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AFCAT Numerical Ability · Percentage

Marks, Pass Marks and Elections

Exam scores as percentages, the pass mark from a shortfall or a surplus, maximum marks from two candidates, overall percentages across papers, and winning margins in a vote.

All Percentage notes

Why this matters

Every question here is one equation once the pass mark or the margin is written down, and most are EASY. The HARD ones give two candidates on either side of the pass mark, or a vote where some people did not vote; the marks go when the margin is taken on the wrong total.

Concept 1 of 5: Scores, questions and negative marking

A score is a part of the maximum marks, so it converts to a percentage like any other part. With negative marking, every wrong answer takes something back, so count right and wrong answers separately.

Definition

  • Percentage scored =marksmaximum×100= \dfrac{\text{marks}}{\text{maximum}} \times 100; maximum =marks×100percentage= \dfrac{\text{marks} \times 100}{\text{percentage}}.
  • Attempted and correct stack: correct == correct share ×\times attempted share ×\times total questions.
  • Negative marking: score == (marks per right ×\times right) −- (penalty ×\times wrong). Questions left blank score nothing.
  • If every question is attempted, wrong == total −- right, so one equation finds the number right.

Score with negative marking

score=m×right−p×wrong\text{score} = m \times \text{right} - p \times \text{wrong}

Worked example

A test has 50 questions with 4 marks for a right answer and 1 mark off for a wrong one. A student attempts every question and scores 130. How many were right, and what percentage of the maximum did the student score?
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Percentage · Marks and Elections

In an exam with 150150 questions, a student attempted 80%80\% and got 75%75\% of attempted questions correct. How many questions did he answer correctly?

Wrong answers cost twice

Moving one answer from right to wrong with +4 and −1 marking drops the score by 5, not 4. That is why the equation becomes 5x5x.
Drill 17 more on scores, questions and negative marking

Concept 2 of 5: The pass mark from one student's score

A student who scores 113 and fails by 22 shows that the pass mark is 135. If the pass mark is 36% of the maximum, the maximum follows.

Definition

  • Failed by ff marks: pass mark == score +f+ f. Passed by ss marks: pass mark == score −s- s.
  • Then maximum =pass mark×100pass percentage= \dfrac{\text{pass mark} \times 100}{\text{pass percentage}}.
  • 'Missed a distinction by 45 marks' works the same way, with the distinction cut-off in place of the pass mark.

Pass mark

pass mark=score+shortfall=p100×maximum\text{pass mark} = \text{score} + \text{shortfall} = \dfrac{p}{100} \times \text{maximum}

Worked example

A candidate needs 36% to pass. She scores 113 marks and fails by 22. Find the maximum marks.
Practice this conceptself-check · 2 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Percentage · Marks and Elections

A student needs 75%75\% marks to get distinction. He scored 540540 marks and missed distinction by 6060 marks. Find maximum marks.

Failing by means add, passing by means subtract

Scored 113 and failed by 22: the pass mark is 135. Subtracting gives 91, and that wrong pass mark leads to a wrong maximum among the options.
Drill 7 more on the pass mark from one student's score

Concept 3 of 5: Two scores either side of the pass mark

One student at 36% falls 33 short of the pass mark; another at 48% is 15 above it. The gap between them is 12% of the maximum and also 33+15=4833 + 15 = 48 marks, so 12% is 48 marks.

Definition

  • Write the pass mark both ways and set them equal: a100M+f=b100M−s\tfrac{a}{100}M + f = \tfrac{b}{100}M - s.
  • So (b−a)%(b - a)\% of MM =f+s= f + s: add the shortfall and the surplus.
  • If both are below the pass mark (both fail), subtract the two shortfalls instead.
  • With the pass percentage given and one score as a percentage, the gap between the two percentages is the shortfall.

Maximum from two candidates

b−a100 M=f+s\dfrac{b - a}{100}\,M = f + s

Worked example

A student who scores 36% fails by 33 marks. Another who scores 48% passes by 15 marks. Find the maximum marks and the pass mark.
Practice this conceptself-check · 2 quick reps

The same idea in a practice question:

Practice questionHard

Example 3 · Percentage · Marks and Elections

A student scores 70% and fails by 30 marks. He scores 80% and passes by 20 marks. What is the maximum marks?

Add the shortfall and the surplus

One student 33 below the pass mark and another 15 above it are 33+15=4833 + 15 = 48 marks apart. Subtracting, 33−1533 - 15, is only right when both students fail.
Drill 5 more on two scores either side of the pass mark

Concept 4 of 5: Overall percentage across papers

An overall percentage is total marks over total maximum. When every paper has the same maximum it is the plain average of the percentages; when papers carry different weights or maxima, each counts by its size.

Definition

  • Overall % =total markstotal maximum×100= \dfrac{\text{total marks}}{\text{total maximum}} \times 100.
  • Equal maxima: the average of the percentages.
  • Weights given (theory 60%, practical 40%): overall =∑(weight×percentage)= \sum (\text{weight} \times \text{percentage}).
  • A missing paper from an aggregate: aggregate ×\times number of papers −- the sum of the others (equal maxima).

Weighted overall percentage

overall=∑wipi∑wi\text{overall} = \dfrac{\sum w_i p_i}{\sum w_i}

Worked example

A student scores 70% in theory, which carries 60% of the weight, and 95% in practical, which carries 40%. Find the overall percentage.
Practice this conceptself-check · 2 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Percentage · Marks and Elections

A student scores 8585% in Math, 7575% in Science and 9090% in English. If the weights are 3:2:13:2:1 respectively, what is his weighted average percentage?

Unequal maxima are not averaged

40 out of 50 is 80% and 66 out of 150 is 44%, but the overall is 106200=53%\tfrac{106}{200} = 53\%, not 62%. The bigger paper counts for more.
Drill 12 more on overall percentage across papers

Concept 5 of 5: Elections: shares, margins and invalid votes

With two candidates, a winner on 58% leaves the loser 42%, so the winning margin is 16% of the votes. Turn the margin into a percentage and it gives the total.

Definition

  • Two candidates: margin == winner % −- loser % of the valid votes.
  • Invalid votes come off first: valid == total ×\times (1 −- invalid share).
  • If only some voters vote, the margin is a share of the votes cast, and the votes cast are a share of the electorate: multiply the shares.
  • Read whether the question wants the electorate, the votes cast, the valid votes or one candidate's votes.

Total from the margin

valid votes=margin×100winner%−loser%\text{valid votes} = \dfrac{\text{margin} \times 100}{\text{winner}\% - \text{loser}\%}

Worked example

In a two-candidate election the winner gets 58% of the valid votes and wins by 1,200 votes. How many valid votes were cast?
Practice this conceptself-check · 2 quick reps

The same idea in a practice question:

Practice questionHard

Example 5 · Percentage · Marks and Elections

In an election, candidate A gets 6060% of votes and wins by 400400 votes. If 2020% of voters did not vote, what is the total number of eligible voters?

The margin is a share of the votes cast

If 80% vote and the margin is 10% of the votes cast, it is 8% of the voters' list. Dividing the margin by 0.1 gives the votes cast, not the list.
Drill 6 more on elections: shares, margins and invalid votes

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