Joint and combined variations

So far we have seen one variable depending on one other. Many real-world relationships involve more than one input. Joint variation handles this: $z$ depends on $x$ and $y$ together, possibly with different kinds of proportion for each.

Concept

Joint variation. $z$ is jointly proportional to $x$ and $y$ (both directly) if

$z = k x y.$

Doubling $x$ doubles $z$; doubling $y$ doubles $z$. Doubling both quadruples $z$.

Example: cost of paint is jointly proportional to wall area and number of coats. $\text{cost} = k \cdot A \cdot c$.

Combined variation. $z$ is directly proportional to $x$ and inversely proportional to $y$:

$z = k \dfrac{x}{y}.$

Example: speed of work per worker. If $W$ workers finish $J$ jobs in $T$ days, then $W \propto J/T$ (more jobs need more workers; more time needs fewer).

The work formula. Total work $\propto$ (workers) $\times$ (time). If the same job takes different worker-times, then for the same job, worker-time is constant.

Many-variable proportion. The general form:

$z = k \cdot \dfrac{x_1^{a_1} \cdot x_2^{a_2} \cdots}{y_1^{b_1} \cdot y_2^{b_2} \cdots}$

For class VIII, you will mostly see two variables. Higher classes use more.

Setting up problems.

1. Read carefully and identify which quantities are directly related to $z$ and which are inversely.
2. Write the equation with $k$.
3. Use a known set of values to solve for $k$.
4. Plug in the new values.

Worked examples

Example 1. $z$ is jointly proportional to $x$ and $y$. When $x = 4, y = 5$, $z = 60$. Find $z$ when $x = 6, y = 8$.

• $60 = k \cdot 4 \cdot 5 \Rightarrow k = 3$.
• $z = 3 \cdot 6 \cdot 8 = 144$.

Example 2. $z$ varies directly with $x$ and inversely with $y$. If $z = 12$ when $x = 4, y = 6$, find $z$ when $x = 7, y = 14$.

• $z = k x / y$. $12 = k \cdot 4 / 6 \Rightarrow k = 18$.
• $z = 18 \cdot 7 / 14 = 9$.

Example 3. If $15$ men can finish a job in $20$ days, how many days will it take $25$ men to finish the same job?

• Worker-days are constant: $15 \cdot 20 = 25 \cdot t$. $t = 300/25 = 12$ days.

Example 4. If $20$ workers can build $5$ km of road in $30$ days, how many days for $30$ workers to build $9$ km?

• Worker-days per km: $20 \cdot 30 / 5 = 120$ worker-days per km.
• For $9$ km: $120 \cdot 9 = 1080$ worker-days total.
• For $30$ workers: $1080 / 30 = 36$ days.

Try it yourself

1. $z$ is jointly proportional to $x$ and $y$. When $x = 3, y = 4$, $z = 60$. Find $z$ when $x = 5, y = 2$.
2. $z$ varies directly with $x$ and inversely with $y$. $z = 30$ at $x = 6, y = 4$. Find $z$ at $x = 4, y = 6$.
3. $24$ workers finish a job in $18$ days. How long for $36$ workers?
4. If $8$ workers can paint $4$ houses in $6$ days, how many days for $12$ workers to paint $9$ houses?
5. $y \propto \dfrac{x}{z}$ with $y = 10$ when $x = 4, z = 2$. Find $y$ at $x = 6, z = 3$.
6. The volume of a gas is inversely proportional to pressure (at fixed temperature). If $V = 2$ L at $P = 3$ atm, find $V$ at $P = 1.5$ atm.
7. The force needed to lift a weight is jointly proportional to the weight and the distance lifted. If $50$ N lifts $5$ kg through $2$ m, what is needed to lift $8$ kg through $3$ m?
8. If $5$ pumps fill a $1000$-L tank in $4$ h, how many pumps for an $1800$-L tank in $3$ h?

Activity / Insight

The gas-law experiment (Boyle's law). Take a syringe with no needle. Block the tip with your finger. Push the plunger gently , the volume of trapped air decreases and you feel resistance increasing. The relationship is $PV = \text{constant}$, exactly the inverse proportion we have studied. This experiment, done by Robert Boyle in $1662$, started the modern science of gases.