A firm that wants to make $q$ units should make them as cheaply as possible. The Least-Cost Mix of Inputs said "spend so the last dollar on each input buys the same output." With isoquants and Optimization we can find the exact mix.
Key idea
Minimize $wL + rK$ subject to $f(K, L) = q$. At an interior solution $$MRTS = \frac{MP_L}{MP_K} = \frac{w}{r}, \quad\text{equivalently}\quad \frac{MP_L}{w} = \frac{MP_K}{r},$$ together with the isoquant $f(K, L) = q$.
Why it works
An isocost line $wL + rK = C$ shows every mix costing $C$; its slope is $-w/r$. Slide isocost lines toward the origin until the last one still touches the isoquant: there they are tangent, so the isoquant's slope $-MRTS$ equals $-w/r$. In per-dollar terms: if $MP_L/w > MP_K/r$, moving a dollar from capital to labor adds output, so the firm could cut cost and keep output the same. Only equal output per dollar leaves nothing to gain. This is the producer's version of the consumer's tangency in Minimizing Spending for a Target Utility.
Worked example: Square-root technology
$Q = \sqrt{KL}$, $w = 4$, $r = 1$, $q = 20$.
- $MRTS = K/L = w/r = 4$, so $K = 4L$.
- Isoquant: $\sqrt{4L\cdot L} = 2L = 20$, so $L = 10$ and $K = 40$.
- Cost: $4(10) + 1(40) = 80$. Labor is dearer, so the firm uses four machines per worker.
Answer: $L = 10$, $K = 40$, cost \$80.
Worked example: Perfect substitutes
$Q = 5L + K$, $w = 5$, $r = 6$, $q = 15$.
- Output per dollar: labor $5/5 = 1$, capital $1/6$.
- Use only labor: $L = 15/5 = 3$, $K = 0$, cost \$15.
- A corner: the MRTS of 5 never equals $w/r = 5/6$.
Answer: $L = 3$, cost \$15.
Worked example: Fixed proportions
$Q = \min\{K, L/2\}$, $w = 5$, $r = 25$, $q = 20$.
- Use exactly what each unit needs: $K = 20$, $L = 40$.
- Cost: $25(20) + 5(40) = 700$. Prices don't change the mix, only the bill.
Answer: \$700.
The same tangency can be found with a Lagrange multiplier: minimize $wL + rK - \mu\,(f(K, L) - q)$. The conditions $w = \mu MP_L$ and $r = \mu MP_K$ give $MP_L/MP_K = w/r$, and the multiplier $\mu$ turns out to be marginal cost: the extra cost of one more unit of output.
Common mistake
Setting the MRTS equal to $r/w$, or spending everything on whichever input has the lower price. The tangency compares labor's marginal product per dollar with capital's, so the wage goes with labor: $MP_L/w = MP_K/r$, or $MRTS = w/r$. And a cheap input is not a bargain if it adds little output; what matters is output per dollar. Check: if labor gets dearer, your answer should use relatively more capital.