Please could somebody give me some advice.
Cog A has 8 teeth and connects with cog B which has 10 teeth. A ratio of 8:10 or 2:5 at its lowest.
If A completes 10 turns, how many turns does B complete?
I get this to be 8 at a ratio of 8:10 or 1:0.8
Is this correct?
What is the least number of turns that A can complete so that B also completes an exact number of turns?
I get this to be 5 at a ratio of 5:4
Is this correct?
B completes 30 turns. On how many occations will both A and B have been back in their starting position at the same time?
I get this to be 6 at a ratio of 2:5
Is this correct?
Any comments gratefully received.
Your calculations are mostly correct, but let's go through each question and see how to arrive at the answers step by step.
1. If Cog A completes 10 turns, we can determine the number of turns completed by Cog B using the ratio of their teeth. Since Cog A has 8 teeth and Cog B has 10 teeth, the ratio is 8:10. To find the number of turns completed by Cog B, we can set up a proportion:
8/10 = x/10
Cross-multiplying, we have:
8 * 10 = 10 * x
80 = 10x
Dividing both sides by 10, we get x = 8.
So if Cog A completes 10 turns, Cog B will complete 8 turns.
2. To find the least number of turns that Cog A can complete so that Cog B also completes an exact number of turns, we need to find the lowest common multiple (LCM) of their tooth numbers. Cog A has 8 teeth, and Cog B has 10 teeth.
To find the LCM, we can list the multiples of each number until we find a common multiple:
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, ...
Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, ...
From the lists, we can see that the lowest common multiple is 40. So Cog A needs to complete 40 turns for Cog B to complete an exact number of turns.
3. To find the number of occasions when both Cog A and Cog B have been back in their starting positions at the same time, we need to find the least common multiple (LCM) of the number of turns completed by each cog.
Cog A completes turns that are multiples of 10 (10, 20, 30, ...), and Cog B completes turns that are multiples of 8 (8, 16, 24, ...). To find the LCM of 10 and 8, we can list their multiples until we find a common multiple:
Multiples of 10: 10, 20, 30, 40, 50, 60, 70, 80, ...
Multiples of 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, ...
From the lists, we can see that the lowest common multiple is 40. So Cog A and Cog B will be back in their starting positions at the same time every 40 turns.
Since B completes 30 turns, we need to find how many times 40 goes into 30. Dividing 30 by 40, we get 0 with a remainder of 30. So Cog A and Cog B will be back in their starting positions at the same time 0 times when B completes 30 turns.
Therefore, the answer to the question is 0 occasions.
I hope this explanation helps! Let me know if you have any other questions.