Final answer:
By understanding the conversion between sexagesimal minutes (60 per degree) and centesimal minutes (100 per degree), we can set up a ratio of 3:5. multiplying both sides of the equation 5M = 3m by the appropriate factors to get denominators of 27 and 50 respectively, proves that M/27 equals m/50.
Step-by-step explanation:
To prove that M/27 = m/50 where M and m are the number of sexagesimal and centesimal minutes of any angle respectively, we need to understand the relationship between these two units of angle measurement.
One degree in the sexagesimal system (used in most common applications) is divided into 60 minutes, hence the term 'sexagesimal minute'. On the other hand, in the centesimal system, one degree is divided into 100 minutes, which are called 'centesimal minutes'. a complete circle contains 360 degrees in both systems. to equate M and m, we can set up a ratio. since there are 60 sexagesimal minutes in a degree and 100 centesimal minutes in a degree, we can express this as a ratio: 60 sexagesimal minutes / 100 centesimal minutes which reduces to 3/5. therefore, for any given angle, 3 sexagesimal minutes (M) are equivalent to 5 centesimal minutes (m). now let's find a common factor to compare M and m: 5M = 3m
To make this equation have the form M/27 = m/50 we multiply both sides by a common factor so that the denominators are 27 and 50 respectively:
5M * 50 = 3m * 27
250M = 81m
When we divide both sides by 250*3, we get: M/27 = m/50
Thus, proving that for any angle the number of sexagesimal minutes over 27 equals the number of centesimal minutes over 50.