Question

Determine whether quantities vary directly or inversely and find the constant of variation. 16 factory workers made 40 pieces in a certain amount of time. How many workers are needed to make (a) 20, (b)25 and (c)100 pieces in the same amount of time, assuming each worker works at the same rate?

Answer 1

Part A)
`8\ workers`

Part B)
`10\ workers`

Part C)
`40\ workers`

Explanation:

we know that

A relationship between two variables, x, and y, represent a proportional variation if it can be expressed in the form
`k=(y)/(x)` or
`y=kx`

Let

x -----> the number of workers

y ----> the number of pieces made

In this problem

The relation between the variables x and y represent a direct variation (proportional variation)

so

For x=16, y=40

Find the value of the constant of proportionality k

`k=(y)/(x)`

substitute the values

`k=(y)/(x)`

`k=(40)/(16)\\k=2.5\ (pieces)/(worker)`

The linear equation is equal to

`y=2.5x`

Part A) How many workers are needed to make 20 pieces

For y=20 pieces

substitute in the linear equation

`20=2.5x`

`x=8\ workers`

Part B) How many workers are needed to make 25 pieces

For y=25 pieces

substitute in the linear equation

`25=2.5x`

`x=10\ workers`

Part C) How many workers are needed to make 100 pieces

For y=100 pieces

substitute in the linear equation

`100=2.5x`

`x=40\ workers`