Before you begin
There is no new mathematics in this quiz. Not one new operation. Everything here is Quiz 18's three questions arriving in the clothes they actually wear: a sale sign, a receipt, a rent notice, a loan. What is new is the translating, and the fact that most real questions want one step more than the percent.
The second step is where people stop too early
A discount problem almost never wants the discount. It wants what you pay. A tax problem wants the total, not the tax. A raise wants the new wage. So the shape of a real percent problem is usually: find the part, then add it on or take it off. Two moves, and the second one is the one that gets forgotten, because by then the interesting work feels done.
The shortcut that removes the second step
If a coat is 30% off, then you pay the other 70%. That is not a trick, it is just what a whole is: the piece taken and the piece left have to make 100% between them. So instead of finding 30% and subtracting, find 70% and stop.
Tax works the same way in reverse. Paying a price plus 8.875% tax means paying 108.875% of it, so multiplying by 1.08875 gives the total directly. A 6% raise means the new wage is 106% of the old, so multiply by 1.06. Once you see it, most of this topic becomes one multiplication.
Percent change: what changed, over what it started at
To find a percent increase or decrease, work out the change, then divide by the original. Quiz 18 spent a whole question on why the original is the whole, and this is where that matters most, because both numbers are sitting right there and only one of them is the base.
Rent going from $1,450 to $1,595 has risen by $145, and $145 divided by $1,450 is 10%. Dividing by $1,595 instead would give 9.09%, which is a real number, correctly computed, and an answer to a question nobody asked.
What percent change does not do
Here is the thing worth carrying out of this quiz. A percent down and the same percent back up do not cancel. They cannot, because the second percent is taken of a different whole.
The same percent, applied twice, was two different amounts of money, because the whole underneath it changed in between. This is not a curiosity. It is how a wage cut works, how a price that goes up and comes back down leaves you worse off, and why a 50% loss needs a 100% gain to undo it.
Simple interest
Borrow money and you pay rent on it. Simple interest is the plainest version: interest = principal × rate × time, with the rate as a decimal and the time in years. Borrow $1,200 at 6% for two years and the interest is 0.06 × 1,200 × 2 = $144, so you repay $1,344. It is Quiz 18's part-and-whole with a clock added: the rate says how much per year, and the time says how many years of it.
The guard
Ten percent is one door right, so build everything from it. A 15% tip is 10% plus half of that again. A 20% tip is 10% doubled. And check the direction before the digits: a discount leaves you paying less than the sticker, a tax leaves you paying more, and an answer on the wrong side of the original is wrong no matter how neat the arithmetic looks.
Three to study before you start
A $64 coat is 25% off. What do you pay? The long way: 0.25 × 64 = $16 off, and 64 − 16 = $48. The short way: 25% off leaves 75% on, so 0.75 × 64 = $48 in a single multiplication. Both are correct and the second is faster and safer, because there is no second step to forget. Guard: a quarter off should leave you paying three quarters, and $48 is three quarters of $64.
What is the total on a $120 purchase with 8.875% New York City sales tax? The long way finds the tax first: 0.08875 × 120 = $10.65, and the total is $130.65. The short way notices that you are paying the price and the tax, which is 100% + 8.875% = 108.875% of the price, so 1.08875 × 120 = $130.65 directly. Guard: 10% of $120 is $12, so the tax should be a little under $12, and the total a little under $132.
A price falls from $250 to $200. The change is $50, and the original is $250, so the decrease is 50 ÷ 250 = 20%. Now put it back. Going from $200 up to $250 is the same $50, but the original is now $200, so the increase is 50 ÷ 200 = 25%. Down twenty, up twenty-five, and the price has only returned to where it was. The dollars were identical both times. The percents were not, because percent always answers compared to what, and the what had changed.
Now you
Work without a calculator, on paper. Questions 4 and 7 are multiple choice: choose the one best answer. Read each question to the end and check what it actually asks for, which is often one step past the percent.
- A pair of boots costs $45 and is 20% off. What do you pay?
- A purchase of $120 is charged 8.875% sales tax. What is the total?
- You leave a 20% tip on a $38 meal. How much is the tip?
- A store takes 30% off everything. Which single calculation gives the sale price of a $70 coat?
- A) 0.30 × 70
- B) 0.70 × 70
- C) 70 ÷ 0.30
- D) 70 + 0.30 × 70
- A rent rises from $1,450 to $1,595. What is the percent increase?
- $1,200 is borrowed at 6% simple interest for 2 years. (a) How much interest is owed? (b) What is repaid altogether?
ReadingA reading rest stop, for the stubborn ones.
Five minutes with the Smithsonian's guide to the deep sea. This time find the part headed Finding Food, and read how little of what falls from above ever arrives at the bottom.
The engraving above is figure 11 from Filhol's plate of 1885, a flask-shaped sea cucumber, not a photograph from the article.
- A pair of boots costs $45 and is 20% off. Brian answers $9. What went wrong?
- A) Nothing; $9 is correct.
- B) His percent is right but he stopped one step early. $9 is the discount; the price is $45 − $9 = $36.
- C) He used 20% when he should have used 80%, so the discount is really $36.
- D) He should have divided $45 by 0.20.
- A price falls from $80 to $68. What is the percent decrease?
- A wage of $20.00 an hour is cut by 20%. A year later it is raised by 20%. Is the worker back at $20.00? Show the two steps and explain what happened.
- A worker's take-home pay is $2,600 a month and the rent is $1,300. (a) What percent of take-home pay is the rent? (b) The rent rises 6%. What is the new rent? (c) What percent of pay is it now?
Check your work
Open the key: after you've finished all ten
Reading your results
| Questions | The skill they test | If they gave trouble |
|---|---|---|
| 1, 4, 7 | Finishing the second step | Read to the end and ask what is being requested. The complement removes the step: 20% off means 80% paid. |
| 2, 3 | Tax and tip | Build from 10%, which is one door right. Tax and tip are added on, so the total is more than the price. |
| 5, 8 | Percent change | Change divided by the original, always. Both numbers are in front of you and only one is the base. |
| 6 | Simple interest | Principal times rate times time, then remember the principal is still owed on top. |
| 9, 10 | When the whole moves | The same percent of a different whole is a different amount. Recompute the base every time it changes. |
Eight or more right: percent is finished, and Quiz 20 opens new country. Five to seven: review the flagged rows and retake this in a few days. Fewer than five: good news: we've found the right ground to work. Carry one habit for a week: every time you see a percent in the wild, on a receipt, a sign, a statement, say out loud what the whole is. Not the arithmetic, just the whole. Most of what goes wrong in this topic is answering a correctly computed question that nobody asked. Then come back to these same ten.
Next in the Deep Sea Series: Quiz 20 — Below Zero. Six quizzes of fractions, decimals and percent have all been about numbers between the whole ones. Now the number line runs the other way, past zero, and the first question is what it even means for there to be less than nothing.
Two thousand percent of an ordinary woodlouse
The difference is size. A normal isopod runs half a centimeter to a centimeter and a half. This one reaches ten centimeters or more, which is somewhere between 670% and 2,000% of ordinary, depending which ordinary you measure against. Quiz 17 said percents can go past a hundred whenever an amount exceeds the whole it is compared to. Here is an animal that does it by a factor of twenty.
And it does something almost no other isopod does. Isopod young normally leave their mother after about a month in the brood pouch and then molt their way to adulthood on their own, shedding their skeleton fifteen times or more. The young of Arcturus baffini do not leave. They cling to those long front antennae and stay there, riding, until they are nearly grown.
One honest note about this one. Joseph Sabine described it in 1824 from Baffin Bay, on a voyage searching for the Northwest Passage, so it came to Filhol's deep-sea plate by a different road than most of its neighbors on the page. That is why the line at the top of this quiz names no zone. The rule in this series is that the label has to be true of the animal shown, and where we cannot say, we do not.
All engravings are in the public domain: Henri Filhol, La vie au fond des mers (Paris: G. Masson, 1885), digitized by the Bibliothèque nationale de France.