Finding the Day of the Week: 150 Days Ago Calculation

This problem asks us to determine the day of the week 150 days ago from today, based on information about a past day.

Step 1: Determine Today's Day

We are told that three days ago it was Friday. Let's count forward from Friday to find today's day:

• Three days ago: Friday
• Two days ago: Saturday
• One day ago: Sunday
• Today: Monday

So, today is Monday.

Step 2: Calculate the Day 150 Days Ago

The days of the week repeat in a cycle of 7 days. To find the day of the week a certain number of days ago (or in the future), we need to determine how many full weeks are in that period and what the remaining number of days is. The remainder after dividing the number of days by 7 tells us how many days forward or backward from the starting day we need to count.

We need to find the day 150 days ago from today (Monday). We divide 150 by 7:

\( \frac{150}{7} \)

We can perform the division:

\( 150 = 7 \times q + r \)

Where \(q\) is the quotient and \(r\) is the remainder.

\( 150 \div 7 \)

\( 150 = 7 \times 21 + 3 \)

So, 150 days is equal to 21 full weeks and 3 remaining days. Going back 150 days is equivalent to going back 3 days from today's day.

Step 3: Count Back 3 Days from Today

Today is Monday. We need to count back 3 days:

• 1 day ago from Monday: Sunday
• 2 days ago from Monday: Saturday
• 3 days ago from Monday: Friday

Therefore, the day of the week 150 days ago from today was Friday.

Let's summarise the calculation:

The day 150 days ago was Friday.

Revision Table: Day Calculation

Understanding how to calculate days of the week involves modular arithmetic with base 7.

Additional Information: Modular Arithmetic and Days

Problems involving days of the week often use the concept of modular arithmetic, specifically modulo 7. Each day of the week can be assigned a number (e.g., Sunday=0, Monday=1, ..., Saturday=6). If today is day \(D\), then \(N\) days in the future will be day \((D + N) \pmod{7}\), and \(N\) days ago will be day \((D - N) \pmod{7}\). When working with subtraction and modulo, if the result is negative, you add 7 to make it positive while keeping the remainder the same.

In this problem:

• Today is Monday. Let's assign Monday the number 1.
• We want to find the day 150 days ago.
• Using modular arithmetic: \((1 - 150) \pmod{7}\)
• \(1 - 150 = -149\)
• We need to find \((-149) \pmod{7}\). We know \(149 = 7 \times 21 + 2\), so \(149 \equiv 2 \pmod{7}\).
• Therefore, \(-149 \equiv -2 \pmod{7}\).
• To get a positive remainder, we add 7: \(-2 + 7 = 5\).
• So, \((1 - 150) \pmod{7} \equiv 5 \pmod{7}\).

If Monday is 1, let's check our numbering system:

• Sunday = 0
• Monday = 1
• Tuesday = 2
• Wednesday = 3
• Thursday = 4
• Friday = 5
• Saturday = 6

The result 5 corresponds to Friday, which matches our previous calculation by counting back.