Lesson 9.2 — Wheels, Prices, Seats, and Other Two-Type Stories
轮子、价格、座位和其他两类问题
A chicken-and-rabbit model is really a story translator. “Heads” can mean models, books, boats, or tickets. “Legs” can mean wheels, cost, seats, or revenue. We will identify the two object types, keep the units consistent, remove any completely known items, and then use a baseline to find both counts.
Translate the story before calculating
A toy shop buys 30 airplane and car models. An airplane model has 3 wheels, a car model has 4 wheels, and all the models have 110 wheels altogether. Identify the role of every number.
| Type | What one object contributes | Unknown count |
|---|---|---|
| Airplane model | 3 wheels | ? |
| Car model | 4 wheels | ? |
| Total objects | 30 models | |
| Weighted total | 110 wheels | |
Check the story roles
Repair an all-airplane baseline
Pretend all 30 models are airplanes. Then replace airplanes with cars until the wheel total reaches 110.
Move the slider to replace airplane models with car models.
Replace one model at a time
30 × 3 = 90
110 − 90 = 20
20 ÷ 1 = 20
Complete the solution
Both baselines must agree
You may start with the lower contribution or the higher contribution. A correct route reaches the same pair of counts.
Why the routes agree
The all-airplane route directly counts the cars, because cars are the upgrades.
The all-car route directly counts the airplanes, because airplanes are the downgrades.
Read the reverse route
Books: remove the known item first
A teacher buys 18 books for 117 yuan. One is a dictionary costing 17 yuan. The other 17 are science books costing 8 yuan each and storybooks costing 4 yuan each. Find the two unknown book counts.
The dictionary stays fixed. Change only the 17 unknown books.
Change storybooks into science books
18 − 1 = 17 unknown books
117 − 17 = 100 yuan for those books.
17 × 4 = 68
100 − 68 = 32
32 ÷ (8 − 4) = 8
Complete the book problem
Use a “known-item gate”
A two-type method needs exactly two unknown types. If one item or group is completely known, remove both its count and its contribution before building the baseline.
The two removals
Removing only the money but not the item count—or only the item count but not the money—creates mismatched totals.
Known-item laboratory
Check the preprocessing rule
Use Mission 4’s fixed purchase: 18 books costing 117 yuan, including one 17-yuan dictionary. These questions do not change when you edit the laboratory.
Seats: large and small boats
Forty-six people use 12 completely full boats. Each large boat holds 5 people, and each small boat holds 3. How many boats of each size are used?
Upgrade small boats to large boats
Complete the boat problem
Revenue: one-way and round-trip tickets
A class buys 99 train tickets for 280 yuan. Count a round-trip ticket as one ticket, even though it covers an outward and a return journey. A one-way ticket costs 2 yuan and a round-trip ticket costs 4 yuan. Find both ticket counts.
Upgrade one-way tickets
Complete the ticket problem
Use one translator for many stories
Once a story has been translated into N, a, b, T, the same engine solves models, books, boats, tickets, and many other situations.
Two-type story laboratory
The translation engine
Transfer to a new story
Twenty-five passes cost either 3 yuan or 7 yuan. The total cost is 91 yuan.
Two-type story workshop
Solve at least six of the eight problems. The checker tests the final answers. Compare your method with worked review after an attempt.
Try a fresh challenge
Try these new questions before opening help. Earn your certificate by completing Missions 2, 4, and 9 and answering all five exit questions correctly. Optional reflections and extra exploration do not affect your score.
Hint
Find the extra wheels above 22 bikes.
Worked solution — open after trying
(62 − 22 × 2) ÷ 2 = 9 cars; 13 bikes give 26 + 36 = 62 wheels.
Hint
Each upgrade adds 3 seats.
Worked solution — open after trying
(78 − 15 × 4) ÷ 3 = 6 large boats; 9 small boats give 42 + 36 = 78 seats.
Hint
Remove the atlas from the cost as well as from the count.
Worked solution — open after trying
130 − 18 = 112 yuan, for 19 books.
Hint
Start with nineteen 4-yuan books.
Worked solution — open after trying
(112 − 19 × 4) ÷ 3 = 12; seven 4-yuan books plus twelve 7-yuan books cost 112 yuan.
Hint
Find both counts before comparing them.
Worked solution — open after trying
(164 − 120) ÷ 2 = 22 return tickets and 18 one-way tickets; 22 − 18 = 4.
Two-Type Story Translator
Lesson checkpoints completed
You practised translating story quantities into consistent units and using baseline differences. Revisit any steps for which you needed solution help.
Lesson 9.2 complete
Teaching notes
The known-item laboratory, general translator, added practice, feedback, and exit ticket are new instructional scaffolds.