Percentage items are set four or five different ways on the paper and revised as four or five different topics, which is the reason they cost people time they do not have. They are one relation wearing costumes.
Every percentage question is part = rate × whole. Two of the three are given and you solve for the third. The only real difficulty is deciding which number in the sentence is the whole, and the phrase “of” almost always points straight at it.
One picture, three questions
Draw the whole as a bar. Shade the rate. The shaded piece is the part. Whichever of the three the question leaves blank, the same picture answers it.
One relation, three questions. Give any two of rate, part and whole and the third is forced.
What is 40% of 130? The whole and the rate are given, so multiply.
Worked example · finding the part
What is 40% of 130?
A130
B78
C520
D52
Three percentage questions share one relationship: part = whole x rate. 40% means 40/100, so 40% of 130 is 130 x 40/100 = 52. Decide which of the three quantities is missing before you calculate; most errors here are answers to a different one of the three questions.
AThis is the number that 52 is 40% of, which answers a different question.
BThis is the remaining part rather than the part the question asks for.
CThe decimal point has been moved one place; check the size of the result against the original number.
52 is what percent of 130? Same bar, same two ends, and now the rate is the blank. Divide the part by the whole. These two items came out of the generator at the same seed, which is why they are built from the same numbers: it is genuinely one question asked from either side.
Worked example · finding the rate
52 is what percent of 130?
A60%
B400%
C20%
D40%
Three percentage questions share one relationship: part = whole x rate. Divide the part by the whole: 52/130 = 0.4, which is 40%. Decide which of the three quantities is missing before you calculate; most errors here are answers to a different one of the three questions.
AThis is the remaining part rather than the part the question asks for.
BThe decimal point has been moved one place; check the size of the result against the original number.
CThis is half the correct figure; check which quantity the rate applies to.
Percent change: the whole is the number you started from
This is where most of the marks are lost, and it is lost on one decision. When a figure moves from 360 to 288, the change is 72 — but 72 is a percentage of what? Of 360, the original. Divide by the number you started from, never the one you ended at.
The same drop, measured two ways. Against the original it is 20%; against the new figure it is 25%, and 25% will be sitting in the options.
Worked example · percent change
The number of households served fell from 360 to 288. By what percent did it decrease?
A120%
B80%
C25%
D20%
A percentage change is the size of the change divided by where you started - never by where you ended. The change is 72, and the starting figure is 360, so the decrease is 72/360 = 20%. If you divide by the finishing figure you get a number that looks reasonable and is wrong, which is why this question is worth slowing down for.
A120% is the finishing figure as a percentage of the starting one, not the change.
B80% is what remains rather than what changed.
CThis divides the change by the finishing figure. A percentage change is always measured against where you started, so the divisor is 360.
Discount and commission are the same sum with a different last step
A discount question and a commission question compute exactly the same thing — a rate of a whole — and then differ in what they ask you to report. Commission wants the part. Discount wants what is left.
That is the entire difference, and it is worth a mark every time, because the part you computed on the way is always one of the four options. Twenty percent off P480 gives you P96 in your hand and P384 on the answer sheet.
Worked example · discount
A wristwatch priced at P480 is sold at a 20% discount. How much does a buyer pay for it?
AP480
BP96
CP384
DP576
A discount question asks for the price paid, not the money saved - and both numbers are sitting there waiting to be chosen. 20% of P480 is P96, so the buyer pays P480 - P96 = P384. The quicker route is to pay 80% of the marked price in one step, which avoids the subtraction altogether.
AThis is the marked price, unchanged. The discount has not been applied at all.
BP96 is the discount itself. The question asks what is actually paid, which is what is left after taking it off.
DThe discount has been added to the marked price rather than taken off it.
Worked example · commission
A sales agent is paid 8% commission on sales. For one month these totalled P12,400. How much commission was earned?
AP11,408
BP9,920
CP99
DP992
Commission is a straight percentage of a total: no subtraction, no second step. 8% of P12,400 is P12,400 x 8/100 = P992. The trap is the option showing what is left after the commission, which answers a question nobody asked.
AThis is what is left after the commission, not the commission.
BThe rate has been read as 80% rather than 8%.
CThe decimal point has moved one place the other way.
A shortcut worth having: for a discount, take the rate you keep rather than the rate you lose. Twenty percent off is eighty percent of, so 0.80 × 480 lands on P384 in one step and never leaves P96 lying around to be picked by mistake.
Ratios are percentages with the fraction written out
A ratio of 7 to 4 is not a new idea. It says the whole comes in seven parts and you are being asked about four of them, which is 4 ÷ 7 of it — the same rate the percentage questions handed you as a number.
Seven parts, four of them shared. Divide the total by the number of parts first; the ratio tells you how many to take.
Worked example · ratio share
For every 7 sacks of palay harvested, a tenant receives 4. At the same ratio, how many does the tenant receive when 511 are harvested?
A894
B219
C292
D296
The ratio is 4 to 7, so find how many times 7 fits into 511: it fits 73 times. Scale the other side by the same figure: 4 x 73 = 292. Name which side of the ratio the question asks for before you multiply; the other side is always on offer.
AThis uses the ratio upside down, which gives more than the whole quantity.
BThis is what is left after the tenant's share, not the share itself.
DThis adds the ratio's first figure to the answer instead of scaling by it.
The three mistakes that cost the most
Dividing by the wrong number in a percent-change item. The base is where the figure started. This single decision separates the right answer from a distractor that is placed in the options precisely because so many people make it.
Answering with the discount instead of the price. You did the arithmetic correctly and reported the wrong quantity. Read the last six words of the question again before you mark anything.
Treating a ratio of 7 to 4 as four-elevenths. That is the mistake for “divided in the ratio 7:4”, where there really are eleven parts. When the wording is “for every 7, receives 4”, seven is the whole. The two wordings look alike and do not mean the same thing, so decide which one you are reading before you divide.
Practising it
Percentage items reward one thing above all: doing enough of them that identifying the whole stops being a decision. That is difficult from a fixed list, because by the second pass you remember that the answer to the P480 one was P384 and never locate the whole at all.
Every example above came from our own bank at a fixed seed, and the percentages and ratios practice generates them fresh rather than drawing from a set, with the misconception behind each wrong option written out — so a mistake tells you which of the three above you just made.