Question

Danica and Darrelle have developed a lucrative business selling handmade silver bracelets. Darrelle makes the more complex bracelets and Danica makes the simpler ones. It takes Darrelle 4 hours to make 3 bracelets and Danica can make 5 bracelets every 2 hours.

a. Convert this scenario into linear equation(s); show both the standard form and the slope-intercept form of your equation(s).
c. Each bracelet Darrelle sells makes $10.50 in profit. Each bracelet Danica sells makes $3.50 in profit. If Darrelle and Danica work the same amount of time, explain each of the steps you will use and then calculate how many hours will it take for the whole business to make at least $500 profit?

Answer 1

Solution-

(a)

Let Darrelle and Danica makes bracelets for t hours.

Now,

Since Darrelle takes 4 hours to make 3 bracelets.

So, Number of bracelet made by Darrelle in 1 hour

= 3/4 = 0.75

And number bracelets made by Darrelle in t hours

n= 0.75t .....(1)

Similarly ,

Since Danica can make 5 bracelets every 2 hours.

So, number of bracelet made by Danica in 1 hours

= 5/2 = 2.5

And number of bracelets made by Danica in t hours

n= 2.5t ...(2)

Hence, the given scenario can be convert into linear equations in slope-intercept form as

Number of bracelets made by Darrelle in t hours

=0.85t and

Number of bracelets made by Danica in t hours

n= 2.5t .

(c)

Since Each bracelet Darrelle sells makes $10.50 in profit.

So,Total profit in dollars by Darrelle in t hours

= 0.75t ×10.50 = 7.875t

(Using equation 1)

Similarly,

Since Each bracelet Danica sells makes $3.50 in profit.

So, Total profit in dollars by Danica in t hours

= 2.5t×3.5 = 8.75t

(Using equation 2)

Now, If Darrelle and Danica work the same amount of time then number of hours will required for the whole business to make at least $500 profit is calculated as

7.875t + 8.75t ≥ 500

16.625t ≥ 500

t ≥ 500/16.625

t ≥ 30.075 hours

t = 31

Now, putting t = 31 in equation(1) we get

Darrelle make n= 0.75×31 = 23.25 brcelts

But fractional bracelets not possible.

So, let us put t = 32 in equation (1) to get

Number of bracelets made by Darrelle in 32 hours =

n = 0.75×32 = 24

Also putting t = 32 in equation (2) to get

Number of bracelets made by Danica in 32 hours

= n =2.5×32 = 80

So, here both are whole number which is possible.

Hence, minimum 32 hours required to make the business to reach atleast 500 dollars .