A watch which gains uniformly is 6 minutes slow at 4 p.m. on...

A watch which gains uniformly is 6 minutes slow at 4 p.m. on a Sunday and $10 \frac{2}{3}$ minutes fast on the following Sunday at 8 p.m. During this period (Day and Time) when was the watch correct? (a) 2.36 a.m. (b) $1.36 \mathrm{a} . \mathrm{m}$. (c) $2.36 \mathrm{p} . \mathrm{m}$. (d) $1.36 \mathrm{p} . \mathrm{m}$.

A watch which gains uniformly is 6 minutes slow at 4 p.m. on a Sunday and $10 \frac{2}{3}$ minutes fast on the following Sunday at 8 p.m. During this period (Day and Time) when was the watch correct?
(a) 2.36 a.m.
(b) $1.36 \mathrm{a} . \mathrm{m}$.
(c) $2.36 \mathrm{p} . \mathrm{m}$.
(d) $1.36 \mathrm{p} . \mathrm{m}$.

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Key Concepts

Uniform Rate of Change

This concept refers to a quantity increasing or decreasing at a constant rate over time. In the context of clocks, it implies that the error or deviation from the correct time changes linearly, making it possible to predict the error at any given moment using a straight-line equation.

Time Arithmetic and Unit Conversion

Solving problems involving clocks often requires working with specific time units and carefully handling conversions between minutes and hours. Accurate time arithmetic is essential to calculate precise intervals and determine key moments like when a defective clock will show the correct time.

Linear Interpolation and Slope Calculation

With uniform changes over time, it becomes necessary to interpolate between known values to find an unknown value. Calculating the slope—or the rate at which the error is changing—allows one to establish a linear equation that can be solved to find the exact moment when the error is zero.

Relative Error and Correction in Timekeeping

Relative error in timekeeping is the difference between the actual time and the clock's displayed time. By understanding and quantifying this error, one can determine when a clock that is gaining or losing time will align with the correct time, thus solving for the correction point.

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