AMC 8 · 2017 · #23
Grade 4 rate-rationumber-theoryProblem
Each day for four days, Linda traveled for one hour at a speed that resulted in her traveling one mile in an integer number of minutes. Each day after the first, her speed decreased so that the number of minutes to travel one mile increased by 5 minutes over the preceding day. Each of the four days, her distance traveled was also an integer number of miles. What was the total number of miles for the four trips?
Pick an answer.
AMC 8 2017 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.
Try it yourself first — the explanation is most useful after you’ve attempted it.
Toolkit + CCSS Solution
Understand
Restated: For four straight days Linda walks for exactly $1$ hour ($= 60$ minutes) each day. On Day $1$ it takes her a whole number of minutes, $m_1$, to cover one mile. Every following day that per-mile time grows by $5$ minutes (so Day $2$ takes $m_1+5$ min/mile, Day $3$ takes $m_1+10$, Day $4$ takes $m_1+15$). On every day the total miles she covers is also a whole number. Find the total miles she walks over the four days.
Givens: Daily travel time = $60$ minutes (one hour); Day $k$ pace = $m_1 + 5(k-1)$ minutes per mile, for $k = 1, 2, 3, 4$; $m_1$ is a positive integer; Daily distance $d_k = 60 / (m_1 + 5(k-1))$ must also be a positive integer; Answer choices: (A) $10$, (B) $15$, (C) $25$, (D) $50$, (E) $82$
Unknowns: The total number of miles $d_1 + d_2 + d_3 + d_4$ Linda walks over the four days
Understand
Restated: For four straight days Linda walks for exactly $1$ hour ($= 60$ minutes) each day. On Day $1$ it takes her a whole number of minutes, $m_1$, to cover one mile. Every following day that per-mile time grows by $5$ minutes (so Day $2$ takes $m_1+5$ min/mile, Day $3$ takes $m_1+10$, Day $4$ takes $m_1+15$). On every day the total miles she covers is also a whole number. Find the total miles she walks over the four days.
Givens: Daily travel time = $60$ minutes (one hour); Day $k$ pace = $m_1 + 5(k-1)$ minutes per mile, for $k = 1, 2, 3, 4$; $m_1$ is a positive integer; Daily distance $d_k = 60 / (m_1 + 5(k-1))$ must also be a positive integer; Answer choices: (A) $10$, (B) $15$, (C) $25$, (D) $50$, (E) $82$
Plan
Primary tool: #6 Guess and Check
Secondary: #2 Make a Systematic List, #8 Analyze the Units
The integer-distance condition forces every per-mile time to be a factor of $60$. There are only $12$ such factors, so we use Tool #2 (Systematic List) to write them all out, then Tool #6 (Guess and Check) to test each candidate value of $m_1$ in order: does $m_1, m_1+5, m_1+10, m_1+15$ keep landing on factors of $60$? Tool #8 (Analyze the Units) underpins the setup — the relationship $\text{distance} = \dfrac{60 \text{ min}}{m \text{ min/mile}}$ has units of miles, which is exactly what the problem asks for.
Execute — Answer: C
4.MD.A.2 Step 1 - Set up the daily distance formula.
- In $60$ minutes at a pace of $m$ minutes per mile, Linda covers $60 / m$ miles.
- For that to be a whole number, $m$ must be a factor (divisor) of $60$.
💡 Dividing minutes by minutes-per-mile leaves miles — a Grade 4 distance/time word-problem move.
4.OA.B.4 Step 2 - List every factor of $60$ in order.
- These are the only legal values for $m_1, m_1+5, m_1+10$, and $m_1+15$.
💡 Writing out every factor pair of $60$ is Grade 4 factor-finding, and a systematic list guarantees we miss nothing.
4.OA.C.5 Step 3 - Check the candidates in order.
- The four paces $m_1, m_1+5, m_1+10, m_1+15$ jump by $5$ each step, so we need four factors of $60$ that line up on an arithmetic ladder with common difference $5$.
- Walk $m_1$ up from $1$:
💡 Building a sequence with a fixed +$5$ rule and testing whether each term lands on a factor of $60$ is exactly Grade 4 "generate a number pattern following a rule."
3.OA.C.7 Step 4 - Convert each pace to a daily distance using the formula from Step 1.
- With $m_1 = 5$, the four paces $5, 10, 15, 20$ min/mile produce distances $12, 6, 4, 3$ miles.
💡 Dividing $60$ by single- and two-digit factors uses Grade 3 fluent multiplication/division facts within $100$.
4.NBT.B.4 Step 5 Add the four daily distances to get the total mileage, then match it to the answer choices.
💡 Adding four small whole numbers fluently is the Grade 4 multi-digit addition standard.
4.MD.A.2 Set up the daily distance formula. In $60$ minutes at a pace of $m$ minutes per 4.OA.B.4 List every factor of $60$ in order. These are the only legal values for $m_1, m_ 4.OA.C.5 Check the candidates in order. The four paces $m_1, m_1+5, m_1+10, m_1+15$ jump 3.OA.C.7 Convert each pace to a daily distance using the formula from Step 1. With $m_1 = 4.NBT.B.4 Add the four daily distances to get the total mileage, then match it to the answ Review
Reasonableness: Sanity-check the four daily distances. At $5$ min/mile Linda is jogging quickly ($12$ mph) for $12$ miles; at $20$ min/mile she is walking ($3$ mph) for $3$ miles. The distances $12, 6, 4, 3$ drop off as the pace slows, which matches the story. Their sum, $25$, is choice (C), and the only other near-by choice ($50$) would require doubling everything — but $m_1 = 10$ gives the sequence $(10, 15, 20, 25)$, and $25$ is not a factor of $60$, so no second solution exists. The answer is uniquely (C).
Alternative: Tool #15 (Reorganize Information) gives a slicker path: arrange the $12$ factors of $60$ as a number line and look for four dots equally spaced by $5$. Only $5, 10, 15, 20$ fit (other candidates fail because $25$, $35$, $45$ are not divisors of $60$). The visual immediately delivers $m_1 = 5$ without scanning every starting value.
CCSS standards used (min grade 4)
3.OA.C.7Fluently multiply and divide within 100 (Dividing $60$ by each pace ($5, 10, 15, 20$) to find the daily distances $12, 6, 4, 3$.)4.MD.A.2Solve word problems involving distances, time, liquid volumes, and money (Setting up $\text{distance} = 60 \text{ min} \div (m \text{ min/mile})$ to turn a per-mile pace into a daily mileage.)4.OA.B.4Find all factor pairs and recognize multiples; determine prime or composite (Listing every divisor of $60$ so we can see which integer paces are even allowed.)4.OA.C.5Generate a number or shape pattern following a given rule (Building the arithmetic sequence $m_1, m_1+5, m_1+10, m_1+15$ for each candidate $m_1$ and checking whether all four terms are factors of $60$.)4.NBT.B.4Fluently add and subtract multi-digit whole numbers (Summing the four daily distances $12 + 6 + 4 + 3 = 25$.)
⭐ This AMC 8 problem only needs Grade 4 factor-finding and pattern-making you already know!
⭐ This AMC 8 problem only needs Grade 4 factor-finding and pattern-making you already know!
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