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Chapter 6: Problem 9

Professor Euclid is paid every other week on Friday. Show that in some month she is paid three times.

Short Answer

Expert verified
Therefore, there are months in which Professor Euclid receives salary three times. So, Propfessor Euclid will always be paid three times in any month that starts on Friday or Saturday, or in February of a leap year if it starts on a Friday.

Step by step solution

01

Understanding the payment cycle

Professor Euclid is paid every other week on Friday. This implies that she gets her salary every fourteen days. Now, let's analyze the month.
02

Breaking down a month

Typically, a month will have 28 to 31 days. The shortest month, February, has 28 days in a common year, and 29 in a leap year. Again, let's consider February being the shortest month with 28 days (4 weeks).
03

Case: Shortest month with Even beginning

If February starts on a Friday, then professor Euclid will get paid on day 1 (1st of February), day 15 (15th of February), and finally, on day 29 of February in a leap year.
04

Case: Longest month with Odd beginning

If we consider months with 31 days, and let's say the month starts on a Saturday, she will get paid on day 7 (7th of the month) and day 21 (21st of the month), and finally, on day 35 of the calendar or which is 1st of next month. As the payment came 3 times between 1 date of a month to the same date of next month.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Payment Cycle
Understanding payment cycles is crucial in determining how often a person receives their salary. In this context, a payment cycle can be defined as the period between consecutive payments. For Professor Euclid, the payment occurs every other week, specifically on Fridays. This means she receives her salary every 14 days.
This kind of cycle repeats continuously, regardless of how many days are in a particular month. Thus, effectively working through how this aligns with the calendar days helps to visualize the payments a person might receive within a single month.
  • 14-day cycle from one Friday to the next
  • Repeats every other week
  • Influenced by the start day of the month
Calendar Math
Calendar math involves using calculations and deductions based on the months and the days of the week. Typically, months in our calendars vary from having 28 to 31 days. February is the shortest month, with 28 days in common years and 29 in leap years. Other months have either 30 or 31 days.
To determine specific occurrences, such as getting paid three times in one month, an awareness of month lengths and start days is vital. An example is when a month starts on a particular weekday, which affects the cycle of payments within that month.
  • February: 28 days, 29 days in a leap year
  • Common month lengths: 30 or 31 days
  • Strategic scheduling: Identifying opportunities for multiple occurrences
Week Cycle Analysis
Week cycle analysis involves examining how weekly patterns, like payments or events, fall within the structure of a month. It includes understanding when certain events overlap or coincide with calendar boundaries.
For Professor Euclid, the week cycle reveals that during months with a certain structure, she can receive her salary three times. This becomes evident with close analysis of months where the cycle restarts. For instance, a month starting on a Friday, combined with the 14-day payment cycle, allows for three payments within that month.
  • Analyzing the starting day of the month
  • Identifying overlaps between cycles and months
  • Aligning payment cycles with calendar weeks: Enhanced planning and scheduling

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