If we see someone in a spacecraft and their clock is running slower than ours do they see ours running even slower?

Question

From what I understand if we see someone's clock running slower than ours they will see ours running slower rather than theirs, but in my physics textbook they ask this question:

A spacecraft zooms past the Earth with a constant velocity. An observer on the Earth measures that an undamaged clock on the spacecraft is ticking at one- third the rate of an identical clock on the Earth. What does an observer on the spacecraft measure about the Earth-based clock’s ticking rate?

(a) It runs more than three times faster than his own clock.

(b) It runs three times faster than his own.

(c) It runs at the same rate as his own.

(d) It runs at one-third the rate of his own.

(e) It runs at less than one-third the rate of his own.

I thought the answer would be it runs one-third the rate of his own, but the correct answer (according to my book) is that it runs less than one-third the rate of his own. Is this correct? I don't know how to use the equations I've been given for special relativity to figure this out.

Answer 1

From what I understand if we see someone's clock running slower than ours they will see ours [...]

Determining and comparing (for clocks whose intications can be counted as "ticks") the ratios

$$ \frac{\hbox{Number } n \hbox{ of consecutive ticks of clock } A}{ \hbox{ Clock } A\hbox{'s Duration } \tau_{A} \hbox{ from the first one of these } n \hbox{ consecutive ticks until the final one}} $$

in order to find out whether

• any one clock ran/ticked at a constant rate, and

• whether two clocks, which were (separately) ticking at constant rates, had been ticking at equal rate(s), or which one had been running/ticking slower than the other

is not accomplished by merely one of them "seeing" the other;
but is a result of measurement.

in my physics textbook they ask this question:

A spacecraft zooms past the Earth with a constant velocity. An observer on the Earth measures

You see: "measures"!

that an undamaged clock on the spacecraft is ticking at one- third the rate of an identical clock on the Earth.

Rubbish! Clocks which were measured having run/ticked at unequal rates are certainly not called "identical"; and not even called "equal".

What does an observer on the spacecraft measure about the Earth-based clock’s ticking rate? [...]

I strongly advise you, and especially your teacher, to get a better textbook that covers the topic of special-relativity properly.