← Sensim Lab Sensim Math

AMC 10 · 2021 · #6

Grade 6 rate-ratio
Archetype Meeting Time with Piecewise Rates
ratefraction-arithmeticratio-proportion identify-subproblemsdimensional-analysis ↑ Prerequisites: rate
📏 Medium solution 💡 2 insights

Problem

Chantal and Jean start hiking from a trailhead toward a fire tower. Jean is wearing a heavy backpack and walks slower. Chantal starts walking at 44 miles per hour. Halfway to the tower, the trail becomes really steep, and Chantal slows down to 22 miles per hour. After reaching the tower, she immediately turns around and descends the steep part of the trail at 33 miles per hour. She meets Jean at the halfway point. What was Jean's average speed, in miles per hour, until they meet?

Pick an answer.

(A)
$~\frac{12}{13}$
(B)
~1
(C)
$~\frac{13}{12}$
(D)
$~\frac{24}{13}$
(E)
~2
View mode:

Toolkit + CCSS Solution

Understand

Restated: Chantal and Jean start together at the trailhead. Chantal hikes at $4$ mph to the halfway point, then $2$ mph up the steep top half to the tower, then turns around and comes back down the steep half at $3$ mph. She meets Jean exactly at the halfway point. During that same time, Jean has been hiking from the trailhead to the halfway point at a constant pace. What was Jean's average speed in miles per hour?

Givens: Let $d$ = distance from trailhead to halfway point; total trail length $= 2d$; Chantal: $4$ mph for first $d$, $2$ mph for next $d$ (up steep), $3$ mph for last $d$ (down steep); Jean: covers distance $d$ at constant unknown speed; They start at the same moment and meet at the halfway point; Choices: (A) $\frac{12}{13}$, (B) $1$, (C) $\frac{13}{12}$, (D) $\frac{24}{13}$, (E) $2$

Unknowns: Jean's average speed in mph until they meet

Understand

Restated: Chantal and Jean start together at the trailhead. Chantal hikes at $4$ mph to the halfway point, then $2$ mph up the steep top half to the tower, then turns around and comes back down the steep half at $3$ mph. She meets Jean exactly at the halfway point. During that same time, Jean has been hiking from the trailhead to the halfway point at a constant pace. What was Jean's average speed in miles per hour?

Givens: Let $d$ = distance from trailhead to halfway point; total trail length $= 2d$; Chantal: $4$ mph for first $d$, $2$ mph for next $d$ (up steep), $3$ mph for last $d$ (down steep); Jean: covers distance $d$ at constant unknown speed; They start at the same moment and meet at the halfway point; Choices: (A) $\frac{12}{13}$, (B) $1$, (C) $\frac{13}{12}$, (D) $\frac{24}{13}$, (E) $2$

Plan

Primary tool: #7 Identify Subproblems

Secondary: #8 Analyze the Units, #1 Draw a Diagram

Chantal's trip splits naturally into three legs at three different speeds — Tool #7 (Identify Subproblems) handles each leg separately, then sums the times. Tool #8 (Analyze the Units) keeps the rate relation $\text{time} = \text{distance} / \text{speed}$ honest. A quick number-line sketch (Tool #1) makes it obvious that Jean has covered only $d$ miles while Chantal has covered $3d$ miles in the same time.

Execute — Answer: A

#1 Draw a Diagram 6.RP.A.3 Step 1
  • Mark the trailhead, the halfway point, and the tower on a number line, distance $d$ apart.
  • Chantal walks $d$ at $4$ mph (flat), $d$ at $2$ mph (steep up), $d$ at $3$ mph (steep down), landing at the halfway point.
  • Jean walks $d$ at an unknown pace.
  • Same time elapsed.
$$\text{trailhead} \xrightarrow{d,\;4} \text{half} \xrightarrow{d,\;2} \text{tower} \xrightarrow{d,\;3} \text{half}$$

💡 A picture turns the wordy trip into three labeled segments and one segment for Jean.

#8 Analyze the Units 6.RP.A.2 Step 2
  • For each leg use $\text{time} = \dfrac{\text{distance}}{\text{speed}}$.
  • Leg 1: $\dfrac{d}{4}$ h.
  • Leg 2: $\dfrac{d}{2}$ h.
  • Leg 3: $\dfrac{d}{3}$ h.
  • Units check: miles $\div$ (miles/hour) $=$ hours.
$$t_1 = \dfrac{d}{4},\;\; t_2 = \dfrac{d}{2},\;\; t_3 = \dfrac{d}{3}$$

💡 Unit rate $r = d/t$ inverts cleanly to $t = d/r$ — Grade 6 rate reasoning.

#7 Identify Subproblems 5.NF.A.1 Step 3
  • Add Chantal's three leg times to get the total time elapsed when they meet.
  • Common denominator $12$: $\dfrac{d}{4} = \dfrac{3d}{12},\;\dfrac{d}{2} = \dfrac{6d}{12},\;\dfrac{d}{3} = \dfrac{4d}{12}$.
  • Sum $= \dfrac{13d}{12}$ hours.
$$T = \dfrac{d}{4} + \dfrac{d}{2} + \dfrac{d}{3} = \dfrac{3d+6d+4d}{12} = \dfrac{13d}{12}$$

💡 Same denominator first, then add — Grade 5 fraction addition.

#7 Identify Subproblems 6.RP.A.3 Step 4
  • Jean covered distance $d$ in the same time $T = \dfrac{13d}{12}$.
  • So Jean's average speed is $\dfrac{d}{T} = d \cdot \dfrac{12}{13d} = \dfrac{12}{13}$ mph.
  • The $d$ cancels — the answer does not depend on the actual length of the trail.
$$s_J = \dfrac{d}{\;13d/12\;} = \dfrac{12}{13}\;\text{mph}$$

💡 Dividing by a fraction = multiplying by its reciprocal; the unknown distance cancels.

#7 Identify Subproblems 4.NF.A.2 Step 5

Match to answer choices: $\dfrac{12}{13}$ is choice (A).

$$\dfrac{12}{13}\;\Rightarrow\;\textbf{(A)}$$

💡 Read the simplified fraction off the list of options.

[1] #1 6.RP.A.3 Mark the trailhead, the halfway point, and the tower on a number line, distance
[2] #8 6.RP.A.2 For each leg use $\text{time} = \dfrac{\text{distance}}{\text{speed}}$. Leg 1: $
[3] #7 5.NF.A.1 Add Chantal's three leg times to get the total time elapsed when they meet. Comm
[4] #7 6.RP.A.3 Jean covered distance $d$ in the same time $T = \dfrac{13d}{12}$. So Jean's aver
[5] #7 4.NF.A.2 Match to answer choices: $\dfrac{12}{13}$ is choice (A).

Review

Reasonableness: Sanity check the size of the answer. Chantal walked $3d$ miles in time $T$ while Jean walked $d$ miles in the same time, so Jean's average speed must be one-third of Chantal's average speed. Chantal's average speed is $\dfrac{3d}{13d/12} = \dfrac{36}{13}$ mph, so Jean's is $\dfrac{12}{13}$ mph — matches. The answer is less than $1$ mph, which fits the story (heavy backpack, slower than every one of Chantal's legs).

Alternative: Tool #6 (Guess and Check) on the answer choices: pick a convenient $d$ such as $d = 12$ miles. Then Chantal's legs take $3,\,6,\,4$ hours, total $13$ hours. Jean covers $12$ miles in $13$ hours, average $\dfrac{12}{13}$ mph — directly (A). Plugging a number is faster than algebra and avoids fraction errors.

CCSS standards used (min grade 6)

  • 6.RP.A.3 Use ratio and rate reasoning to solve real-world and mathematical problems (Treating each hiking leg as a rate problem and computing Jean's overall mph.)
  • 6.RP.A.2 Understand the concept of a unit rate and use rate language (Converting speed into time per mile via $t = d / r$ for each leg.)
  • 5.NF.A.1 Add and subtract fractions with unlike denominators (Adding $\frac{d}{4} + \frac{d}{2} + \frac{d}{3}$ via the common denominator $12$.)
  • 4.NF.A.2 Compare two fractions with different numerators and different denominators (Selecting the matching choice $\frac{12}{13}$ from the five options.)

⭐ This AMC 10 problem only needs Grade 6 rate reasoning you already know — time = distance / speed for each of Chantal's three legs, add the times with a common denominator $12$ to get $\frac{13d}{12}$ hours, then Jean's speed is $\frac{d}{13d/12} = \frac{12}{13}$ mph.

⭐ This AMC 10 problem only needs Grade 6 rate reasoning you already know — time = distance / speed for each of Chantal's three legs, add the times with a common denominator $12$ to get $\frac{13d}{12}$ hours, then Jean's speed is $\frac{d}{13d/12} = \frac{12}{13}$ mph.

More like this

Same archetype — closest grade level first.