Two machines, Y and Z, work at constant rates producing identical item

Hey awsome bunuel, i went a bit long in deducing the actual relation between the Rates.

Now atleast i know what to think when i see such question and dont need to go on the practice pad .

rate z/rate y=2/3
time z/time y=3/2
3/2*9=13 1/2 minutes

Y- 3 items
Z- 2 items
Let the batch be of 27 items, therefore Y would take 1 min to produce 3 units and 9 mins to produce 27 items. (9 items taken is consistent with the time mentioned in the question for Y to produce the batch of items)
Time required by X to produce 27 items is 27/2 (as in 1 min X produces 2 units) = 13.5 mins or 13 1/2 mins.
E

Given

• Two machines, Y and Z, work at constant rates producing identical items.
• Machine Y produces 3 items at the same time Machine Z produces 2 items.
• Machine Y takes 9 minutes to produce a batch of items.

To Find

• Minutes it takes for machine Z to produce the same number of items.

Approach and Working Out

• Speed ratio –

o \(\frac{Y}{Z }\) = \(\frac{3}{2 }\)

• Time ratio –

o \(\frac{Y}{Z }\) = \(\frac{2}{3 }\)
• Time for Y = 9 minutes.

• Time for Z = \(\frac{3}{2 }\)× 9 minutes

o = 13 ½ minutes

Correct Answer: Option E

Given: Two machines, Y and Z, work at constant rates producing identical items. Machine Y produces 3 items in the same time Machine Z produces 2 items.
Asked: If machine Y takes 9 minutes to produce a batch of items, how many minutes does it take for machine Z to produce the same number of items?

Machine Y takes time to produce a batch of items = 9 mins
Machine X will take time to produce a batch of items = 9*3/2 = 27/2 = 13 1/2

IMO E

__ | rate | time | work |
Y)__ 1.5r___ 9____ 1
Z)___ r___ 13.5___ 1

Working behind the matrix table:
It is given that Y produces 3 items in the same time Z produces 2
=> The rate of Y is 1.5 times faster than that of Z

I've assumed the batch size as 1 for simplicity.
In the stem, we're given:
1.5r*9=1
=> r= 1/13.5

Plugging the value of r in (Z):
1/r = 13.5

Use W= nRT formula

R1*T1 = R2*T2

3*9 = 2*T2

T2= 13.5

|W |R |T |
Y|27|3 |9 |
Z|27|2 |X |

T = W/R = 27/2
T = 13.5 Min

Work = Rate * Time
Machine Y produces 3 items in the same amount of time in which machine Z produces 2 items.
Let time be t, and Rate of Y be y and rate of Z be z.
First step is to get the relation in the rates.

For Y : Work = Rate * Time
3items = yt
For Z : Work = Rate * Time
2 items = zt
Using t common in above we get:
3/y = 2/z
or
3z = 2y. We got a equations of Y and Z rates.

Now, machine Y takes 9 minutes to produce a batch of items (lets call it B)

For Y : Work = Rate * Time
B = y.9
minutes taken by machine Z (lets call it T)to produce the same number B
For Z : Work = Rate * Time
B = zT
Equating above

9y = zT.
We already know 3z = 2y. Putting y = 3z/2
9*3z/2 = zT
T = 13.5 minutes.

x=time Z and Y take
B=batch of items

\(\frac{1}{y}=\frac{3}{x}\) and \(\frac{1}{z}=\frac{2}{x}\)

\(\frac{1}{z}=\frac{2}{x}\)
x=2z

\(\frac{1}{y}=\frac{3}{x}\)
\(\frac{1}{y}=\frac{3}{2z}\)
\(y=\frac{2z}{3}\)

now y takes 9 minutes in a batch

\(\frac{1}{y}=\frac{B}{9}\)
replace the y with 2z/3

\(\frac{3}{2z}=\frac{B}{9}\) multiply by \(\frac{2}{3}\) to get

\(\frac{1}{z}=\frac{2B}{27}\)

so Z takes 27 min in producing 2 Batch, so producing 1 batch takes 27/2=13,5