AMC 10 · 2023 · #1
Grade 6 rate-ratioProblem
Cities A and B are 45 miles apart. Alicia lives in A and Beth lives in B. Alicia bikes towards B at 18 miles per hour. Leaving at the same time, Beth bikes toward A at 12 miles per hour. How many miles from City A will they be when they meet?
Pick an answer.
AMC 10 2023 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: Two bikers start at the same time from cities $A$ and $B$, which are $45$ miles apart, and pedal toward each other. Alicia (from $A$) goes $18$ mph; Beth (from $B$) goes $12$ mph. How far from $A$ are they when they meet?
Givens: Distance between $A$ and $B$ is $45$ miles; Alicia leaves $A$ at $18$ mph heading toward $B$; Beth leaves $B$ at $12$ mph heading toward $A$; They start at the same time; Answer choices: (A) $20$, (B) $24$, (C) $25$, (D) $26$, (E) $27$
Unknowns: Distance from city $A$ to the meeting point (in miles)
Understand
Restated: Two bikers start at the same time from cities $A$ and $B$, which are $45$ miles apart, and pedal toward each other. Alicia (from $A$) goes $18$ mph; Beth (from $B$) goes $12$ mph. How far from $A$ are they when they meet?
Givens: Distance between $A$ and $B$ is $45$ miles; Alicia leaves $A$ at $18$ mph heading toward $B$; Beth leaves $B$ at $12$ mph heading toward $A$; They start at the same time; Answer choices: (A) $20$, (B) $24$, (C) $25$, (D) $26$, (E) $27$
Plan
Primary tool: #1 Draw a Diagram
Secondary: #8 Analyze the Units
The problem describes positions on a road and motion along that road — Tool #1 (Draw a Diagram) is the natural lead. A simple labeled segment from $A$ to $B$ with arrows showing the two bikers makes the structure obvious: the gap of $45$ miles closes at the combined speed $18 + 12 = 30$ mph. Tool #8 (Analyze the Units) is the verification companion — tracking miles, mph, and hours through every multiplication makes sure the final number is in miles, which is what the problem asks for. Algebra (Tool #13) would also work but the diagram-plus-units path is faster and more visual for a Grade 6 rate concept.
Execute — Answer: E
6.RP.A.2 Step 1 - Draw a $45$-mile segment with $A$ on the left and $B$ on the right.
- Put Alicia at $A$ with a rightward arrow labeled $18$ mph; put Beth at $B$ with a leftward arrow labeled $12$ mph.
- The picture shows the gap between them shrinking from both ends at the same time.
💡 Drawing the road with two arrows turns a word problem into a closing-gap picture — the Grade 6 "unit rate" idea that miles-per-hour is how fast the gap shrinks.
4.OA.A.3 Step 2 - From the diagram, the two arrows close the gap at the same time, so their speeds add.
- Compute the combined closing speed.
💡 When two things move toward each other, the gap shrinks at the sum of their speeds — a Grade 4 multi-step word-problem addition.
6.RP.A.3 Step 3 - Find the time until they meet by dividing the $45$-mile gap by the $30$-mph closing speed.
- Track units: miles divided by miles-per-hour leaves hours.
💡 Distance $\div$ rate $=$ time — the Grade 6 rate-reasoning that miles cancel and leave hours, exactly the unit the next step needs.
6.RP.A.3 Step 4 - The meeting point is wherever Alicia has biked to in $1.5$ hours.
- Multiply her speed by the time.
- The question asks for distance from $A$, which is exactly Alicia's traveled distance.
💡 Rate $\times$ time $=$ distance — the same Grade 6 unit-rate move in reverse; mph $\times$ hr cancels to mi, the requested unit.
6.RP.A.2 Draw a $45$-mile segment with $A$ on the left and $B$ on the right. Put Alicia a 4.OA.A.3 From the diagram, the two arrows close the gap at the same time, so their speeds 6.RP.A.3 Find the time until they meet by dividing the $45$-mile gap by the $30$-mph clos 6.RP.A.3 The meeting point is wherever Alicia has biked to in $1.5$ hours. Multiply her s Review
Reasonableness: Cross-check from Beth's side. In $1.5$ hours Beth covers $12 \times 1.5 = 18$ miles from $B$ toward $A$. Alicia's $27$ miles plus Beth's $18$ miles is $27 + 18 = 45$ miles — the full distance, so they really do meet at that spot. Also, since Alicia is the faster rider, the meeting point should sit past the midpoint ($22.5$ mi) on the $B$ side; $27 > 22.5$ confirms this.
Alternative: Tool #13 (Convert to Algebra): let $t$ be the meeting time. Alicia's position from $A$ is $18t$ and Beth's position from $A$ is $45 - 12t$. Setting them equal gives $18t = 45 - 12t \Rightarrow 30t = 45 \Rightarrow t = 1.5$, then $18 \cdot 1.5 = 27$ miles. Same answer (E), more symbol-pushing.
CCSS standards used (min grade 6)
4.OA.A.3Solve multi-step word problems using four operations with whole numbers (Adding the two speeds $18 + 12 = 30$ to get the combined closing speed at which the gap between Alicia and Beth shrinks.)6.RP.A.2Understand the concept of a unit rate and use rate language (Interpreting "$18$ miles per hour" as a unit rate — the number of miles Alicia covers each hour — so the diagram has a precise meaning.)6.RP.A.3Use ratio and rate reasoning to solve real-world and mathematical problems (Using distance $=$ rate $\times$ time twice: once to get $t = 45/30 = 1.5$ hr, then $d_A = 18 \times 1.5 = 27$ mi.)
⭐ This AMC 10 problem only needs Grade 6 unit rates you already know — speeds add when two riders close in, then distance equals speed times time!
⭐ This AMC 10 problem only needs Grade 6 unit rates you already know — speeds add when two riders close in, then distance equals speed times time!
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