Basic doubt surrounding cross multiplication

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dddanny2006: Master | Next Rank: 500 Posts; Posts: 209; Joined: Thu Jan 12, 2012 12:59 pm

by dddanny2006 » Mon Feb 03, 2014 12:23 pm

Twelve identical machines,running continuously at the same constant rate,take 8 days to complete a shipment.How many additional machines,each running at the same constant rate,would be needed to reduce the time required to complete a shipment by 2 days?

A.2 B.3 C.4 D.6 E.9

C

I got to the answer by some uncanny ways,but I wanted to know about a basic concept.

Why cant we setup a cross multiplication proportion below--

12 machines---------------------------8days

1 machine------------------------------How many days?

=8/12 days

But this is wrong,1 machine should take 96 days to complete a shipment.When can/cant setup proportions?

According to the question the rate for 12 machines is 1/8

(1/8)<----------------------->12 machines
? <------------------------>1 machine

=1/96

Hence I came to the conclusion that 1 machine would take 96 days to complete 1 shipment.

Please explain to me the area in bold.Why didnt it work?

I know the answer to the problem.Its 4 more machines
[spoiler][/spoiler][/i]

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Source: Beat The GMAT — Problem Solving | Mon Feb 03, 2014 12:23 pm

GMATGuruNY: GMAT Instructor; Posts: 15539; Joined: Tue May 25, 2010 12:04 pmFollowed by: 1906 members; Location: New York, NY; Thanked: 13060 times; GMAT Score:790

by GMATGuruNY » Mon Feb 03, 2014 12:51 pm

dddanny2006 wrote:Twelve identical machines,running continuously at the same constant rate,take 8 days to complete a shipment.How many additional machines,each running at the same constant rate,would be needed to reduce the time required to complete a shipment by 2 days?

A.2 B.3 C.4 D.6 E.9

C

Approach 1:

(number of machines)(number of days) = (number of machines)(number of days).

The number of machines is INVERSELY PROPORTIONAL to the number of days.
As the number of machines increases, the number of days must decrease, so that the same amount of work is done in each case.

Since 12 machines take 8 days, and we want to know the number of machines required to complete the job in 6 days, we get:
(12)(8) = (x)(6)
x = 16.

Thus, the number of machines must increase from 12 to 16 -- an increase of 4 machines.

The correct answer is C.

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by GMATGuruNY » Mon Feb 03, 2014 12:56 pm

dddanny2006 wrote:Twelve identical machines,running continuously at the same constant rate,take 8 days to complete a shipment.How many additional machines,each running at the same constant rate,would be needed to reduce the time required to complete a shipment by 2 days?

A.2 B.3 C.4 D.6 E.9

C

Approach 2:

Let the rate per machine = 1 unit per day.
Thus, the rate for 12 machines = 12 units per day.
In 8 days, the amount of work produced = r*t = 12*8 = 96 units.
To produce 96 units in 2 fewer days -- in other words, in 6 days -- the required rate per day = w/t = 96/6 = 16 units per day.
To increase the rate from 12 units per day to 16 units per day -- an increase of 4 units per day -- 4 additional machines are required.

The correct answer is C.

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dddanny2006: Master | Next Rank: 500 Posts; Posts: 209; Joined: Thu Jan 12, 2012 12:59 pm

by dddanny2006 » Mon Feb 03, 2014 1:08 pm

Thans for that Mitch.It did make sense.For clarity sake,can you set it up in the form of cross multiplication please?

For example,this way

A<-------------------->B
C<-------------------->D

and then we do cris cross multiplication.

GMATGuruNY wrote:
dddanny2006 wrote:Twelve identical machines,running continuously at the same constant rate,take 8 days to complete a shipment.How many additional machines,each running at the same constant rate,would be needed to reduce the time required to complete a shipment by 2 days?

A.2 B.3 C.4 D.6 E.9

C
Approach 1:

(number of machines)(number of days) = (number of machines)(number of days).

The number of machines is INVERSELY PROPORTIONAL to the number of days.
As the number of machines increases, the number of days must decrease, so that the same amount of work is done in each case.

Since 12 machines take 8 days, and we want to know the number of machines required to complete the job in 6 days, we get:
(12)(8) = (x)(6)
x = 16.

Thus, the number of machines must increase from 12 to 16 -- an increase of 4 machines.

The correct answer is C.

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by GMATGuruNY » Mon Feb 03, 2014 3:00 pm

dddanny2006 wrote:When can/cant setup proportions?[/b][/i]

A direct proportion is defined as follows:
xâ‚�/yâ‚� = xâ‚‚/yâ‚‚.

We can set up a direct proportion for x and y if the following is true:
As the value of x doubles, the value of y also doubles.
For example:
If a machine can produce 2 tires in 5 hours, how many hours would it take the machine to produce 20 tires?
Here, if the machine works for twice the time, it will produce twice the number of tires.
Thus, we can set up the following proportion:
(2 tires)/(5 hours) = (20 tires)/(x hours).
2x = 100
x = 50 hours.

An INVERSE proportion is defined as follows:
xâ‚�yâ‚� = xâ‚‚yâ‚‚.

We can set up an inverse proportion for x and y if the following is true:
As the value of x doubles, the value of y decreases by 1/2.
In the problem that you posted, if the number of machines doubles, the job will take 1/2 the time to complete.
Thus, we can set up an inverse proportion, as I did in my first post above.

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by [email protected] » Mon Feb 03, 2014 3:00 pm

Hi dddanny2006,

While cross-multiplication isn't the easiest way to solve this problem, here's how it would work.

First off, it's worth noting that we're dealing with a "rate" question. Based on the starting info...

Rate x Time = Total Output)
(12 machines) x (8 days) = 96 machine-days of work

To use cross-multiplication, you CAN'T put 12 and 8 "on the same side"; it would have to be written this way:

X/8 = 12/Y

X = days
Y = machines

Now, cross-multiplying, we'd get:

(X)(Y) = 96

The prompt wants us to complete the job in 2 fewer days, so X = 6 days

(6)(Y) = 96

Y = 16 hours

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Rich

Contact Rich at [email protected]

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