Gears X and Y are interconnected within and power

1 min:
4 inches-> 3000 rev
1 inch-> 3000* 4 rev
10 inch -> 3000*4/10 = 1200 rev

4 inch:
1 min -> 3000 rev
1 sec-> 3000/60 = 50 rev

10 inch:
1 min -> 1200 rev
1 sec -> 1200/60 = 20 rev

Diff: 50-20 = 30

Ans: "E"

Let y be the number of revs / sec by gear Y. The assumption is revs are inversely proportional to the diameter of the wheel. ---------(1)
3000 revs/sec = 50 revs / sec
From (1) 50 * 4 = y * 10
y = 20
Difference = 50 - 20 = 30 revs / sec

Doesn't this state that the diameters of X and Y are 4 and 10, respectively? So shouldn't we be using 2 and 5 for the radii?

First convert the frequency from revolutions per minute to revolutions per second.

\(\frac{3000}{60 seconds}=\frac{50}{second}\)

Since the diameter is directly proportional to the length of the circumference, and the circumference is inversely proportional to the frequency, the diameter is inversely proportional to the frequency

\(\frac{1}{4}:50=\frac{1}{10}:X \Rightarrow X=20\)

The difference of the two frequencies is \(50-20=30\)

Answer E

Simple question. I dont think this can be 700 questions especially when compared to several 700 levels questions that I saw in GMAT club.

the solution

4 inches gear makes- 3000
10 inches gear makes- X

apply chain rule.

you will 4X3000/10 = 1200 (number of revs that 10 inch gear makes)


Finally divide 3000 and 1200 by 60, you will get 50 and 20 respectively answer= 30.


The only that I doubt in this question is, Is there a valid reason to think that a larger gear makes fewer revolutions.

Hi shriramvelamuri,

This prompt is based on multi-shape Geometry, which is a category that consistently ranks at the 'higher end' of the difficulty spectrum on the GMAT. If you think that it's easy, then that's great, but most Test Takers would NOT find this question to be so simple.

The prompt also involves rates; since the larger gear has a larger circumference, it has to turn through "more space" to complete a full rotation. By definition it will have fewer revolutions because its rate is based on the rate of the smaller gear (which has a smaller circumference and has to turn through "less space" to complete a full rotation).

GMAT assassins aren't born, they're made,
Rich

Note that we are talking about the RATIO of the radii. The ratio of the radii will be the same as the ratio of the diameters. If the diameter of one circle is twice the diameter of another, then its radius will also be twice the radius of the other.

what is the chain rule here?I am messing up the calculation when i use the normal unitary method to find the number of revolutions made by 10 inch wheel and ending up with (3000*10)/4.

You will multiply 3000 by 4/10 because a larger gear will make fewer revolutions. Since you want to reduce 3000, you will multiply it by 4/10 (a number smaller than 1).

tricky one!
X makes 3000/60 -> 50 revolutions
Circumference is 2pi * r => circumference for gear X is 8pi and for Y is 20pi
now we know that gear x makes 50 revolutions per second, or a total distance of 8pi*50, and this is the total distance gear y makes. to find how many revolutions gear y makes, divide 8pi*50/20pi -> simplify pi's, 10,s and 2 with 4, and get 20. Gear Y makes 20 revolutions. Difference is thus 50-20=30.

P.S. the proportion way of solving the problem is very interesting!!!

gear X makes 50 rps
ratio of gear X to gear Y circumferences=4⫪/10⫪= 2/5
(2/5)(50)=20
50-20=30 rps difference between gear X and Y

My reasoning if it helps anyone:

The stem says the two gears are interconnected. This means that the distance covered by both gears will be equal to each other.

Gear 1 travels 4pi units every revolution and rotates at \(\frac{50 revolutions}{second}\)
Gear 2 travels 10pi units every revolution and rotates a \(\frac{x revolutions}{second}\)

We can divide out pi and set the equation as such:

\(4 * 50 = 10 *x\)
\(x = 20\)
Thus difference between the two speeds can be solved \(50 - 20 = 30\)

both the gears would cover the same distance.
the smaller one will cover perimeter of gear*no of revolutions = 4pi*3000
the larger one will also cover 12000pi hence 10pi*y=12000
y=1200
difference = 1800
1800/60 = 30

another way to do this is to remember that for two interconnected gears ratio of their diameter = reciprocal ratio of their revolutions per time unit as if the diameter is greater, a lesser number of revolutions will be completed in the same time if the gears are interconnected.
4/10=y/3000
y=1200
diff = 1800/60 = 30