Question

x is directly proportional to y and inversely proportional to z. if x= 1/2 when y = 3/4 and z=2/3, find x when y= 7/8 and z = 7/9

Answer 1

Answer:

1/2

Step-by-step explanation:

x is directly proportional to y (this means y will go on top because of the directly part) and inversely proportional to z (this means z will go on bottom due to the inversely part).

There is a constant k such that:

$x=k \cdot \frac{y}{z}$.

Contant means it will never change.  It will not care what (x,y,z) you use, it will remain the same.

We will use the first point to find k and then that k will still be there no matter what (x,y,z) they give you.

We have (1/2 , 3/4 , 2/3) is on our graph of the equation:

$x=k \cdot \frac{y}{z}$.

Insert the numbers:

$\frac{1}{2}=k \cdot \frac{\frac{3}{4}}{\frac{2}{3}}$

Multiply both sides by $\frac{2}{3}$:

$\frac{1}{2}\cdot \frac{2}{3}=k \cdot \frac{3}{4}$

Simplify left hand side:

$\frac{1}{3}=k \cdot \frac{3}{4}$

Multiply both sides by 4:

$\frac{4}{3}=k \cdot 3$

Multiply both sides by 1/3  (or you can say divide by 3):

$\frac{4}{9}=k$

So k=4/9 no matter the (x,y,z).

$x=\frac{4}{9} \cdot \frac{y}{z}$

We are asked to find x given y=7/8 and z=7/9.

Input these numbers:

$x=\frac{4}{9} \cdot \frac{\frac{7}{8}}{\frac{7}{9}}$

Change the division to multiplication:

$x=\frac{4}{9} \cdot \frac{7}{8} \cdot \frac{9}{7}$

I see a 7 on top and bottom that I can cancel:

$x=\frac{4}{9} \cdot \frac{1}{8} \cdot \frac{9}{1}$

I see a 9 on top and bottom that I can cancel:

$x=\frac{4}{1} \cdot \frac{1}{8} \cdot \frac{1}{1}$

Let's go ahead and multiply and reduce more later if we can.

Multiply straight across on top.

Multiply straight across on bottom.

$x=\frac{4}{8}$

Divide top and bottom by 4:

$x=\frac{1}{2}$