Chapter 9 · Lesson 5 · Grade 5

Scores, Penalties, and Blank Answers

得分题、扣分题和不作答

Correct, wrong, and blank responses are three different “types.” Learn how their point contributions combine, how score bounds reveal hidden counts, and when a score does—or does not—determine a unique answer pattern.

Count the response types. Track what each type contributes. Verify the score.

10 interactive missionsoriginal problems included in fullObjective grading onlyAutosaves in this browser

Mission 1

See correct, wrong, and blank as three response types

A score is a weighted total. The number of questions and the number of points are different kinds of quantities.

Not complete

A complete scoring rule

Each question is exactly one of correct, wrong, or blank. Add the stated contributions to obtain the score, starting from zero; do not replace a negative total with zero.

Response type	Contribution	Meaning
Correct	+4	Adds four points
Wrong	−1	Removes one point
Blank	0	Changes nothing

Score = 4 × correct − 1 × wrong + 0 × blank

Two totals must stay separate

A test can have:

25 questions

and a student can earn:

78 points

You cannot subtract 25 from 78, because questions and points are different units.

Chapter connection: response types play the same role as chickens, rabbits, or room sizes. Each type contributes a different amount.

Check the language of the model

Which quantity is a weighted total?

What does a blank answer contribute in this rule?

Which equation counts every question?

Can a score be negative when wrong answers lose points?

Mission 2

Start from an all-correct baseline

A baseline makes every question the same type. Then each replacement changes the score by a fixed amount.

Not complete

All 25 correct

25 × 4 = 100 points

The actual score is 78, so the response pattern must lose:

100 − 78 = 22 points

Replacement costs

Changing one correct answer into a wrong answer changes:

+4 → −1, a loss of 5

Changing one correct answer into a blank changes:

+4 → 0, a loss of 4

Try replacement counts

Wrong answers2

Blank answers3

20correct

22points lost

78score

25responses

CorrectWrongBlank

This pattern has 25 responses and scores 78 points.

Read the baseline

All-correct score

Points lost from the all-correct baseline to the target score 78

Loss when correct becomes wrong

Loss when correct becomes blank

Mission 3

Use score bounds to pin down the number correct

For a fixed correct count, the remaining questions can be wrong or blank. That creates a possible score range.

Not complete

Too few correct

With 19 correct, even making every other question blank gives the highest possible score:

19 × 4 = 76 < 78

So the student must have at least 20 correct answers.

Too many correct

With 21 correct, making all four remaining questions wrong gives the lowest possible score:

21 × 4 − 4 = 80 > 78

So the student cannot have 21 or more correct answers. With each additional correct answer, even the minimum score increases: one −1 is replaced by +4.

Correct-count range explorer

Assumed correct count20

75minimum score

80maximum score

78target score

Yespossible?

5060708090100

Use both bounds

Minimum possible correct count

Why are 19 correct answers impossible?

Why are 21 correct answers impossible for the target score 78?

Therefore, exact correct count

Mission 4

Complete the 78-point problem: find wrong and blank answers

Now that the correct count is known, the remaining five responses can be separated by how they affect the score.

Not complete

Complete problem

A mathematics test has 25 multiple-choice questions. A correct answer earns 4 points, a wrong answer loses 1 point, and a blank answer earns 0 points. A student scores 78 points. How many questions were correct, wrong, and blank?

Twenty correct answers give20 × 4 = 80

→

The actual score is78

Only wrong answers deduct points. The score is 2 points below 80, so exactly two of the five remaining responses are wrong. The other three are blank.

Move the wrong-answer slider

Wrong among the remaining 52

20correct

2wrong

3blank

78score

Record and verify the solution

Correct answers

Wrong answers

Blank answers

Verification score

With w wrong answers and b blanks, which all-correct replacement equation also matches?

Mission 5

Test whether a score has one solution, several, or none

With blanks allowed, total questions and score can sometimes leave more than one response pattern. A systematic integer search prevents guessing.

Not complete

Feasible-combination explorer

Questions

Points for correct

Penalty magnitude for wrong

Target score

Blank answers are allowed

1valid patterns

Uniquestatus

20least correct

20most correct

Correct	Wrong	Blank	Question check	Score check

An ambiguous score

Ten questions use +4, −1, and 0. A score of 20 can come from:

5 correct, 0 wrong, 5 blank6 correct, 4 wrong, 0 blank

So the score alone does not always determine all three counts.

When all questions are answered

If blanks are forbidden, only correct and wrong remain. The total question count and the score usually determine one pair—or prove the score impossible.

Read the explorer

Answer for the fixed scenarios named below. Changing the explorer’s inputs does not change these questions.

Valid patterns for 25 questions, +4/−1/0, score 78

Valid patterns for 10 questions, +4/−1/0, score 20

If blanks are forbidden in the 10-question, +4/−1, score-20 case, correct answers

Why can blanks create ambiguity?

Mission 6

Guided practice: three students answer every question

When there are no blanks, an all-wrong baseline creates a score ladder with equal steps.

Not complete

Complete original practice

Three students each answer all 10 questions. A correct answer earns 10 points, and a wrong answer loses 3 points. Xiaoming scores 87, Xiaohong scores 74, and Xiaohua scores 9. How many questions did they answer correctly altogether?

All-wrong baseline

10 × (−3) = −30

Changing one wrong answer to correct changes the score by:

10 − (−3) = 13

So every possible score is one step of 13 above −30.

Score ladder

Correct answers9

1wrong

87score

−30all-wrong baseline

13step size

Recover each correct count

All-wrong score

Score increase for one wrong-to-correct replacement

Xiaoming correct

Xiaohong correct

Xiaohua correct

Correct altogether

Mission 7

Split a total and a difference, then decode each score

Sometimes the individual scores are hidden, but their sum and difference reveal them first.

Not complete

Complete practice problem

Jiaojiao and Tiantian each answer all 10 questions. A correct answer earns 20 points, and a wrong answer loses 12 points. Together they score 208 points. Jiaojiao scores 64 points more than Tiantian. How many questions does each student answer correctly?

First recover the individual scores

Jiaojiao = (208 + 64) ÷ 2 = 136Tiantian = (208 − 64) ÷ 2 = 72

Then use the all-wrong baseline

10 × (−12) = −120

Each wrong-to-correct replacement adds:

20 − (−12) = 32

Score comparison

Jiaojiao

136

Tiantian

Correct = (score + 120) ÷ 32

Complete both layers of reasoning

Jiaojiao's score

Tiantian's score

Wrong-to-correct score step

Jiaojiao correct

Tiantian correct

Difference in correct answers

Mission 8

Choose the right strategy and reject impossible scores

Different clues suggest different first moves. Units, step sizes, and whole-number conditions are powerful error checks.

Not complete

All questions answeredUse an all-wrong baseline. Every replacement raises the score by correct points + penalty magnitude.

Blanks allowedUse score bounds, then list whole-number triples if needed.

Two hidden scoresSplit their total and difference before decoding each score.

Score-step feasibility

Ten answered questions use +6 and −2.

All-wrong baseline = −20Step size = 6 − (−2) = 8

Test the proposed score of 32:

32 − (−20) = 52

But 52 is not divisible by 8, so 32 is impossible under this rule.

Whole-number and range checks

Counts must be nonnegative whole numbers.
Counts must add to the total number of questions.
The reconstructed score must equal the stated score.
With no blanks, the score must land on the score ladder.

Error detective

If all questions are answered, best first baseline

With blanks allowed and too few clues, what should you do?

For +6/−2, score step size

Is score 32 possible for 10 answered questions under +6/−2?

Why must a decimal response count be rejected?

Mission 9

Scores and response types workshop

Solve at least six of eight. Open hints only after you have tried.

Not complete

1. All answered

Fifteen questions use +4 correct and −1 wrong. A student scores 45. How many are correct?

Hint

All wrong is −15. Each replacement adds 5.

2. Calculate a score

Twenty questions use +5, −2, and 0. A student has 15 correct, 3 wrong, and 2 blank. Find the score.

Hint

Use 5 × correct − 2 × wrong.

3. Another score ladder

Twelve answered questions use +8 correct and −2 wrong. A student scores 56. How many are correct?

Hint

All wrong is −24; each replacement adds 10.

4. Three response types

Ten questions use +4, −1, and 0. A student has 6 correct, 2 wrong, and 2 blank. Find the score.

Hint

Blank answers add zero.

5. Worked example

In the 25-question test scoring +4 for correct, −1 for wrong, and 0 for blank, a student scores 78. How many answers are wrong?

Hint

First use score bounds to find the correct count. Then compare its points with the actual score to find the penalty.

6. Three-student practice

For scores 87, 74, and 9 under +10/−3 with 10 answered questions, how many correct answers are there altogether?

Hint

The three correct counts are found from (score + 30) ÷ 13.

7. Two hidden scores

Two scores total 208 and differ by 64. What is the higher score?

Hint

Add the total and difference, then divide by 2.

8. Feasibility

Ten answered questions use +10 correct and −3 wrong. Can a student score 75?

Hint

Possible scores are −30 plus a whole-number multiple of 13.

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. Twelve questions are all answered. Correct earns 6 points; wrong loses 2. What is the all-wrong baseline? Include the minus sign.

Hint

Twelve wrong answers each contribute −2.