Find right and wrong counts from total score

3.OA.D.83.OA.A.3 · take · grade 3

Archetype: Two-Category Counts from a Total · step in a 3-type progression

In a math quiz contest, you earn $4$ points for each question you get right and lose $1$ point for each question you get wrong. If Ava solved $50$ questions in this contest and ended up with $155$ points, how many questions did Ava get wrong?

Show solution

Understand

Ava answered 50 quiz questions. Each correct answer is worth +4 points and each wrong answer is -1 point. Her final score was 155 points. We must find how many questions she got wrong.

Givens

Total questions answered: 50
Right answer: +4 points
Wrong answer: -1 point
Final score: 155 points

Unknowns

The number of questions Ava got wrong

Constraints

Right count plus wrong count equals 50
Both counts are whole numbers from 0 to 50

Plan

#6 Guess and Check · also uses: #9 Solve an Easier Related Problem#11 Work Backwards

Start from the easier 'all correct' case to get a clean total, then see how each wrong answer changes the score by a fixed amount. The fixed drop per wrong answer lets us divide to find the count.

Execute

#9 Solve an Easier Related Problem 3.OA.A.3

If all 50 answers were correct, Ava would earn 4 points each.

50 \times 4 = 200

Pretending everything is right gives a simple multiplication to anchor the count.

#6 Guess and Check 3.OA.B.5

Turning one right answer into a wrong one loses the +4 it would have earned and adds a -1 penalty, so the score drops by 5 for each wrong answer.

4 + 1 = 5

Each mistake costs the missed 4 points plus a 1-point penalty, a steady 5 each time.

#11 Work Backwards 3.OA.D.8

Her actual score is 155, which is 45 below the perfect 200.

200 - 155 = 45

The gap from a perfect score is the total damage all the wrong answers caused.

#6 Guess and Check 3.OA.A.3

Each wrong answer costs 5 points, and the total loss is 45, so divide to get the number of wrong answers.

45 \div 5 = 9

Sharing the 45 lost points into groups of 5 tells how many mistakes there were.

Answer: 9 questions

Review

If 9 are wrong, 41 are right: 41 times 4 = 164, minus 9 = 155, which matches the final score exactly. The counts 41 and 9 add to 50.

Convert to a single-unknown equation (tool 13): with w wrong, right is 50 - w, so 4 times (50 - w) minus w = 155, giving 200 - 5w = 155 and w = 9.

Standards · min grade 3

3.OA.A.3 Solve multiplication and division word problems within 100 — Finding the perfect score and dividing the lost points into groups of 5
3.OA.B.5 Apply properties of operations as strategies to multiply and divide — Reasoning that each wrong answer changes the score by a fixed 5 points
3.OA.D.8 Solve two-step word problems using four operations within 100 — Combining the perfect score and the score gap across multiple steps

💡 Pretend every answer was right, then count how far the score fell; each mistake costs the same amount, so just divide!