Cascaded Decade Counters — Overall Divide Ratio If three decade (MOD-10) counters are cascaded, by what factor is the input frequency divided at the final output?

Introduction / Context:
Counters are ubiquitous for frequency division, timebase generation, and event counting. When multiple counters are cascaded (the output of one drives the clock of the next), the overall divide ratio equals the product of individual moduli. Decade counters, by definition, divide by 10 each cycle.

Given Data / Assumptions:

• Three identical decade counters (each MOD-10).
• Cascaded ripple or synchronous configuration; modulus multiplication still applies.
• Only one clock input applied to the first stage.

Concept / Approach:
Each stage reduces the frequency by its modulus. Therefore, the net division is the product of the moduli: 10 * 10 * 10. The final output toggles at one-thousandth of the input frequency, a common arrangement in digital clocks and frequency meters.

Step-by-Step Solution:
Stage 1: divide by 10 → f1 = fin / 10.Stage 2: divide by 10 again → f2 = f1 / 10 = fin / 100.Stage 3: divide by 10 again → f3 = f2 / 10 = fin / 1000.Hence, overall division = 1,000.

Verification / Alternative check:
Consider a 1 MHz input. After three cascaded decades: 1 MHz → 100 kHz → 10 kHz → 1 kHz. That is a 1,000:1 reduction.

Why Other Options Are Wrong:

• 10 or 100: Too small; ignore cumulative effect of three stages.
• 20: Not a power-of-ten divisor for decade chains.

Common Pitfalls:
Forgetting that cascaded counters multiply moduli, or confusing decade (base-10) with binary (base-2) counters when estimating total division.

Final Answer:
1,000