# If a varies jointly as b and c, and a = -10, when b = 12 and c = 2, how do you find a when b = 8 and c = ½?

Apr 27, 2017

$a = - \frac{5}{3} = - 1 \frac{2}{3}$

#### Explanation:

There are two ways to find the answer, finding the constant of proportionality or just setting up a proportion.

Finding the constant of proportionality :

$a = k b c$

Plug in the first set of numbers to find $k$:

$- 10 = k \cdot 12 \cdot 2$

$- 10 = 24 k$

$k = - \frac{10}{24} = - \frac{5}{12}$

Now use $k$ to find $a$:

$a = - \frac{5}{12} \cdot \frac{8}{1} \cdot \frac{1}{2} = - \frac{40}{24} = - \frac{5}{3} = - 1 \frac{2}{3}$

Using proportions:

$\frac{{a}_{1}}{{a}_{2}} = \frac{{b}_{1} {c}_{1}}{{b}_{2} {c}_{2}}$

$\frac{- 10}{{a}_{2}} = \frac{12 \cdot 2}{\frac{8}{1} \cdot \frac{1}{2}}$

$\frac{- 10}{{a}_{2}} = \frac{24}{4}$

Use the cross product $\frac{a}{b} = \frac{c}{d}$: $\text{ } a \cdot d = b \cdot c$

$- 10 \cdot 4 = 24 \cdot {a}_{2}$

$- 40 = 24 \cdot {a}_{2}$

${a}_{2} = - \frac{40}{24} = - \frac{5}{3} = - 1 \frac{2}{3}$