Minute hand overtakes hour hand every 70 minutes (true time): How much does the clock gain or lose in a day?

Introduction / Context:On a perfect clock, consecutive coincidences (minute hand overtakes hour hand) occur every 720/11 minutes ≈ 65 5/11 min. If the observed interval (by true time) is 70 minutes, the clock’s mechanism runs slow by a constant factor.

Given Data / Assumptions:

• Perfect interval T₀ = 720/11 min.
• Observed true-time interval T = 70 min.
• Uniform scale factor r on hand speeds (same for both hands).

Concept / Approach:Relative speed scales by r, so T = T₀ / r ⇒ r = T₀ / T = (720/11) / 70 = 72/77. Hence the clock runs at 72/77 of real time (slow). In 24 true hours, indicated time = 24 × r hours; daily loss = 24 − 24r.

Step-by-Step Solution:1) r = 72/77.2) Indicated in 24 h true = 24 × 72/77 = 1728/77 h.3) Daily loss = 24 − 1728/77 = (1848 − 1728)/77 = 120/77 h = (120 × 60)/77 = 7200/77 minutes.

Verification / Alternative check:Since T > T₀, the clock must be slow ⇒ a loss (not a gain). The magnitude matches 7200/77 ≈ 93.5 min/day.

Why Other Options Are Wrong:“Gain” contradicts T > T₀; 7300/77 and 7300/78 use wrong bases.

Common Pitfalls:Using (T − T₀) proportion instead of scaling the rate; forgetting both hands scale equally.

Final Answer:7200/77 min loss