AMC 10 · 2007 · #9
Grade 7 rate-ratioPick an answer.
The problem gives no actual numbers, only relationships, so Tool #4 (Introduce a Variable) is the way in: name Yan's distance from home h, his distance from the stadium s, and his walking speed w. Tool #1 (Draw a Diagram) fixes the order home-Yan-stadium on a line, which is what makes the home-to-stadium distance equal h+s. Tool #8 (Analyze the Units) supplies the one physics fact needed: time = distance ÷ speed, so each route's time is a length over a speed. Tool #13 (Convert to Algebra) then turns 'both routes take the same time' into a single equation, and because every term carries the same w, the unknown speed cancels and only the ratio survives.
Draw the line and name the pieces
Naming the two pieces makes both routes writable.
Giving each unknown length and speed a letter lets both routes be written as expressions instead of mystery numbers.
6.EE.B.6Introduce A VariableWrite each route's time
The second route's ride covers the whole road.
Every leg of a trip contributes its own length-over-speed, and biking's larger speed sits in the denominator so that leg costs less time.
6.RP.A.3Analyze The UnitsSet the two times equal
Setting the times equal makes the walking speed cancel.
Equal travel times is the single fact the whole problem hangs on, so writing it as an equation captures everything at once.
7.EE.B.4Convert To AlgebraClear the speed and simplify
Simplifying leaves one clean relation.
Because the same speed multiplies every leg, it factors out entirely, which is why the answer never depends on how fast Yan actually walks.
Because the same speed multiplies every leg, it factors out entirely and the answer never depends on it.
▸ Why?
Each leg's time is its length divided by its speed, so every term carries the same walking speed underneath.
▸ Why?
A factor shared by both sides of an equation scales them together, so dividing it away changes nothing.
Read off the ratio
Reading it off gives 3/4, choice (B).
An equation between two multiples of h and s is really a statement about their ratio, so dividing turns it straight into the answer.
6.RP.A.3Introduce A VariableWhen a trip has no real numbers, name the distances and the speed with letters, write each time as distance over speed, and the unknown speed will cancel to leave just the ratio you want.
- Draw the line and name the pieces
- Write each route's time
- Set the two times equal
- Clear the speed and simplify
- Read off the ratio