Chapter 9 · Lesson 9.6

Expected versus Actual Totals and Reverse Processes

预期总量、实际总量与逆向过程

Some problems hide the answer in what was lost, damaged, removed, or added to everyone. Learn to compare a normal total with the actual total, rebuild an earlier state from repeated changes, and remove a common bonus before separating two types.

Build the normal picture. Measure the change. Work backward carefully.

Grade 5 enrichment10 interactive missionsNo this lesson requiredObjective assessment

Mission 1

Compare a normal total with the actual total

When every item is normal, the total is easy to predict. A damaged or changed item may remove both its normal contribution and an extra penalty.

Not complete

Expected total

In every payment problem on this page, “actual payment” means the final amount after all damage compensation has been deducted. Each item is either intact or damaged.

Suppose N items would normally earn p units each.

Expected total = N × p

This is the total if nothing changes.

Loss caused by one damaged item

If a damaged item earns no normal payment and also causes a penalty c, then one damaged item lowers the payment by:

p + c

It loses its normal payment and adds a penalty.

Core model: number changed = (expected total − actual total) ÷ loss caused by one changed item.

Watch a simple damage model

There are 20 packages. Each intact package earns 5 units. Each damaged package earns nothing and costs an extra 3-unit penalty.

Damaged packages2

100expected total

8loss per damaged item

16total loss

84actual total

Check the model

Expected total for 20 packages at 5 units each

Loss caused by one damaged package

If the actual total is 84, total loss

Number damaged

Why does one damaged item cause a loss of 8?

Mission 2

Broken bottles: compare perfect delivery with actual pay

All information needed for this investigation is included below.

Not complete

Complete problem

A truck carries 2,000 glass bottles. The carrier earns 0.2 yuan for each bottle that arrives intact. For a broken bottle, the carrier receives no freight payment for that bottle and must also pay 1 yuan in compensation. After subtracting compensation, the carrier’s net payment is 379.6 yuan. How many bottles were broken?

Expected
400 yuan

Missing
20.4 yuan

Actual
379.6 yuan

All intact2000 × 0.2 = 400

→

Total loss400 − 379.6 = 20.4

→

One broken bottle0.2 + 1 = 1.2

Test a possible number of broken bottles

Broken bottles17

20.4payment lost

379.6actual payment

1.2loss per broken

Matchtarget check

Record the solution

Expected payment if all arrive intact

Total payment loss

Loss caused by one broken bottle

Broken bottles

Mission 3

Damaged tea sets: decimals and compensation

The same model works when the normal payment and the penalty are both decimals.

Not complete

Complete problem

A transport team carries 1,998 glass tea sets. The contract pays 1.6 yuan for each set that arrives intact. For every damaged set, the team receives no freight payment for that set and must pay 18 yuan in compensation. After subtracting compensation, the team receives 3,059.6 yuan. How many tea sets were damaged?

Build the perfect-delivery total

1998 × 1.6 = 3196.8

This is the expected payment if every tea set arrives intact.

Build the loss per damaged set

1.6 + 18 = 19.6

Each damaged set loses its 1.6-yuan freight payment and adds 18 yuan compensation.

Difference: 3196.8 − 3059.6 = 137.2. Divide that loss by 19.6 for each damaged set.

Complete the calculation

Expected payment

Loss per damaged set

Total payment loss

Damaged sets

Which verification is correct?

Mission 4

Use a general expected-versus-actual laboratory

Change the numbers and let the model check whether a whole-number answer is possible.

Not complete

Damage-loss calculator

Number of items

Normal payment per intact item

Extra penalty per damaged item

Actual payment

400expected total

1.2loss per changed item

20.4total difference

17changed-item count

The numbers produce a valid whole-number count.

Practice-lamp preset

Three hundred lamps would earn 0.4 unit each if intact. Each damaged lamp earns no normal payment and also incurs a 3.6-unit penalty. The actual total is 112.

Expected 120; loss per damaged lamp 4; total loss 8

Feasibility checks

A valid count must be a nonnegative whole number and cannot exceed the total number of items.

If the actual total is above the expected total, this particular damage model does not fit.

Read the laboratory

Use the named, fixed presets, not any edited input values.

Changed items in the bottles preset

Changed items in the tea-set preset

Changed lamps in the practice preset

Correct general loss per damaged item

Mission 5

Reverse repeated removals to rebuild the starting amounts

A final state may be easier to understand by adding every removed amount back.

Not complete

Complete red-and-white ball problem

A box originally contains red and white glass balls. The number of red balls is 2 more than three times the number of white balls. In every round, 15 red balls and 7 white balls are removed. After several rounds, 53 red balls and 3 white balls remain. How many red balls were in the box at the beginning?

Initial red53 + 15kadd back 15 per round

must equal
3 × white + 2

Initial white3 + 7kadd back 7 per round

53 + 15k = 3(3 + 7k) + 2

Work through the equation

Expand the right side: 53 + 15k = 11 + 21k. Subtract 11 and 15k from both sides: 42 = 6k. Therefore k = 7 rounds. Add back to check: red = 53 + 105 = 158; white = 3 + 49 = 52; and 158 = 3 × 52 + 2.

Test the number of rounds

Rounds completed7

158initial red

52initial white

1583 × white + 2

Matchrelation check

Complete the reverse process

Number of rounds

Initial white balls

Initial red balls

Which step rebuilds the initial red count?

Mission 6

Use a general reverse-process laboratory

The same reasoning works whenever two quantities obey an initial relationship and both change by fixed amounts each round.

Not complete

Reverse repeated changes

At the beginning, quantity A obeys A = mB + c. Each round removes fixed amounts from A and B. The final amounts are known.

Multiplier m

Extra c

Removed from A each round

Removed from B each round

Final A

Final B

7rounds

158initial A

52initial B

Validwhole-number check

The initial relationship is satisfied.

Practice-counter preset

Initially, A = 2B + 4. Each round removes 4 from A and 1 from B. Finally A is 8 and B is 5.

8 + 4k = 2(5 + k) + 4

Read both presets

Use the original ball problem from Mission 5 and the practice-counter statement above. Edited laboratory values do not change these questions.

Rounds in the worked example-balls preset

Rounds in the practice-counter preset

Initial A in the practice-counter preset

Initial B in the practice-counter preset

Mission 7

Remove a common temporary bonus before separating two types

When the same bonus is added to everyone, subtract that common part first. Then the ordinary two-type difference becomes visible.

Not complete

Complete monkey problem

There are 35 monkeys, some large and some small. These 35 workers exclude the king, who supervises but does not pick peaches. Each worker picks for all eight hours. Without the monkey king, one large monkey picks 15 kg of peaches per hour and one small monkey picks 11 kg per hour. Whenever the monkey king supervises, every monkey picks 12 kg more per hour. The monkeys work for 8 hours, but the king supervises only during the first hour and last hour. Together they pick 4,400 kg. How many small monkeys are there?

Hour 1
+12 each

Hour 2
ordinary

Hour 3
ordinary

Hour 4
ordinary

Hour 5
ordinary

Hour 6
ordinary

Hour 7
ordinary

Hour 8
+12 each

Common bonus35 × 12 × 2 = 840

→

Ordinary contribution4400 − 840 = 3560

→

All-small baseline35 × 11 × 8 = 3080

Each large monkey instead of a small monkey adds: (15 − 11) × 8 = 32 kg. The ordinary total is 480 kg above the all-small baseline.

Test the number of large monkeys

Large monkeys15

20small monkeys

3560ordinary 8-hour total

840common bonus

4400complete total

Complete the monkey calculation

Common temporary bonus

Ordinary contribution after removing the bonus

All-small ordinary baseline

Large monkeys

Small monkeys

Why remove the 840 first?

Mission 8

Choose the right transformation and repair common errors

The arithmetic is often short after the story is transformed into the right model.

Not complete

Expected vs actual

Damage, breakage, or penalty

Build the perfect total, then divide the missing amount by the loss from one changed item.

Reverse process

Repeated removals with a final state

Add each removed amount back for every round, then enforce the initial relationship.

Remove common bonus

Everyone receives the same temporary increase

Subtract the common addition before comparing the two ordinary types.

Ordinary baseline

Two types with different normal contributions

Assume every object is the low type, then repair the total one replacement at a time.

Strategy and error detective

Broken goods with lost freight and compensation

Two ball colors are removed in fixed batches and final counts are known

Every worker gets the same two-hour productivity bonus

A damaged item loses 0.5 normal pay and adds a 2-unit penalty. Loss per item

To rebuild an earlier amount after removals, should you add back or subtract again?

Adding the same bonus rate to both types changes the difference between their per-hour rates.

Mission 9

Independent workshop

Solve at least six of the eight questions. Try first, then use a hint; worked solutions are available for checking your reasoning.

Not complete

1. Four hundred parcels earn 0.5 yuan each if intact. Each damaged parcel earns nothing and incurs a 2-yuan penalty. Actual payment is 190 yuan. How many parcels were damaged?

Hint

Compare perfect payment with actual payment.

Worked solution — open after trying

(400 × 0.5 − 190) ÷ (0.5 + 2) = 4.

2. One hundred plates earn 2 yuan each if intact. A broken plate earns nothing and incurs a 3-yuan penalty. What is the payment loss per broken plate, in yuan?

Hint

Include both the lost payment and the penalty.

Worked solution — open after trying

2 + 3 = 5 yuan per plate.

3. For those 100 plates, actual payment is 175 yuan. How many plates broke?

Hint

Calculate the payment if every plate arrived intact.

Worked solution — open after trying

(200 − 175) ÷ 5 = 5 broken plates.

4. Sixty lamps earn 1 yuan each if intact. A damaged lamp earns nothing and incurs a 4-yuan penalty. Actual payment is 45 yuan. How many lamps were damaged?

Hint

Compare the perfect payment with the actual payment, then find the full loss per damaged lamp.

Worked solution — open after trying

(60 − 45) ÷ (1 + 4) = 3.

5. Initially A = 3B + 2 counters. Each round removes 5 A and 1 B. Finally A = 12 and B = 6. How many rounds occurred?

Hint

Add back: 12 + 5k = 3(6 + k) + 2.

Worked solution — open after trying

12 + 5k = 20 + 3k; 2k = 8, so k = 4.

6. In that counter problem, what was the initial B count?

Hint

Add back one B for every round.

Worked solution — open after trying

6 + 4 = 10. Initial A = 12 + 20 = 32; 32 = 3 × 10 + 2.

7. Eight workers work 2 hours. In addition to the boxes counted by the two-hour ordinary rate, each packs 3 bonus boxes once during the day. How many bonus boxes are packed altogether?

Hint

The bonus is once per worker, not per hour.

Worked solution — open after trying

8 × 3 = 24 boxes.

8. Those eight workers pack 88 boxes including the bonus. Faster workers pack 5 boxes/hour; slower workers pack 3 boxes/hour. How many faster workers are there?

Hint

Remove the common bonus you found in Question 7, then use an all-slower baseline for two hours.

Worked solution — open after trying

(88 − 24 − 8 × 2 × 3) ÷ (2 × (5 − 3)) = 4 faster workers. Check: 4 × 10 + 4 × 6 + 24 = 88.

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.

Not complete

1. Two hundred cups earn 0.5 yuan each if intact. A broken cup earns nothing and incurs a 1.5-yuan penalty. Actual payment is 92 yuan. How many cups broke?

Hint

Find the perfect payment and the full loss per broken cup.

Worked solution — open after trying

(200 × 0.5 − 92) ÷ (0.5 + 1.5) = 4 cups.

2. Initially red counters = 2 × blue counters + 3. Each round removes 5 red and 2 blue. Finally there are 9 red and 5 blue. How many rounds occurred?

Hint

Add back: 9 + 5k = 2(5 + 2k) + 3.

Worked solution — open after trying

9 + 5k = 13 + 4k, so k = 4.

3. In that counter problem, how many red counters were there initially?

Hint

Add back five reds for each round.

Worked solution — open after trying

9 + 5 × 4 = 29 reds; blue = 5 + 2 × 4 = 13, and 29 = 2 × 13 + 3.

4. Ten workers each work 3 hours. Faster workers pack 8 boxes/hour; slower workers pack 5 boxes/hour. In addition to the boxes counted by the three-hour ordinary rate, each worker packs 2 bonus boxes once that day. Total: 206 boxes. How many boxes are the common bonus?

Hint

The bonus happens once per worker, not once per hour.

Worked solution — open after trying

10 × 2 = 20 bonus boxes.

5. In that worker problem, how many faster workers are there?

Hint

Remove the bonus, then use an all-slower baseline.

Worked solution — open after trying

(206 − 20 − 10 × 3 × 5) ÷ (3 × (8 − 5)) = 4. Check: 4 × 24 + 6 × 15 + 20 = 206.

Optional explanation: Why does your method work? Check both totals.

✓

Lesson checkpoints completed

Change-Process Detective

You completed the checkpoints on payment losses, reverse processes and common bonuses. Revisit any steps for which you needed solution help.

Lesson 9.6 complete

Teaching notes

The student lesson is self-contained.The general laboratories, practice-package and practice-lamp examples, feedback, and assessment items are additional instructional scaffolds.

” All data and context needed to solve every activity are included here.

← Previous: Lesson 9.5 Next: Lesson 9.7 →