Question

What time does a 12-hour clock read

a) 80 hours after it reads 11:00?

b) 40 hours before it reads 12:00?

c) 100 hours after it reads 6:00?

Short answer

1. 7:00
2. 8:00

c. 10:00

Step-by-step solution

Concept of Division Algorithm 

• Let \(a\) be an integer and \(d\) be a positive integer.
• Then there are unique Integers\(q\)and\(T\)with\(0 \le \;r < d\)such that\(a = dq + r\).
• \(q\)is called the quotient and\(T\)is called the remainder.
• \(\begin{array}{*{20}{l}}{q = a{\rm{ div }}d}\\{\;r = a{\rm{ mod }}d}\end{array}\) a

(a) Finding the time that 12-hour clock reads 80 hours after it reads 11:00

\(\begin{array}{l}a = 80\\d = 12\end{array}\)

Let us determine the quotient and remainder

\(\begin{array}{c}a = 80\\ = 72 + 8\\ = 6 \cdot 12 + 8\\ = 6\;d + 8\end{array}\)

The quotient is then the coefficient of\(d \Rightarrow q = 6\)

The remainder is the constant:\(r = 8\;({\rm{with}}\;0 \le 8 < 12)\)

\(80\;\bmod \;12 = 8\)

The remainder is 8, thus 80 hours after it reads 11:00 is the same time as 8 hours after it reads 11:00, 8 hours after 11:00 is 7 o’clock.

\(11:00 + 8:00 = 19:00 \Rightarrow 7:00\)

Hence the time is 7.00

 (b) Finding the time that 12-hour clock reads 40 hours after it reads 12:00

\(\begin{array}{l}a = 40\\d = 12\end{array}\)

Let us determine the quotient and remainder

\(\begin{array}{c}a = 40\\ = 36 + 4\\ = 3 \cdot 12 + 4\\ = 3\;d + 4\end{array}\)

The quotient is then the coefficient of\(d \Rightarrow q = 3\)

The remainder is the constant:\(r = 4\;({\rm{with}}\;0 \le 4 < 12)\)

\(40\;\bmod \;12 = 4\)

The remainder is 4, thus 40 hours before it reads 12:00 is the same time as 4 hours before it reads 12:00, 4 hours before 12:00 is 8 o’clock.

\(12:00 - 4:00 = 8:00\)

Hence the time is 8.00

(c) Finding the time that 12-hour clock reads 100 hours after it reads 6:00 

\(\begin{array}{l}a = 100\\d = 12\end{array}\)

Let us determine the quotient and remainder

\(\begin{array}{c}a = 100\\ = 96 + 4\\ = 8 \cdot 12 + 4\\ = 8\;d + 4\end{array}\)

The quotient is then the coefficient of\(d \Rightarrow q = 8\)

The remainder is the constant:\(r = 4\;({\rm{with}}\;0 \le 4 < 12)\)

\(100\;\bmod \;12 = 4\)

The remainder is 4, thus 100 hours after it reads 12:00 is the same time as 4 hours after it reads 6:00. 4 hours after 6:00 is 10 o’clock.

Hence the time is 10.00