Zeno takes 20 min longer to decorate a dozen cupcakes than it takes Lia. When they work together it takes them 10.5 min. How long would each take to do a job alone?

Answer 1

Erin, Raymond’s answer is correct, although his verification calculation are confused and wrong ( at the top he says, correctly, that Zeno’s time for a dozen is 35 minutes, but near the bottom he says the rate is one dozen in 1/35 of a minute. ) 

Zeno takes 20 min longer to decorate a dozen cupcakes than it takes Lia. When they work together it takes them 10.5 min. How long would each take to do a job alone?

As with any word problem we have to name all the quantities clearly.

Let Rz be the rate that Zeno can make a dozen cupcakes, in dozens per minute

Let Rl be the rate that Lia can make a dozen 

cupcakes, in dozens per minute

Also you need to know the fundamental formula, as Raymond said,

Work = rate x time

From this, we can calculate expressions for time:

Let Tz be the time it takes for Zeno to decorate 1 dozen cupcakes, in minutes

Let Tl be the time it takes for Lia to decorate 1 dozen cupcakes, in minutes

1 = Rz * Tz (one dozen takes Zeno Tz minutes, working at Rz dozens per minute)

1 = Rl * Tl (one dozen takes Lia Tl minutes, working at Rl dozens per minute.)

From these we can derive

Rz = 1/Tz

Rl = 1/Tl

Now translate  the other expressions into algebra:

Tz = Tl + 20 (the time it takes Zeno to decorate a dozen is 20 minutes longer than the time Lia takes to do a dozen).

Also,

1 = (Rz + Rl) * 10.5 (the time it takes both of them to decorate is 10.5 minutes

1 = (1/Tz + 1/Tl) * 10.5

But we have another expression for Tz, we will substitute it in

1 = (1/(Tl + 20) + 1/Tl)* 10.5

1/10.5 = (1/(Tl + 20) + 1/Tl)

Put over common denominator

1/10.5 = (Tl + (Tl + 20)) / (Tl*(Tl+20))

1/10.5 = (2Tl + 20)/ (Tl*(Tl+20))

Remove the denominators by cross multiplying

(Tl*(Tl+20)) = 10.5*(2Tl + 20)

Multiply out

Tl^2 + 20 Tl = 21 Tl + 210

Combine like terms

Tl^2 - Tl - 210 = 0

Factor

(Tl -15) (Tl + 14) = 0

Ignore spurious negative answer

Tl = 15

From this, compute Tz

Tz = Tl + 20 = 35

Now to verify, we need to plug these values into all the equations.

The rates are

Rz = 1/Tz = 1/35

Rl = 1/Tl = 1/15

And now the combined effort

1 = (Rz + Rl) * 10.5

= (1/35 + 1/15) * 10.5

= (3/105 + 7/105) * 10.5

= (10/105) * 10.5 = 1

Answer 2

Z can do a dozen in 35 minutes. L can do a dozen in 15 minutes

Z=20+L that's time = d/r=1/r d=rt distance (total production) = rate of speed times time.

Together they take 10.5 minutes = time "distance" or total production= 12 cupcakes=1 dozen

Z=time to do 12 cupcakes for Z=1/r=(20+L)=1/r L=time to do 12 cupcakes for L=1/r₂

12 or 1 dozen = rt = speed times time = r(10.5)+r₂(10.5)=(r+r₂)10.5 r+r₂=1/10.5=1/L+1/Z=1/L+1/(20+L)

Multiply both (10.5)L(20+L) to get

L(20+L)=10.5(20+L)+10.5(L)

20L+L²=10.5(20)+10.5L+10.5L = 210+21L

L²-L-210=0 Use quadratic equation, ignore negative square root

L=1/2 + sqr841/2 = 1/2 + 29/2 = 30/2 = 15 minutes

Z=L+20=35 minutes

That's the answer L=15, Z=35

What follows isn't needed, just additional calculations, to make sure it's correct.

Z can do a dozen cupcakes in 1/35 minutes = 12/35=.343 cupcakes per minute

L can do a dozen in 12/15=4/5=.8 cupcakes per minute

Together they can do a dozen in (12/35 cupcakes/minute) x 10.5 + (4/5)cupcakes/minute) x 10.5

= 10.5 (12/35+4/5) = 10.5(12/35+28/35)=10.5(40/35)=2.1(40/7)=.3(40)=12cupcakes

L can do 4/5 per minute To do 12 requires 12/(4/5)=60/4=30/2=15 minutes

Z can do 12/35 per minute. To do 12 requires 12/(12/35) = 35 minutes