Percent and Percent Change

Percent questions are the most common single shape in Arithmetic Reasoning, and they have the highest error rate of anything on the subtest. That combination is why this is the first lesson: no other topic returns as many points per hour.

Three questions, one operation

Almost everything here is one of three forms.

The first two are mechanical. The third is where the points are lost.

The denominator rule

Enlarge

Zoom in

A percent change is the difference divided by the value you started from.

A price rising from $40 to $50 is a $10 change. Ten divided by forty is 0.25, so that is a 25 percent increase.

Ten divided by fifty is 0.20, and 20 will be among your four choices. It is there because the test writers know exactly which value people reach for. The same trap works in reverse: a fall from $50 to $40 is a 20 percent decrease, not 25, because now fifty is where you started.

This asymmetry surprises people. The same ten dollars is a different percentage depending on which direction you travel, and that is not a trick. It follows from what the word "change" measures.

Converting between the three forms

You will be asked to move between fractions, decimals and percents constantly, and doing it by instinct is faster than doing it by rule.

Knowing these cold turns a calculation into a recall. Twenty-five percent off is a quarter off, and a quarter of 60 is 15, which takes no arithmetic at all.

Percent off, then tax, in that order

Discount questions stack two operations, and the order matters.

A $80 jacket at 25 percent off is $60. Six percent tax on $60 is $3.60, so the total is $63.60. Applying the tax first gives $84.80 and then the discount gives $63.60 as well - the order happens not to matter for the final total, because multiplication commutes.

Where order does matter is a discount on a discount. Thirty percent off, then a further 10 percent off, is not 40 percent off. It is 0.70 times 0.90, which is 0.63, so 37 percent off. The second discount applies to the already-reduced price. This is a favorite of test writers because the intuitive answer is wrong and is always among the choices.

A shortcut worth having

To increase a number by a percent, multiply by one plus the decimal. To decrease, multiply by one minus it.

• Up 15 percent: multiply by 1.15
• Down 15 percent: multiply by 0.85
• Up 8 percent: multiply by 1.08

One operation instead of two, and no chance of forgetting to add the change back on - which is its own common error.

Mixtures

A mixture question tracks one ingredient. Work out how much of it there is before and after, and set the new amount over the new total.

A 120-pound mixture is 25 percent sand. How much sand must be added to make it 40 percent sand?

There are 30 pounds of sand now. Add x pounds of sand, and the mixture becomes (30 + x) sand out of (120 + x) total. Set that equal to 40 percent: 30 + x = 0.4(120

• x), so 30 + x = 48 + 0.4x, and 0.6x = 18, giving x = 30 pounds. Check it: 60 of 150 is 40 percent.

Mixture questions are among the hardest on the subtest (all five are difficulty 3). If one is taking too long, work from the answer choices instead: add each choice and see which gives the target percent.

What you can skip

Across the 120 questions on this topic:

• Successive percent changes by formula. Two changes in a row come up; multiply the growth factors (up 10 percent is x 1.10, down 10 percent is x 0.90) rather than memorizing anything more.

Where people lose points

Using the new value as the denominator. Covered above, and it is worth more than every other error on this topic combined.

Answering the wrong question. The problem gives a price and a percent increase and asks for the new price. Having correctly calculated the increase, writing down $10 instead of $50 is remarkably easy. Read the last sentence again before choosing.

Treating successive percents as additive. Thirty then ten is 37, not 40.

Rounding a third early. If a step gives 33.33, carry it. Rounding to 33 and then multiplying by a large number can land nearer a wrong choice than the right one.

Work one in under a minute

1. Name what is being asked. "They want the percent increase." Plain words.
2. Identify the original value. Write it down. This single habit prevents the most common error on the subtest.
3. Estimate. Ten dollars on forty is about a quarter, so about 25 percent.
4. Calculate, then check the answer against your estimate.

Where this leads

Percent is the foundation for several later topics. Unit rate and scaling, discounts and markup, and simple interest are all percent operations wearing different clothes, and the composite score questions on the money topics assume you can do this without thinking.