 4 : Suppose y varies jointly with x and z. If y = 36 when x = 4 and z = 3, find y when x = 12 and z = 36. Problem 5 : Suppose x varies directly as y and z. If y = 3 and z = 4, then x = 24. Find the value of x when y = 7 and z = 4.

16. Joint Variation Word Problems

Joint Variation Word Problems. JOINT VARIATION WORD PROBLEMS. Problem 1 : z varies directly with the sum of squares of x and y. z = 5 when x = 3 and y = 4. Find the value of z when x = 2 and y = 4. ... Problems on Solving Logarithmic Equations. Read More. SAT Math Preparation Videos (Part - 2) Jun 24, 24 01:38 AM.

17. Pre-Calculus

This video will show you how to solve a joint variation problem. Remember that in joint variation there are three variables that are all connected, so we ne...

18. Exploring Combined Variation Examples and Joint Variation ...

Here, the variable z varies jointly with x and y, and k is the constant of variation. Here are some examples of joint variation. Here, l is the rectangle's length, w its width, and the constant of variation k is 1. Here, l and w are the length and the width of the base, respectively, while h is the pyramid's height.

19. Joint Variation

Step 4: Use the equation in step 3 and the information in the problem to answer the question. While solving the word problems, remember including the units in the final answer. Joint Variation Problems with Solutions. Problem 1: The area of a triangle varies jointly as the base and the height. Area = 12m² when base = 6m and height = 4m.

20. Joint Variation and Combined Variation

Summary. The joint variation equation is: y varies jointly with x and z. This means y varies with x and z. y=k⋅x⋅z. Alas the relationship is more complicated than a direct relation or inverse relation. Really, joint variations are combinations of both of these. An example of a joint variation is the area of a triangle: A=12bh.

21. Variation Word Problems

Purplemath. It's one thing to be able to take the words for a variation equation (such as " y varies directly as the square of x and inversely as the cube root of z ") and turn this into an equation that you can solve or use. It's another thing to extract the words from a word problem. But, because the lingo for variation equations is so ...

22. Problems on Variation

Solved examples on application problems on joint variation are explained here step-by-step with detailed explanation: 10. The weight of a sphere varies jointly as the cube of its radius and the density of the material of which it is made. The radii of two spheres are as 17 : 8 and the densities of materials as 3 : 4. ... Worked out Examples on ...

23. Solving Joint Variation Problems

math joint variation problems