# A sample of 144 firecrackers contained 8 duds. How many duds would you expect in a sample of 2016...

## Question:

A sample of 144 firecrackers contained 8 duds. How many duds would you expect in a sample of 2016 firecracker?

## Linear Models & Real Life Problems

The majority of real life problems might be modeled using linear equations. Since we often deal with direct proportionality between two measures, we might write a linear equation y = mx relating the number of objects y to the number of objects x by the proportionality constant m.

Let's assume that we are modelling the dependence of the number o firecrackers, f, on the number of duds, d. Since this problem is considering direct proportionality, we may state that the linear equation for this model would be f(d) = md.

Given initial conditions of f = 144 and d = 8, we will find the proportionality constant m:

{eq}m = \frac{f(d)}{d} = \frac{144}{8} = 18. {/eq}

That said, the linear model describing f(d) would be f(d) = 18d.

We may solve this model for the given number of firecrackers, f = 2016:

{eq}d = \frac{f(d)}{m} = \frac{2016}{18} = 112. {/eq}

The designed model states that a sample of 2016 firecrackers would contain a total of {eq}\boxed{112} {/eq} duds. 