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In a particular machine, there are 2 gears that interlock; One gear is 06 Nov 2020, 08:58
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PrashantPonde
Each gear has life to 6,000,000,000 revolutions. As both gears are interlocked, smaller gear will have more revolutions and will wear out before the larger one.

(1) The diameter of the larger gear is twice the diameter of smaller gear.
INSUFFICIENT: This means - the circumference of the larger gear is twice that of smaller gear. I.e. 1 revolution of larger gear = smaller gear rotates two times.
Even if we know relative ratio for diameters, we don’t know what rate at they are revolving per minute.
- If the gears are rotating at extreme high speed, they will wear out soon -> hence difference of their life (in minutes) will be smaller.
- If the gears are rotating at extreme slow speed, they will last lot longer -> hence difference of their life (in minutes) will be much larger.
Hence we cannot conclude how many days the gears will last long.

(2) The smaller gear revolves 600 times per minute.
INSUFFICIENT: This information is clearly insufficient. This tells the revolution rate only for smaller gear. No info about larger gear (speed or relative diameter ratio)

Combining (1) & (2)
SUFFICIENT: As we know smaller gear has 600 RPM and 1/2 the diameter than larger one -> the larger gear rotate at half the speed i.e. 300 RPM and hence we can find out how many days they can last long.

We can stop at this point (no need to calculate further during actual exam) but just for sake of curiosity lets calculate further:
No of days larger gear lasts longer than smaller gear = \(\frac{6,000,000,000}{24*60*60 minutes} * (\frac{1}{300}-\frac{1}{600}) = 115.74 days.\)

Hence choice(C) is the answer.
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@Bunuel/@VeritasKarishma/@Chetan2u (any other experts),

Can anyone please let me know, how to derive the above highlighted formula?

Thanks!
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In a particular machine, there are 2 gears that interlock; One gear is 22 Dec 2020, 21:48
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PrashantPonde
Each gear has life to 6,000,000,000 revolutions. As both gears are interlocked, smaller gear will have more revolutions and will wear out before the larger one.

(1) The diameter of the larger gear is twice the diameter of smaller gear.
INSUFFICIENT: This means - the circumference of the larger gear is twice that of smaller gear. I.e. 1 revolution of larger gear = smaller gear rotates two times.
Even if we know relative ratio for diameters, we don’t know what rate at they are revolving per minute.
- If the gears are rotating at extreme high speed, they will wear out soon -> hence difference of their life (in minutes) will be smaller.
- If the gears are rotating at extreme slow speed, they will last lot longer -> hence difference of their life (in minutes) will be much larger.
Hence we cannot conclude how many days the gears will last long.

(2) The smaller gear revolves 600 times per minute.
INSUFFICIENT: This information is clearly insufficient. This tells the revolution rate only for smaller gear. No info about larger gear (speed or relative diameter ratio)

Combining (1) & (2)
SUFFICIENT: As we know smaller gear has 600 RPM and 1/2 the diameter than larger one -> the larger gear rotate at half the speed i.e. 300 RPM and hence we can find out how many days they can last long.

We can stop at this point (no need to calculate further during actual exam) but just for sake of curiosity lets calculate further:
No of days larger gear lasts longer than smaller gear = \(\frac{6,000,000,000}{24*60*60 minutes} * (\frac{1}{300}-\frac{1}{600}) = 115.74 days.\)

Hence choice(C) is the answer.
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Can somebody explain the leap in logic here?
Why does twice the circumference equate to a halving of the speed?

Bunuel?
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Sneha2021
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In a particular machine, there are 2 gears that interlock; One gear is 26 Oct 2021, 10:29
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CEdward

In case of revolutions or circular motion, 1 rev = 2πr (time taken for 1 revolution)
Time is also directly proportional with diameter (T ∝ D), So if diameter is doubled, time taken will be doubled. A larger wheel will take more time to complete a revolution (2πr) than the smaller wheel.

Time is inversely proportional to speed for constant work (T = 1/S)

If time is doubled, speed is half (speed is inversely proportional to D and T)
From statement 1 & statement 2 - If speed of smaller gear is 600 RPM, then speed of larger will be half i.e 300 RPM

Also question gives a hint "the larger gear is guaranteed to last how many days longer than the smaller gear" means larger gear will take more time and therefore rotate at a slower speed than the smaller gear

CEdward
PrashantPonde
Each gear has life to 6,000,000,000 revolutions. As both gears are interlocked, smaller gear will have more revolutions and will wear out before the larger one.

(1) The diameter of the larger gear is twice the diameter of smaller gear.
INSUFFICIENT: This means - the circumference of the larger gear is twice that of smaller gear. I.e. 1 revolution of larger gear = smaller gear rotates two times.
Even if we know relative ratio for diameters, we don’t know what rate at they are revolving per minute.
- If the gears are rotating at extreme high speed, they will wear out soon -> hence difference of their life (in minutes) will be smaller.
- If the gears are rotating at extreme slow speed, they will last lot longer -> hence difference of their life (in minutes) will be much larger.
Hence we cannot conclude how many days the gears will last long.

(2) The smaller gear revolves 600 times per minute.
INSUFFICIENT: This information is clearly insufficient. This tells the revolution rate only for smaller gear. No info about larger gear (speed or relative diameter ratio)

Combining (1) & (2)
SUFFICIENT: As we know smaller gear has 600 RPM and 1/2 the diameter than larger one -> the larger gear rotate at half the speed i.e. 300 RPM and hence we can find out how many days they can last long.

We can stop at this point (no need to calculate further during actual exam) but just for sake of curiosity lets calculate further:
No of days larger gear lasts longer than smaller gear = \(\frac{6,000,000,000}{24*60*60 minutes} * (\frac{1}{300}-\frac{1}{600}) = 115.74 days.\)

Hence choice(C) is the answer.

Can somebody explain the leap in logic here?
Why does twice the circumference equate to a halving of the speed?

Bunuel?
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In a particular machine, there are 2 gears that interlock; One gear is 17 Aug 2022, 22:54
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ThatDudeKnows how is speed related to circumference here and why does it lead to halving? I easily think speeds can be different for two gears as well even though one has half the diameter as another and they are connected avigutman
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In a particular machine, there are 2 gears that interlock; One gear is 18 Aug 2022, 13:36
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Elite097
ThatDudeKnows how is speed related to circumference here and why does it lead to halving? I easily think speeds can be different for two gears as well even though one has half the diameter as another and they are connected avigutman
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Elite097 Measuring the speed of a rotating gear is tricky because it's not actually moving anywhere - it stays in place.

The typical way to measure the speed of rotation, therefore, is using units of rpm (revolutions per minute): measuring how many full circles (360 degrees) the gear does in 1 minute. But, what is a "full circle", or, revolution? If you mark a point on the top of the circumference of the gear, a full revolution is complete when that point comes back to the top of the gear. In other words, the whole circumference of the gear would have passed through the top. For example: The second hand of a conventional analog clock rotates at 1 rpm.

Another way to measure the speed of rotation is to measure the movement of the point you marked at the top of the gear. That point will travel the full circumference of the gear in every revolution. So, if a gear with diameter d rotates at a speed of 6 rpm for example, the point you marked will travel a distance of 6*(pi*d) per minute. What if you double the circumference? How would that impact the point's speed, if you keep the gear's rpm the same? Conversely, if you want the point's speed to remain unchanged, what must you do to the gear's rpm when you double its circumference?

In this question, we have 2 gears that interlock. What does that mean? Here's a visualization. If one gear has twice the circumference of the other, it must rotate half as many times per minute. Why? Well, if you mark a little dot every inch along each of the circumferences, the big gear will have twice as many such dots. However, since the gears interlock, the dots on the two gears must be moving at the same speed. If your gear has twice as many dots, and those dots are moving at the same speed as the dots on my small gear, then your dots have to travel twice the distance and will take twice as long to complete their journey of a full revolution. Therefore, your gear must be doing half as many revolutions per minute as my gear.
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In a particular machine, there are 2 gears that interlock; One gear is 26 Nov 2024, 16:57
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