Reverse Percentage, the Question Students Read Backwards

Most percentage questions run forwards: here is the original, take 15 percent off. A reverse percentage question runs the other way: here is the price after 15 percent was taken off, find the original. That single change of direction is where a lot of otherwise capable students come unstuck, because they reach for the move they know, adding 15 percent back, and it gives the wrong answer.

The reason it is wrong is worth seeing clearly, because the fix is one disciplined question.

Why adding the percentage back fails

Suppose a jacket is reduced by 15 percent and now sells for 68 dollars. The tempting move is to add 15 percent of 68 back on. But the 15 percent was taken off the original price, not off 68, so 15 percent of 68 is the wrong size. The discount and the add-back are percentages of different amounts.

The fix is to ask one question before any arithmetic: what percentage of the original is the figure I have been given?

Worked example A, a discount

A jacket sells for 68 dollars after a 15 percent discount. Find the original price.

After 15 percent is taken off, what remains is 85 percent of the original. So 68 dollars is 85 percent of the original, which means:

original = 68 / 0.85 = 80 dollars.

Checking forwards, 85 percent of 80 is 68, so the original price was 80 dollars. Notice the number you divide by is 0.85, not 1.15, because 68 is what is left, not the original.

Worked example B, removing GST

A bill comes to 109 dollars including 9 percent GST. Find the amount before GST.

Here the 9 percent was added on, so 109 dollars is 109 percent of the pre-GST amount:

pre-GST = 109 / 1.09 = 100 dollars.

Checking forwards, 100 dollars plus 9 percent is 109 dollars. The common error is taking 9 percent of 109 and subtracting it, which uses the wrong base.

A third, with an increase

A price after a 12 percent increase is 224 dollars. The 224 is 112 percent of the original, so original = 224 / 1.12 = 200 dollars. Up or down, the routine is identical: name the figure as a percentage of the original, then divide.

The teaching point

The whole of reverse percentage turns on one question: the figure I was given is what percentage of the original? Answer that, divide, and you are done. The error is always a base error, taking a percentage of the wrong amount. This is the same "what does one unit represent" discipline GPA trains in Primary with the bar model, carried up into Secondary percentage, and it covers the related staples too: percentage increase and decrease, simple and compound interest, and profit and loss all turn on the same "percentage of what base" question.

If your child can do percentages forwards but reads the reverse version backwards, that is a precise, common, fixable gap. GPA's programmes work on exactly this base-naming discipline, from PSLE through O-Level, and a short diagnostic consult will show you where the reading flips.

Every figure on this page was checked independently before publishing.