I am currently doing aops introduction to algebra book and I can't figure out how to solve joint proportion or some ppl call it joint variation. I don't which values should i substitute for the formula, z=kxy. I don't know which value should i substitute into z for the word problems. I can get through direct proportion and inverse proportion fine but when i got to the word problems in joint variation, i started to struggle. I couldn't figure out which value's to substitute in this word problem:
Five woodchucks would chuck eight pieces of wood in two hours if a woodchuck could chuck wood. How much wood would one woodchuck chuck if one woodchuck would chuck wood for one day?
and why does this next problem have two variables in it's numerator?
Five chickens eat 10 bags of scratch in 20 days. How long does it take 18 chickens to eat 100 bags of scratch?
can someone please try explain this to me in simple math...??
Problem 1
Let $$\begin{align}n &= \text{no. of woodchucks} \\ w &= \text{wood chopped} \\ t &= \text{time taken}\end{align}$$
Case 1: $ n = 5, \ w = 8, \ t = 2$
Case 2: $ n = 1, \ w = ?, \ t = 24$
Logically, $t\propto n $ and $t\propto w $
$$\implies \ t=knw \ \ \ \ .\ .\ .\ (1)$$
where $k$ is a constant.
Substituting values given in case 1, $k = \dfrac{8}{10}$
Now we put the value of k in case 2 and calculate $w$ using $\rm eqn\ 1$
We get $w = \dfrac{24\times 8}{10} = 19.2$ pieces of wood
Problem 2
Let $$\begin{align}c &= \text{no. of chicken} \\ b &= \text{bags} \\ d &= \text{days}\end{align}$$
Case 1: $ c = 5, \ b = 10, \ d = 20$
Case 2: $ c = 18, \ b = 100, \ d = ?$
Logically, $d\propto b $ and $d\propto \dfrac{1}{c} $
$$\implies \ d=\dfrac{kb}{c} \ \ \ \ .\ .\ .\ (1)$$
where $k$ is a constant.
Substituting values given in case 1, $k = 10$
Now we put the value of $k$ in case 2 and calculate $d$ using $\rm eqn\ 1$
We get $d = \dfrac{100\times 10}{18} = \dfrac{500}{9}$ days