Lesson 9.3 — Use Differences and Hidden Relationships

Mission 1

Read the relationship before choosing an operation

A difference clue compares two complete quantities. First name exactly what is being compared.

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Difference between group totals“The chickens’ total legs are 26 more than the rabbits’ total legs.”

Difference between rates“Team B repairs 100 metres more per day than Team A.”

Difference between counts“There are 40 more 8-jiao pencils than 4-jiao pencils.”

Do not subtract unlike units. A difference of 26 legs is not 26 animals. A difference of 100 metres per day is not 100 days. Keep the unit attached to every number.

Check the meaning of the clues

In the animal statement, what does 26 measure?

In the repair statement, what does 100 measure?

In the pencil statement, what does 40 compare?

Which habit should come first?

Mission 2

Neutralize a difference to create equality

There are 100 chickens and rabbits. Each chicken has two legs and each rabbit four. The chickens have 26 more legs altogether than the rabbits. When one group’s total contribution is larger, set aside enough of that group to remove the stated difference.

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The starting relationship

chicken-leg total = rabbit-leg total + 26

Each chicken contributes 2 legs. Setting aside one chicken removes 2 legs from the larger total.

26 ÷ 2 = 13 chickens

After those 13 chickens are set aside, the remaining chicken-leg total equals the rabbit-leg total.

Difference-removal stagesinteractive

Begin with a 26-leg gap between the two group totals. The bars show the relationship, not a scale drawing.

Complete the transformation

Chickens set aside

Animals left in the equal-contribution core

Remaining chicken count : rabbit count

Why is the ratio 2 : 1?

Mission 3

Solve the complete 100-animal problem

There are 100 chickens and rabbits altogether. The chickens’ total number of legs is 26 greater than the rabbits’ total number of legs.

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Test possible rabbit counts

Rabbit count29

71chickens

29rabbits

142chicken legs

116rabbit legs

Difference: 26 legs — the condition is matched.

Reasoning chain

26 ÷ 2 = 13 extra chickens100 − 13 = 87 animals in the equal core87 ÷ (2 + 1) = 29 rabbits100 − 29 = 71 chickens

The 13 set-aside chickens do not disappear. Add them back to the 58 chickens inside the equal core.

Verify both totals

Rabbits

Chickens

Chicken-leg total

Rabbit-leg total

One hint

The two leg totals should differ by 26, while the two animal counts should add to 100.

Mission 4

Reveal the hidden ratio structure

The same solution can be seen as 13 set-aside chickens plus equal-contribution bundles. These 13 are extra beyond two chickens for every rabbit; they are not the difference between the original chicken and rabbit counts.

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One equal-contribution bundle

C
2C
2R
4

Two chickens contribute:

2 + 2 = 4 legs

One rabbit contributes:

4 legs

So every bundle keeps the two group totals equal.

Build the whole population

Equal bundles29

13extra chickens

71total chickens

29rabbits

100all animals

Twenty-nine bundles plus 13 extra chickens make exactly 100 animals.

Algebra bridge: the relationship can be written as C = 2R + 13. This does not replace the visual reasoning; it records it compactly.

Read the hidden structure

Equal bundles needed

Rabbits represented by those bundles

Chickens inside the bundles

Total chickens after adding the 13 extras

Mission 5

Remove a rate bonus before finding the base rate

Two road crews repair a 4,200-metre road, each at a constant daily rate. Crew B repairs 100 metres more per day than Crew A. Crew A works alone for 3 days, then both crews work together for 6 days.

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Work timeline9 A-days · 6 B-days

Crew A

day 1

2

3

4

5

6

7

8

9

Crew B

not working

A rate +100

+100

Crew A contributes 9 base-rate days. Crew B contributes 6 base-rate days plus six 100-metre bonuses.

Test Crew A’s daily rate

Crew A rate240 m/day

360015 base-rate days

600B’s bonus distance

4200total repaired

340B rate

The model matches 4,200 metres.

1 · Count base periods9 + 6 = 15 team-days

2 · Remove the bonus6 × 100 = 600 m

3 · Find A’s rate(4200 − 600) ÷ 15 = 240

4 · Restore B’s bonus240 + 100 = 340

Complete the repair model

Total days Crew A works

Total days Crew B works

B’s total bonus distance

Crew A rate (m/day)

Crew B rate (m/day)

Mission 6

Remove an extra count before forming equal pairs

A school buys 4-jiao and 8-jiao pencils for 68 yuan altogether. There are 40 more 8-jiao pencils than 4-jiao pencils. Find both counts.

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Money units: 1 yuan = 10 jiao, so 68 yuan = 680 jiao. Use jiao for every price and total in this problem.

Pencil-count model40 extra high-price pencils

The sample cards show equal pairs plus a separate group of 40 extra 8-jiao pencils. The calculations use every pencil.

Keep the 40-pencil difference

4-jiao pencils30

304-jiao pencils

708-jiao pencils

680total jiao

matchedtarget check

The counts match the 680-jiao total.

680 jiao total

−

40 × 8 = 320 jiao

=

360 jiao for equal pairs

Why equal pairs appear

After setting aside the 40 extra 8-jiao pencils, the two remaining pencil counts are equal.

one pair costs 4 + 8 = 12 jiao360 ÷ 12 = 30 pairs

Restore the extra count

4-jiao pencils = 308-jiao pencils = 30 + 40 = 7030×4 + 70×8 = 680 jiao

Complete the count-difference reasoning

68 yuan in jiao

Cost of the 40 extra 8-jiao pencils

Cost left for equal pairs

Number of equal pairs

4-jiao pencils

8-jiao pencils

Mission 7

Turn a travel-time split into a two-type model

A trip of 460 kilometres takes 9 hours of travel, with no stopping or transfer time included. Part is by car at 40 km/h and the rest is by train at 60 km/h. How long is each part?

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Change car-hours into train-hours

Train hours5 h

4car hours

5train hours

460kilometres

matchedtarget check

Use an all-car baseline

9 × 40 = 360 km

The real trip is:

460 − 360 = 100 km farther

Changing one hour from car to train adds:

60 − 40 = 20 km

Therefore:

100 ÷ 20 = 5 train-hours

Complete the time split

All-car baseline distance

Extra distance needed

Extra distance per train-hour

Train time (hours)

Car time (hours)

Mission 8

Choose the transformation that matches the clue

The stories look different, but each one becomes easier after removing a carefully identified difference.

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Group-total differenceSet aside enough larger-group contribution to make the totals equal.

Rate bonusRemove bonus rate × bonus periods from the total work.

Count differenceSet aside the extra high-type items, then form equal low-high pairs.

Speed bonusStart with all low-speed time, then upgrade selected time blocks.

Never lose the multiplier

A per-unit difference must be multiplied by the number of units that receive it.

total bonus = bonus per unit × number of bonus units

Examples:

6 workdays × 100 m/day
40 extra pencils × 8 jiao/pencil
5 train-hours × 20 km/hour extra

Check after transforming

Restore the bonus or difference.
Rebuild the original total.
Check counts, time, and units.
Reject negative or impossible quantities.

Error detective

Why subtract 600 metres in the crew problem?

Why subtract 320 jiao in the pencil problem?

Why set aside 13 chickens?

Which move is invalid?

Mission 9

Hidden-relationship workshop

Solve at least six of the eight problems. Write only the requested number.

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1. Sixty chickens and rabbits. Chicken-leg total exceeds rabbit-leg total by 24. Rabbits?

Hint

Set aside 24 ÷ 2 chickens, then use a 2 : 1 core.

2. Ninety chickens and rabbits. The two groups have equal total legs. Rabbits?

Hint

Equal leg totals mean two chickens for every rabbit.

3. Crew A works alone 2 days, then both crews work 5 days. B repairs 80 m/day more. Total 3,280 m. A’s rate?

Hint

A works 7 days; B works 5. Remove 5 × 80 first.

4. Four premium fans and six standard fans cost 1,320 yuan. Premium is 30 yuan more. Standard price?

Hint

Remove four 30-yuan bonuses, then divide by ten fans.

5. A 340-km trip uses only constant speeds of 40 km/h and 60 km/h, with 7 hours of travel and no stops. Hours at 60 km/h?

Hint

Compare with seven hours entirely at 40 km/h.

6. Eight adult and twelve child tickets cost 620 yuan. Adult tickets cost 15 yuan more. Child price?

Hint

Remove eight adult-price bonuses.

7. Fifty 2-unit and 4-unit objects. The 2-unit group total exceeds the 4-unit group total by 10. Number of 4-unit objects?

Hint

Use 2L − 4H = 10 and L + H = 50, or neutralize five low objects.

8. Each higher-type object contributes a bonus of 12 units above the lower-type contribution. Which count multiplies 12 to find the total bonus?

Hint

Ask which objects actually receive the bonus.

Mission 10

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.

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1. There are 84 chickens and rabbits. The chickens have 18 more legs altogether than the rabbits. How many chickens should we set aside to equalize the leg totals?

Hint

Each chicken accounts for 2 of the extra legs.