2017 AMC 8 Problem 23

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To help you master 2017 AMC 8 Problem 23, this page includes:

The original AMC problem to try, plus the correct answer

Clear, step-by-step solutions using different methods

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2017 AMC8 Problem 23

Solution

Solution 1

Divisibility and Factor Listing Method

Express the daily distance as a fraction: 60 minutes divided by the minutes per mile. Since the distance must be an integer, the minutes per mile is strictly limited to the factors of 60. By listing these factors, we can easily spot the only sequence of four numbers that step up by 5 each day.

To guarantee that a fraction $\frac{Numerator}{Denominator}$ evaluates to an integer, the denominator must be a factor (or divisor) of the numerator. Here, the time per mile must perfectly divide the 60 total minutes.

Step 1: Identify the relevant quantities and their relationship

Let $m_{i}$ be the number of minutes it takes to travel one mile on day $i$ . Since Linda travels for exactly one hour each day, we convert this total time to minutes. Her daily distance $d_{i}$ in miles is the total time divided by the time it takes to travel a single mile.

Total Time = 1 hour = 60 minutes

d_{i} = \frac{60}{m _{i}}

Step 2: Determine the mathematical constraints

The problem states that the minutes per mile and the daily distance in miles are both positive integers. Because of our distance fraction, this means the denominator must divide the numerator evenly without leaving a remainder.

m_{i} \in Z^{+}

d_{i} \in Z^{+}

m_{i} must be a factor of 60

Step 3: List all possible factors

To find the valid values for the time per mile, we write out all the positive integers that divide 60 evenly.

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

Step 4: Use the daily change to find the sequence of times

We are given that the minutes per mile increases by 5 each day. We search our organized list of factors for an arithmetic sequence of four numbers separated by exactly 5. By checking the differences between the factors, we find exactly one valid progression.

m_{i + 1} = m_{i} + 5

m_{1} = 5

m_{2} = 10

m_{3} = 15

m_{4} = 20

Step 5: Calculate the distances and the final total

Substitute these specific time values back into our initial fraction to find the daily distances. Finally, add these four daily distances together to answer the question.

d_{1} = \frac{60}{5} = 12

d_{2} = \frac{60}{10} = 6

d_{3} = \frac{60}{15} = 4

d_{4} = \frac{60}{20} = 3

Total Distance = 12 + 6 + 4 + 3 = 25

Correct answer: C. $25$
The total number of miles for the four trips is 25, which corresponds to choice C. Final Check: We can reverse verify the properties. The daily distances are 12, 6, 4, and 3 miles. All are integers. The daily travel times for those distances at exactly 60 minutes are 5, 10, 15, and 20 minutes per mile. These are all integers and they neatly increase by 5 each day. The sum of the distances is indeed 25.

Adapted Problems to Practice

1 / 10

What Formulas Do You Need for This Problem?

Alternative Distance Formula:

Distance = \frac{Total Time}{Time per Unit Distance}

Units must be consistent. For instance, if the total time is measured in minutes and the rate is given in minutes per mile, the resulting distance calculates cleanly into miles. The problem specifically provides the total travel time (60 minutes) and defines speed based on the time required to travel one unit of distance, making this formulation directly applicable.

Divisor (Factor) Constraint:

An integer

x

is a divisor of an integer

y

if the fraction

\frac{y}{x}

evaluates to an integer.

$x$ and $y$ are integers with $x \neq = 0$ . Because the distance formula yields a fraction $\frac{60}{m _{i}}$ that is required by the problem to evaluate to an integer, $m_{i}$ is mathematically restricted to being a divisor of 60.

Key Takeaway: Remember This Strategy for Next Time

A problem features quantities that must be integers, linked by division (e.g., total distance = total time / rate).

Try: Translate the division into a divisibility constraint, and physically write out all factors of the constant numerator.

Why: Listing factors explicitly reduces an infinite set of possible numbers down to a small, finite list. This makes it incredibly easy to test remaining conditions, such as finding a specific sequence or verifying a sum.

A problem expresses speed as "time per unit distance" (like minutes per mile) instead of the standard "distance per time" (like miles per hour).

Try: Use the alternative, rearranged formula: Distance = Total Time / Time per Unit Distance.

Why: This perfectly mirrors how the information is given. It avoids the messy fractions that can appear if you artificially force the problem into the standard $D = R \times T$ formula by converting "minutes per mile" into "miles per minute".

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2017 AMC 8 Problem 23

In This Page:

2017 AMC8 Problem 23