Lesson 3: Joint Proportion

Property

Joint proportion occurs when one variable is directly proportional to the product of two or more other variables:

where $k$ is the constant of proportionality, and $x$ varies jointly with $y$ and $z$.

Property

In a joint proportion equation, the relationship between any two variables can be determined by holding all other variables constant. If two variables are on opposite sides of the equals sign (e.g., one on the left, one in the numerator on the right), they are directly proportional. If they are on the same side of the equation, they are inversely proportional. For an equation $A = kBC$, this means $A \propto B$ and $A \propto C$, but $B \propto \frac{1}{C}$.

Examples

• Given the formula for the volume of a cylinder, $V = \pi r^2 h$. If the radius $r$ is held constant, the volume $V$ is directly proportional to the height $h$.
• In the Ideal Gas Law, $PV = nRT$. If the amount of gas $n$ and the temperature $T$ are held constant, the pressure $P$ and volume $V$ are on the same side of the equation, so they are inversely proportional.
• The force of gravity between two objects is given by $F = G\frac{m_1 m_2}{d^2}$. If the masses $m_1$ and $m_2$ are constant, the force $F$ is inversely proportional to the square of the distance $d$.

Explanation

Joint proportion describes how one variable depends on two or more other variables. By mentally "freezing" all variables except two, you can analyze their specific relationship. This technique simplifies the complex joint relationship into a more familiar direct or inverse proportion. Examining the positions of the two variables in the formula reveals whether they increase together (direct) or one increases as the other decreases (inverse).

Property

For a joint proportion relationship where a set of variables is constant across two states (initial and final), we can set up a ratio equation to solve for an unknown quantity. If quantity $A$ is constant, then $A_{initial} = A_{final}$. For example, if $k = \frac{PV}{T}$ is a constant, then for any two states 1 and 2:

Examples

• The pressure P of a gas is directly proportional to its temperature T and inversely proportional to its volume V. A gas has a pressure of 100 kPa at a temperature of 300 K and a volume of 2 L. If the temperature is increased to 360 K and the volume is increased to 3 L, the pressure changes according to $\frac{P_1V_1}{T_1} = \frac{P_2V_2}{T_2}$.

• The force of gravity F between two objects is jointly proportional to their masses, m1 and m2, and inversely proportional to the square of the distance d between them. Initially, the force is F0. When one mass is doubled (m1 → 2m1) and the distance is halved (d → d/2), the new force changes according to $F = G\frac{m_1m_2}{d^2}$, so $\frac{Fd^2}{m_1m_2} = G$.

Explanation

This skill involves applying the principles of joint proportion to solve for a missing value. By understanding that the constant of proportionality $k$ remains the same, we can equate the expressions for two different scenarios. This creates an equation relating the initial and final states of the variables. You can then isolate and solve for the unknown variable by substituting the known values and performing algebraic manipulation.

Property

Work problems can be modeled using joint proportion. The amount of work done ($W$) is jointly proportional to the number of workers ($N$) and the time ($T$) they work. This relationship is expressed by the equation:

where $k$ is the constant of proportionality, representing the rate of work per worker per unit of time.

Examples

• If 6 workers can build 4 cars in 2 days, how many days would it take 8 workers to build 6 cars?

First, find the constant $k$: $4 = k(6)(2)$, so $k = \frac{4}{12} = \frac{1}{3}$.
Then, solve for the unknown time $T$: $6 = \frac{1}{3}(8)(T)$, which gives $T = \frac{18}{8} = 2.25$ days.

• A group of 4 painters can paint 12 walls in 3 hours. How many walls can 2 painters paint in 5 hours?

First, find the constant $k$: $12 = k(4)(3)$, so $k = \frac{12}{12} = 1$.
Then, solve for the unknown walls $W$: $W = 1(2)(5)$, which gives $W = 10$ walls.

Explanation

This skill applies the concept of joint proportion to solve practical "work" problems. First, you must identify the variables, such as work completed, number of workers, and time. Use the given information to set up a proportion and solve for the constant of proportionality, $k$. Once you have the value of $k$, you can use the formula with a new set of conditions to solve for the unknown quantity.