AMC 8 · 2017 · #23

Grade 4 rate-rationumber-theory

Archetype Small-Domain Constrained Search

ratefactorssequences-arithmeticdivisibility-rules systematic-enumerationguess-and-check ↑ Prerequisites: ratefactorssequences-arithmetic

📏 Medium solution 💡 3 insights

Problem

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?

$\textbf{(A) }10\qquad\textbf{(B) }15\qquad\textbf{(C) }25\qquad\textbf{(D) }50\qquad\textbf{(E) }82$

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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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

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

#8 Analyze the Units 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 $60$.

$$d_k = \dfrac{60 \text{ min}}{m_k \text{ min/mile}} = \dfrac{60}{m_k} \text{ miles}$$

💡 Dividing minutes by minutes-per-mile leaves miles — a Grade 4 distance/time word-problem move.

#2 Make a Systematic List 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 $m_1+15$.

Factors of $60$: $1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60$

💡 Writing out every factor pair of $60$ is Grade 4 factor-finding, and a systematic list guarantees we miss nothing.

#6 Guess and Check 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 $1$:

$$m_1{=}1\!:\,(1,6,11,16)\,\text{✗}\;\; m_1{=}2\!:\,(2,7,12,17)\,\text{✗}\;\; m_1{=}3\!:\,(3,8,13,18)\,\text{✗}\;\; m_1{=}4\!:\,(4,9,14,19)\,\text{✗}\;\; m_1{=}5\!:\,(5,10,15,20)\,\text{✓}$$

💡 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."

#8 Analyze the Units 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.

$$d_1=\tfrac{60}{5}=12,\;\; d_2=\tfrac{60}{10}=6,\;\; d_3=\tfrac{60}{15}=4,\;\; d_4=\tfrac{60}{20}=3$$

💡 Dividing $60$ by single- and two-digit factors uses Grade 3 fluent multiplication/division facts within $100$.

#8 Analyze the Units 4.NBT.B.4 Step 5

Add the four daily distances to get the total mileage, then match it to the answer choices.

$$12 + 6 + 4 + 3 = 25 \text{ miles} \;\Rightarrow\; \textbf{(C)}$$

💡 Adding four small whole numbers fluently is the Grade 4 multi-digit addition standard.

[1] #8 4.MD.A.2 Set up the daily distance formula. In $60$ minutes at a pace of $m$ minutes per

[2] #2 4.OA.B.4 List every factor of $60$ in order. These are the only legal values for $m_1, m_

[3] #6 4.OA.C.5 Check the candidates in order. The four paces $m_1, m_1+5, m_1+10, m_1+15$ jump

[4] #8 3.OA.C.7 Convert each pace to a daily distance using the formula from Step 1. With $m_1 =

[5] #8 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.7 Fluently 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.2 Solve 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.4 Find 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.5 Generate 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.4 Fluently 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!

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: C

Review

CCSS standards used (min grade 4)

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