AMC 10 · 2009 · #10
Easy mode Grade 5A digital clock shows the time as an hour and minutes, like 4:25. It has one glitch: every time a digit should be a 1, it shows a 9 instead. So 1:16 comes out as 9:96. Every other digit shows correctly. Over one whole day, for what fraction of the time does the clock show the right time?
Pick an answer.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: A 12-hour digital clock shows the hour and minute, but every time a digit should be a $1$ it shows a $9$ instead. Over a full day, find the fraction of the time the clock happens to display the correct time.
Givens: The clock shows hours $1$ through $12$ and minutes $00$ through $59$.; Any digit that should be a $1$ is displayed as a $9$.; All other digits display correctly.; For example, 1:16 is shown as 9:96.
Unknowns: The fraction of the day during which the displayed time is correct.
Understand
Restated: A 12-hour digital clock shows the hour and minute, but every time a digit should be a $1$ it shows a $9$ instead. Over a full day, find the fraction of the time the clock happens to display the correct time.
Givens: The clock shows hours $1$ through $12$ and minutes $00$ through $59$.; Any digit that should be a $1$ is displayed as a $9$.; All other digits display correctly.; For example, 1:16 is shown as 9:96.
Plan
Primary tool: #7 Identify Subproblems
Secondary: #16 Change Focus / Count the Complement, #2 Make a Systematic List
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.
Execute — Answer: A
4.NBT.A.2 Step 1 Reframe: correct means no digit is 1
- The only digit the clock gets wrong is $1$ (it swaps it for $9$).
- Every other digit is shown correctly.
- So the whole display is right exactly when not a single digit in it is a $1$.
- That turns a messy 'when is it correct' question into a clean digit-checking question.
💡 If the machine only ever mangles the digit 1, then avoiding every 1 guarantees a perfect display.
4.NF.A.1 Step 2 Count the correct hours
- The hours are $1,2,3,\dots,12$.
- List the ones that contain no $1$: $2,3,4,5,6,7,8,9$.
- The hours $1,10,11,12$ all contain a $1$, so they are wrong.
- That leaves $8$ good hours out of $12$, a fraction of $\frac{8}{12}=\frac{2}{3}$.
💡 Just cross out every hour that has a 1 somewhere and count what survives.
3.OA.A.1 Step 3 Count the correct minutes
- A minute has a tens digit ($0$–$5$) and a ones digit ($0$–$9$).
- For the minute to be correct, neither digit may be a $1$.
- The tens digit can be $0,2,3,4,5$: that is $5$ choices.
- The ones digit can be $0,2,3,4,5,6,7,8,9$: that is $9$ choices.
- Any good tens digit pairs with any good ones digit, so there are $5\times 9 = 45$ correct minutes out of the $60$ in an hour, a fraction of $\frac{45}{60}=\frac{3}{4}$.
💡 Every allowed tens digit can sit next to every allowed ones digit, so multiply the choices.
5.NF.B.4 Step 4 Combine the two subproblems
- A time is correct only when the hour is good and the minute is good.
- The good minutes are the same $\frac{3}{4}$ share inside every hour, so within the $\frac{2}{3}$ of the day that has a good hour, three-quarters of the minutes are also good.
- Multiply the two fractions: $\frac{2}{3}\times\frac{3}{4}=\frac{6}{12}=\frac{1}{2}$.
- Since AM and PM are identical, this is the fraction for the whole day, so the answer is $\textbf{(A)}\ \frac{1}{2}$.
💡 Correct hour and correct minute must both happen, so the fractions multiply.
4.NBT.A.2 The only digit the clock gets wrong is $1$ (it swaps it for $9$). Every other di 4.NF.A.1 The hours are $1,2,3,\dots,12$. List the ones that contain no $1$: $2,3,4,5,6,7, 3.OA.A.1 A minute has a tens digit ($0$–$5$) and a ones digit ($0$–$9$). For the minute t 5.NF.B.4 A time is correct only when the hour is good and the minute is good. The good mi Review
Reasonableness: Sanity-check with raw counts: there are $12\times 60 = 720$ possible times, and $8\times 45 = 360$ 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.
Alternative: Count the correct times directly instead of using fractions. Correct hours: $8$ of them ($2$ through $9$). Correct minutes: $5$ tens-digit choices times $9$ ones-digit choices $= 45$. Multiply to get $8\times 45 = 360$ correct times, divide by the $720$ total, and again get $\frac{1}{2}$.
CCSS standards used (min grade 5)
4.NBT.A.2Read and write multi-digit whole numbers and compare using symbols (Recognizing that the display is correct exactly when none of its digits is a 1.)4.NF.A.1Explain why a fraction is equivalent to another fraction (Reducing the counts of correct outcomes, such as $\frac{8}{12} = \frac{2}{3}$ and $\frac{45}{60} = \frac{3}{4}$.)3.OA.A.1Interpret products of whole numbers as total number of objects in groups (Multiplying 5 tens-digit choices by 9 ones-digit choices to count the 45 correct minutes.)5.NF.B.4Apply and extend understanding of multiplication to multiply a fraction by a fraction (Multiplying the correct-hour fraction by the correct-minute fraction to get $\frac{2}{3}$ x $\frac{3}{4}$ = $\frac{1}{2}$.)
⭐ 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.
⭐ 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.
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