Pass and Fail Percentage: Formulas, Rules and Questions

Pass and Fail Percentage problems relate marks scored to the minimum marks required for passing. These questions commonly ask for pass marks, total marks, marks by which a candidate passes or fails, or the percentage shortfall. The main method is to express the difference in marks as a percentage of the total marks and form an equation.

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What is Pass and Fail Percentage?

Pass percentage is the percentage of total marks required to pass an examination. A candidate fails when the marks scored are below this threshold and passes when the marks scored are equal to or above it.

If the maximum marks are M and the passing marks are P, then passing percentage = (P ÷ M) × 100. If a candidate scores S marks, the candidate passes when S ≥ P and fails when S < P. The marks needed to pass are P − S for a failed candidate, while S − P is the excess marks for a passed candidate.

Pass and Fail Percentage Formula & Tricks

Important Formulas

Passing Percentage
Passing percentage = (Passing marks ÷ Maximum marks) × 100

Use this when the passing marks and maximum marks are known.

Passing Marks
Passing marks = (Passing percentage ÷ 100) × Maximum marks

This gives the minimum score required to pass.

Total Marks when a Candidate Fails
Total marks = Marks by which the candidate fails ÷ (Pass percentage − Candidate's percentage) × 100

If a candidate scores x% and fails by f marks, total marks = 100f ÷ (pass percentage − x).

Total Marks when Candidates Pass and Fail
Total marks = (Marks by which one passes + Marks by which another fails) ÷ (Difference in their percentages) × 100

If one candidate scores y% and passes by p marks, while another scores x% and fails by f marks, total marks = 100(p + f) ÷ (y − x).

Percentage Shortfall
Percentage shortfall = (Passing marks − Marks scored) ÷ Maximum marks × 100

For a failed candidate, this measures the marks short of the passing score as a percentage of maximum marks.

Quick Tricks

Combine the pass and fail differences

When one candidate passes by p marks and another fails by f marks, the difference between their marks is p + f. This avoids finding either score separately before forming the percentage equation.

Example: A candidate passing by 30 marks and another failing by 20 marks have a score difference of 50 marks.
Use percentage difference as the denominator

The difference in marks equals the difference in percentages applied to the total marks. Keep percentage values in the numerator as whole numbers and multiply by 100.

Example: If the percentage difference is 10 and the mark difference is 50, total marks = 50 × 100 ÷ 10 = 500.

Pass and Fail Percentage Concepts

Finding Pass Marks from Total Marks

Pass marks are obtained by applying the required pass percentage to the maximum marks.

For maximum marks M and pass percentage r, pass marks = rM ÷ 100. If the result is not a whole number, the examination rule may require the next whole mark, but the stated rounding rule must be followed.

Example: For a 40% pass mark in an examination of 750 marks, pass marks = 40 × 750 ÷ 100 = 300 marks.

Candidate Fails by a Given Number of Marks

If a candidate scores x% and fails by f marks, the passing percentage exceeds the candidate's percentage by f expressed over the total marks.

Let the total marks be M and the pass percentage be r. The candidate's score is xM ÷ 100, while the pass marks are rM ÷ 100. Therefore, (r − x)M ÷ 100 = f, so M = 100f ÷ (r − x).

Example: A student scores 35% and fails by 30 marks when the pass percentage is 40%. Total marks = 100 × 30 ÷ (40 − 35) = 600.

Candidate Passes by a Given Number of Marks

If a candidate scores x% and passes by p marks, the candidate's score exceeds the pass marks by p.

For total marks M and pass percentage r, xM ÷ 100 − rM ÷ 100 = p. Hence, M = 100p ÷ (x − r), provided x is greater than r.

Example: If a candidate scores 55% and passes by 30 marks at a pass percentage of 50%, total marks = 100 × 30 ÷ (55 − 50) = 600.

One Candidate Passes and Another Fails

When one candidate passes by p marks at y% and another fails by f marks at x%, total marks are found from their combined mark difference.

The difference in their scores is p + f, and the difference in their percentages is y − x. Thus, (y − x)% of the total marks equals p + f, giving total marks = 100(p + f) ÷ (y − x).

Example: If a candidate scoring 50% fails by 20 marks and another scoring 60% passes by 30 marks, total marks = 100 × (20 + 30) ÷ (60 − 50) = 500. The pass marks are 60% of 500 − 30 = 270 marks.

Pass and Fail Percentage Video Lessons

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Pass-Fail Marks Percentage Problems

Learn to solve percentage-based pass and fail marks problems by calculating required marks, pass percentages, failed marks, and related exam-style quantities.

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Practice Pass and Fail Percentage Questions

Practise published questions related to this topic.

1A shopkeeper marks an article 30% above its cost price. He then allows a discount of 20% on the marked price and still makes a profit of ₹32. Find the cost price of the article.→ 2Find the rate of discount when the marked price is ₹11,280 and the selling price is ₹9,588.→ 3A salesman marks his goods 5% above the cost price and gives a discount of 40% on them. Find his loss percentage.→ 4An article is marked for ₹2,000. A 25% discount is offered on the marked price, followed by a second discount of 10% on the new price. What is the final selling price of the article?→ 5A person allows a 10% discount for cash payment from the marked price of a toy and still he makes a 10% gain. What is the cost price of the toy which is marked at ₹880?→ 6A = Largest two digit number exactly divisible by both 2 and 11. B = Largest three digit number exactly divisible by both 6 and 13. What is the value of B - A?→ 7A person sold an article at a loss of 10%. Had he sold it for ₹60 more, he would have gained 5%. Find the cost price of the article.→ 8The product of two numbers is 300 and their LCM is 150. Find their HCF.→ 9A three-digit number is such that the sum of the number and its square root is 132 . Find the number.→ 10A shopkeeper sells an item for ₹225.6 after giving two successive discounts of 90% and 50% on its marked price. Had he not given any discount, he would have earned a profit of 41%. What is the cost price (in ₹) of the item?→

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Quick Revision Notes

Pass and Fail Percentage Revision Points

Use the following relations to solve pass and fail percentage questions.

  • Pass marks = (Pass percentage ÷ 100) × Maximum marks.
  • A failed candidate's shortfall = Pass marks − Marks scored.
  • If x% fails by f marks, total marks = 100f ÷ (Pass percentage − x).
  • If x% passes by p marks, total marks = 100p ÷ (x − Pass percentage).
  • If y% passes by p marks and x% fails by f marks, total marks = 100(p + f) ÷ (y − x).
  • A candidate passes at exactly the passing marks; a score below the passing marks is a failure.
  • Percentage shortfall is measured as shortfall marks ÷ maximum marks × 100, not as 100 minus the pass percentage.

Pass and Fail Percentage FAQs

What is the pass percentage formula?

Pass percentage = (Passing marks ÷ Maximum marks) × 100. For example, 240 passing marks out of 600 gives 240 ÷ 600 × 100 = 40%.

How do you find total marks if a student scores 30% and fails by 40 marks at a 40% pass mark?

Total marks = 100 × 40 ÷ (40 − 30) = 400 marks. The pass marks are 40% of 400, or 160 marks, and the student's score is 120 marks.

How do you find total marks when one student passes by 25 marks and another fails by 15 marks?

If their percentages are y% and x%, respectively, total marks = 100(25 + 15) ÷ (y − x). For y = 60 and x = 50, total marks = 100 × 40 ÷ 10 = 400.

What is the difference between failing by marks and percentage shortfall?

Failing by marks is the numerical difference between pass marks and marks scored. Percentage shortfall expresses this difference over maximum marks: (Pass marks − Marks scored) ÷ Maximum marks × 100.

A student scores 48% and passes by 16 marks when the pass percentage is 44%. What are the total and passing marks?

The percentage difference is 4%. Total marks = 100 × 16 ÷ 4 = 400. Passing marks = 44% of 400 = 176 marks, and the student's score is 192 marks.

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