# Joint and combined variation

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##### Intros
###### Lessons
1. Introduction to joint and combined variation
2. Review: direct variation vs. inverse variation
3. What is a joint variation?
4. What is a combined variation and how is it different from a joint variation?
5. How to solve a variation problem?
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##### Examples
###### Lessons
1. Identifying Types of Variations
Determine whether each equation represents a direct, inverse, joint, or combined variation. Name the constant of variation.
1. $xy = 17$
2. $p = 5q$
3. $b = \frac{3ac}{4}$
4. $m = \frac{n}{8}$
5. $e = \frac{5f}{7g}$
2. Translating Variation Statements Into Equations
Translate the following statements, and then classify the variations.
1. $x$ varies jointly as $y$ and the square of $z$.
2. The speed of a race car varies directly as the distance and inversely as the time.
3. Solving Variation Problems
Find the missing variables.
1. $a$ varies directly with $b$ and $c$. If $a=336$ when $b=4$ and $c=7$, find $a$ when $b=2$ and $c=11$.
2. $p$ varies directly as $q$ but inversely as $r$. If $p=14$ when $q=2$ and $r=5$, find $q$ when $p=105$ and $r=18$.
4. Word Problems of Variations
The volume of a cylinder varies jointly as the height and the square of its radius. A cylinder with an 9 cm height and 6 cm radius has a volume of 1018 cm3.
1. Find the constant of variation $k$. Round your answer to 2 decimal places.
2. What is the volume of a can that has a 7 cm height and 3 cm radius?
5. The time required to process a shipment at Mamazon varies directly with the number of orders being made and inversely with the number of workers. If 1344 orders can be processed by 7 workers in 12 hours, how long will it take 125 workers to process 20,000 items?
###### Topic Notes

In this lesson, we will learn:

• Identifying Types of Variations
• Translating Variation Statements Into Equations
• Solving Variation Problems
• Word Problems of Variations

• Joint variation is a direct variation, but with two or more variables. It has the equation $y=k \cdot x \cdot z$ where $k$ is the constant of variation and $k \neq 0$.
• A combined variation is formed when we combine any of the variations together (direct, inverse and joint). In most cases, we combine direct and inverse variations to form a combined variation. i.e. $y$ varies directly with $x$ and inversely with $z$ $(y = k \cdot \frac{x}{z})$
• Steps to solving a variation problem:
1. Write the general variation formula of the problem.
2. Find the constant of variation $k$.
3. Rewrite the formula with the value of $k$.
4. Solve the problem by inputting known information.