## Machines $$X$$ and $$Y$$ produced identical bottles at different constant rates. Machine $$X,$$ operating alone for

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### Machines $$X$$ and $$Y$$ produced identical bottles at different constant rates. Machine $$X,$$ operating alone for

by Gmat_mission » Sun Nov 28, 2021 6:45 am

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Machines $$X$$ and $$Y$$ produced identical bottles at different constant rates. Machine $$X,$$ operating alone for $$4$$ hours, filled part of a production lot; then Machine $$Y,$$ operating alone for $$3$$ hours, filled the rest of this lot. How many hours would it have taken Machine $$X$$ operating alone to fill the entire production lot?

(1) Machine $$X$$ produced $$30$$ bottles per minute.
(2) Machine $$X$$ produced twice as many bottles in $$4$$ hours as Machine $$Y$$ produced in $$3$$ hours.

Source: GMAT Paper Tests

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### Re: Machines $$X$$ and $$Y$$ produced identical bottles at different constant rates. Machine $$X,$$ operating alone for

by [email protected] » Sun Nov 28, 2021 2:58 pm

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## Global Stats

Gmat_mission wrote:
Sun Nov 28, 2021 6:45 am
Machines $$X$$ and $$Y$$ produced identical bottles at different constant rates. Machine $$X,$$ operating alone for $$4$$ hours, filled part of a production lot; then Machine $$Y,$$ operating alone for $$3$$ hours, filled the rest of this lot. How many hours would it have taken Machine $$X$$ operating alone to fill the entire production lot?

(1) Machine $$X$$ produced $$30$$ bottles per minute.
(2) Machine $$X$$ produced twice as many bottles in $$4$$ hours as Machine $$Y$$ produced in $$3$$ hours.

Source: GMAT Paper Tests
Given: Machines X and Y produced identical bottles at different constant rates. Machine X, operating alone for 4 hours, filled part of a production lot; then Machine Y, operating alone for 3 hours, filled the rest of this lot.
Let x = Machine X's RATE (in terms of the fraction of the job completed PER HOUR)
Let y = Machine Y's RATE (in terms of the fraction of the job completed PER HOUR)

So, in 4 hours, the portion of the job completed by Machine X = 4x
Likewise, in 3 hours, the portion of the job completed by Machine Y = 3y

Since the entire job is completed after both machines perform their individual tasks, we can write: 4x + 3y = 1 (1 represents the entire job completed)

Target question: How many hours would it have taken Machine X operating alone to fill the entire production lot?

Statement 1: Machine X produced 30 bottles per minute.
Since we're given no information about Machine Y, it is impossible to answer the target question with certainty
Statement 1 is NOT SUFFICIENT

Statement 2: Machine X produced twice as many bottles in 4 hours as Machine Y produced in 3 hours.
In 4 hours, the portion of the job completed by Machine X = 4x
In 3 hours, the portion of the job completed by Machine Y = 3y
From statement 2, we can write: 4x = (2)(3y)
Simplified to get: 4x = 6y

We now have the following system of equations:
4x + 3y = 1
4x = 6y

Since we COULD solve this system of equations for x and for y, we COULD determine Machine X's RATE, which means we COULD determine how many hours it would have taken machine X to fill the entire production lot on its own.
Of course, we would never waste valuable time on test day performing such tedious calculations. We need only determine that we COULD answer the target question.
Since we COULD answer the target question with certainty, the combined statements are SUFFICIENT