*Ray is a licensed engineer in the Philippines. He loves to write about mathematics and civil engineering.*

## What Is Joint Variation?

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 variables.

For instance, John Ray and his friends decide to go on a picnic, with John Ray himself driving the car. Two distinct factors shall determine the distance traveled by John Ray and his friends. These are the speed by which John Ray drives the car and the time he spends driving the car. Table 1 shows the distance traveled at various speeds, while Table 2 shows the distance traveled at various times when John Ray drives his car with an interval of 50 kilometers.

Distance | Speed |
---|---|

70 km | 35 kph |

80 km | 40 kph |

100 km | 50 kph |

150 km | 75 kph |

200 km | 100 kph |

Distance | Speed |
---|---|

50 km | 1 hour |

100 km | 2 hours |

150 km | 3 hours |

200 km | 4 hours |

250 km | 5 hours |

From the tables presented, we can see that the distance traveled by John Ray and his friends is both directly proportional to the speed by which John Ray drives his car and the time he drives his car. For this specific example, the equation interprets as k = d/rt where d is the distance traveled, r is the speed, t is the time, and k is the proportionality constant.

If we get the situation where d_{1} is the distance traveled when traveling at a speed of r_{1} for t_{1} hours, and d_{2} is the distance traveled when traveling at a speed of r_{2} for t_{2} hours, we can derive the following equations.

k = d_{1} / (r_{1} t_{1})

k = d_{2} / (r_{2} t_{2})

By using the multiplication property of equality, we can drive the given equation below.

d_{1} / d_{2} = (r_{1}/r_{2}) (t_{1}/t_{2})

**Combined Variation**

Combined variation is a kind of variation which involves both direct and inverse variations. It is a variation in which the quotient of the product of two variables and a variable is a constant.

We can use the example of John Ray and his friends on a picnic stated earlier. This time, we can say that the money they are going to spend on gasoline varies directly or is directly proportional to the amount of gasoline consumed. While on the other hand, the share of money each one must pay varies inversely or is inversely proportional to the number of persons included in the division of the gas expense. Therefore, the cost c varies directly to the amount of gasoline g and inversely as to the number of person p. Therefore, it can be represented as the equation (c/g) p = k or cp = kg. It can also be written as c = kg/p or c_{1}/c_{2} = (g_{1}/g_{2}) (p_{1}/p_{2}).

Below are some examples to solve "joint variation" problems using the joint variation formula.

## Example 1: Finding an Equation of Joint Variation

Find an equation of variation where a varies jointly as b and c, and a = 30 when b = 2 and c =3.

**Solution**

Write the joint variation equation that resembles the general joint variation formula y = kxz. Let a = y, x = b, z = c.

y = kxz

a = kbc

30 = k (2) (3)

k = 5

**Final Answer**

Therefore, a = 4bc is the required equation of joint variation.

## Example 2: Joint Variation for Area of a Rectangle

The area of a rectangle varies jointly as the length and the width, and A = 64 square inches when l = 12 inches and w = 2 inches. Find the area of the rectangle whose length is 15 inches and whose width is 4 inches.

**Solution**

Write the variation equation where the area of the rectangle varies jointly as its length and width.

A = klw

64 = k (12) (2)

k = 2.667

Therefore, when l = 15 inches and w = 4 inches, the new area will be shown.

A = klw

A = 2.667 (15) (4)

A = 160.02 square inches

**Final Answer**

The rectangle area, whose length is 15 inches and whose width is 4 inches is equal to 160.02 square inches.

## Example 3: Translating Joint Variation Equations Into Words

Translate each of the following words into equations.

- k = A / bh
- k = I / Prt
- V = klwh

**Solution and Answer**

- The equation k = A/bh means that A varies jointly as b and h where A is the rectangle area, b is the base, and variable h is the height.
- The equation k = I/Prt means that the variable I varies jointly as P, r, and t where letter I represents the interest, P is the principal amount invested, r is the rate, and t is the time in years.
- The variation equation V = k (lwh) translated to words is that the volume of the rectangular prism is directly proportional to its length, width, and height.

## Example 4: Translating Words Into Mathematical Statements

Translate each statement into a mathematical statement. Use k as the constant of variation.

- The kinetic energy E of a moving object varies jointly as the mass m of the object and the square of the velocity v.
- The acceleration A of a moving object varies directly as the distance d it travels and varies inversely as the square of the time t it travels.
- The heat H produced by an electric lamp varies jointly as the resistance R and the square of the current C.
- The time t required to travel is directly proportional to the temperature T and inversely proportional to the speed v.
- The lateral surface area A of a cylindrical jar varies jointly as the diameter d and the height h of the jar.

**Solution and Answer**

Translate each statement by following the pattern of the joint variation description. Recall that joint variation says 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, then y varies jointly as x and z.

- k = E / mv
- A = kd / t
- k = H / R C2
- t = kT / v
- k = A / dh

## Example 5: Joint Variation for Area of a Triangle

What is the constant of variation in the area of a triangle equation?

**Solution**

The area of a triangle varies jointly as its base (b) and its altitude (h). Since the area of a triangle is expressed as A = ½ b h, thus the following equations:

y = kxz

A = ½ (b) (h)

k = ½

**Final Answer**

Therefore, the constant of variation for the area of the triangle is ½.

## Example 6: Pressure's Joint Variation Equations

The pressure (P) of a gas varies jointly as its density (d) and its absolute temperature (t). If dry air has a density of 0.95 kg/m3 and an absolute temperature of 293.15 Kelvin, what is its pressure given that k or the gas constant is equal to 287 J/kg/K?

**Solution**

Using the general form of "joint variation," express the pressure equation in terms of k, t, and d. Then, substitute the given values directly to the equation.

y = kxz

P = kdt

P = (287) (0.95) (293.15)

P = 79, 927.35 Pascals

P = 80,000 Pa

**Final Answer**

The pressure of the dry air given that k or the gas constant is equal to 287 J/kg/K** **is 80,000 Pa.

## Example 7: Joint Variation for the Maximum Safe Load of a Beam

The maximum safe load if a rectangular beam varies jointly as the width and the square of the depth and inversely as the length. A beam 0.2 meters wide, 0.45 meters deep, and 6 meters long has a maximum safe load of 75 kg. Find the maximum safe load for a beam of the same material, which is 0.3 meters wide, 0.5 meters deep, and 10 meters long.

**Solution**

Create a joint variation formula describing the maximum safe load for the rectangular beam mentioned. Given that the width is equal to w = 0.2 meters, the depth is d = 0.45 meters, and the length is l = 6 meters long, find the value of k for the beam’s maximum safe load. Note that MSL means maximum safe load.

Maximum safe load = k(w)(d)^{2} / l

MSL = k(w)(d)^{2} / l

k = (MSL) (l) / (w) (d)^{2}

k = (75) (6) / (0.2) (0.45)^{2}

k = 11111.1

Find the maximum safe load for the beam of the same material using the value of k obtained.

MSL = k(w)(d)^{2} / l

MSL = (11111.1) (0.3) (0.5)^{2} / 10

MSL = 83.33 kg

**Final Answer**

The maximum safe load for a beam of the same material is equal to 83.33 kilograms.

## Example 8: Right Circular Cylinder’s Joint Variation

The volume of a right circular cylinder is 60 cubic centimeters when the radius of the base is 5 centimeters, and the height of the cylinder is 8 centimeters. Find the volume of a similar right cylinder whose diameter of the base is 16 centimeters and 40 centimeters tall.

**Solution**

The altitude (h) of a right circular cylinder varies directly as the volume (V) and inversely as the square of the length of the radius (r) of the circular base of the cylinder. Given that h = 8 cm, r = 5 cm, and V = 60 cm^{3}, and If k is the variation constant, this may result in the following joint variation equation.

h = kV / r^{2}

k = hr^{2} / V

k = (8) (5)^{2} / 60

k = 3.33

Using the obtained value of k, which is k = 3.33, find the volume of the similar right cylinder of dimensions d = 16 cm and h = 40 cm.

h = kV / r^{2}

V = hr^{2} / k

V = (40) (8)^{2} / 3.33

V = 768.77 cubic centimeters

**Final Answer**

The volume of the similar right cylinder is equal to 768.77 cubic centimeters.

## Example 9: Joint Variation Problem

Say, c varies jointly as a and b. If c = 45 when a = 15 and b = 14, find c when a = 21 and b = 8.

**Solution**

Express the given words into a mathematical statement using the joint variation formula.

k = c / ab

k = 45 / (15) (14)

k = 0.214

c = kab

c = (0.214) (21) (8)

c = 35.95

**Final Answer**

The value of variable c when a = 21 and b = 8 is equal to 35.95 units.

## Example 10: Joint Variation Example

A quantity x varies directly with the square of y and inversely with the square root of z. If x = 8, y = 4, and z = 10, find x when y = 2 and z = 30.

**Solution**

Start by writing an equation to show the relationship between the multiple variables mentioned. Then, substitute x = 8, y = 4, and z = 10 to find the value of the constant k.

x = ky^{2} / √z

8 = k (4)^{2} / √10

k = 1.58

Now, substitute the value of the constant into the equation for the relationship. To find the value of x when y = 2 and z = 30, we will substitute values for y and z into the equation.

x = ky^{2} / √z

x = 1.58 (2)^{2} / √30

x = 1.154

**Final Answer**

The value of x when y = 2 and z = 30 is equal to 1.154.

*This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.*

**© 2021 Ray**