AMC 10 · 2009 · #19

Grade 5 counting

Archetype Overlap / Complement Counting └ Sub-archetype At-Least-One Complement: 1 − P(None)

complementary-countingdigit-countingfraction-arithmetic identify-subproblems ↑ Prerequisites: complementary-counting

📏 Medium solution 💡 2 insights

Problem

A 12-hour digital clock shows the hour and minute, but every time a digit should be a 1 it shows a 9 instead — for example, 1:16 is displayed as 9:96. Over a full day, find the fraction of the time the clock happens to display the correct time.

Pick an answer.

(A)

$\ \frac 12$

(B)

$\ \frac 58$

(C)

$\ \frac 34$

(D)

$\ \frac 56$

(E)

$\ \frac {9}{10}$

AMC 10 2009 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

The display has two independent parts: the hour and the minute. A time is correct only when the hour is correct AND the minute is correct, and the set of correct minutes is the same in every hour. So split the problem: find the fraction of hours with no 1, find the fraction of minutes with no 1, then multiply the two fractions.

1STEP 1

Reframe: correct means no digit is 1

The clock spoils only the digit 1, so the display is right exactly when not a single digit in it is a 1.

display correct ⇔ no digit equals 1

2STEP 2

Count the correct hours

Hours 1, 10, 11, 12 contain a 1, so only 2 through 9 work: 8 good hours out of 12, a fraction of $\frac{2}{3}$ .

good hours = {2,3,4,5,6,7,8,9} → 8/12 = 2/3

3STEP 3

Count the correct minutes

Tens digit 0,2,3,4,5 (5 ways) times ones digit 0,2,…,9 (9 ways) gives 45 good minutes out of 60, a fraction of $\frac{3}{4}$ .

5 × 9 = 45 good minutes → 45/60 = 3/4

4STEP 4

Combine the two subproblems

Hour and minute must both be good, and the same $\frac{3}{4}$ sits inside every hour, so $\frac{2}{3}\times\frac{3}{4}=$ $\frac{1}{2}$ — choice (A).

2/3 × 3/4 = 6/12 = 1/2

Answer

1/2

Sanity-check with raw counts: there are 12 × 60 = 720 possible times, and 8 × 45 = 360 of them have no 1. That is exactly

\frac{360}{720}=

\frac{1}{2}

, matching the multiplied fractions. Half is also believable: a 1 is a fairly common digit, so it is reasonable that it spoils about half of all readings.

💡Key takeaway

Split a two-part display into its parts, find the fraction right in each part, then multiply the fractions to get the fraction right overall.

Reframe: correct means no digit is 1
Count the correct hours
Count the correct minutes
Combine the two subproblems

Reframe: correct means no digit is 1

Count the correct hours

Count the correct minutes

Combine the two subproblems

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