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2009 AMC 10B Problem 19

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2009 AMC 10B Problem 19AMC 10 Step by Step
Name:                             Date: September 13, 2026
Basic Counting · difficulty 3 of 5 · about 4 min · position P16-20

A particular 1212 -hour digital clock displays the hour and minute of a day. Unfortunately, whenever it is supposed to display a 11 , it mistakenly displays a 99 . For example, when it is 1:16 PM the clock incorrectly shows 9:96 PM. What fraction of the day will the clock show the correct time?

  1. A)

    12\frac 12

  2. B)

    58\frac 58

  3. C)

    34\frac 34

  4. D)

    56\frac 56

  5. E)

    910\frac 9{10}

Answer

A

Key insight

The hour and minute are independent, so multiply the fraction of hours without a 1 (8 of 12) by the fraction of minutes without a 1 (45 of 60).

Solution

The time is shown correctly exactly when neither the hour nor the minute contains the digit 11. Every minute of the day is equally long, and each hour is paired with all 6060 minutes, so the fraction of correct minutes is the product of two independent fractions.

Hours: the display cycles through 1,2,…,121, 2, \ldots, 12. The hours containing a 11 are 1,10,11,121, 10, 11, 12, so 88 of the 1212 hours are fine: fraction 812=23\frac{8}{12} = \frac23.

Minutes: the minute is a two-digit string from 0000 to 5959. The tens digit can be 0,2,3,4,50, 2, 3, 4, 5 (55 of 66 options) and the units digit anything except 11 (99 of 1010 options), so 5⋅9=455 \cdot 9 = 45 of the 6060 minutes are fine: fraction 4560=34\frac{45}{60} = \frac34.

Combined: 23⋅34=12\frac23 \cdot \frac34 = \frac12.

The answer is (A) 12\boxed{\textbf{(A)}\ \frac12}.

Why this works

When an outcome requires two independent conditions, the fraction of good outcomes multiplies. Counting digit-avoiding strings position by position is a standard move, but the hour display is not a uniform digit string, so list its bad values (1,10,11,121, 10, 11, 12) explicitly rather than applying a formula.

The trap

Counting the hours as 0 through 11 or 1 through 24, or forgetting that both the tens and units digit of the minutes can be 1.

Common mistakes

  • Counting the hours as 0 through 11 or 1 through 24, or forgetting that both the tens and units digit of the minutes can be 1.
  • Counting minutes with a 11 as 6+10=166 + 10 = 16 instead of 1515, double-counting 1111.

Techniques

Set up the equation/formula and compute; no special trick needed