Keeping one checked gamma triangle shows lab time scaling with proper time. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Four proper seconds become five lab seconds

Gamma 5/4 multiplies proper time 4 s into lab time 5 s.

454=54\cdot\frac{5}{4}=5
Time dilationGamma scales a moving clock interval in the lab.speedquiet leglimitbetagammaproper timelab time

Eight proper seconds become ten lab seconds

Doubling the proper interval to 8 s doubles the lab interval to 10 s because gamma is unchanged.

854=108\cdot\frac{5}{4}=10
Time dilationGamma scales a moving clock interval in the lab.speedquiet leglimitbetagammaproper timelab time

Twelve proper seconds become fifteen lab seconds

The third row keeps the same triangle and changes only the moving-clock reading. Proper time 12 s gives lab time 15 s.

1254=1512\cdot\frac{5}{4}=15
Time dilationGamma scales a moving clock interval in the lab.speedquiet leglimitbetagammaproper timelab time