Deep Sea Series · Quiz 18 · The Three Percent Questions

The Guide

Before you begin

Quiz 17 was translation. This one is the work. Here is the good news, stated first: there is only one relationship in this entire topic, and every percent question you will ever be asked is that one relationship with a different piece left blank. Three blanks, three questions, and once you can tell which is which you already know how to do all three.

Three names for the three numbers

Every percent situation has exactly three quantities in it.

The whole: the thing being measured against. The original price, the full total, the amount you started with.
The part: the piece of it you are talking about. The discount, the tip, the tax, the amount withheld.
The rate: the percent itself.

And they are tied together by one sentence, which is worth writing on the top of your scratch paper every single time:

The part is the rate times the whole. Nothing else in this topic is new. If the part is missing, multiply. If either of the other two is missing, divide.

One bar, two rulers

Here is the picture that holds all three at once. Draw the whole as a bar. Underneath it runs a ruler in amounts, from 0 up to the whole. Above it runs a second ruler in percent, from 0% up to 100%. The two rulers are the same length because they measure the same bar, which is the entire idea of percent.

Drawn to scale: the shading is exactly 15% of the bar. Every percent question hands you two of these three marks and asks for the third.

The three questions, in one set of numbers

Look at what happens when the same three numbers are asked about three different ways. This is the whole quiz in one table.

What is missing	The question sounds like	What you do	Answer
the part	What is 15% of 60?	multiply: 0.15 × 60	9
the whole	9 is 15% of what number?	divide: 9 ÷ 0.15	60
the rate	9 is what percent of 60?	divide: 9 ÷ 60	15%

Notice that two of the three are divisions, and notice which number goes on top in each. When the rate is missing, the part is divided by the whole, and the answer comes out as a decimal that you then walk two doors left. When the whole is missing, the part is divided by the rate, and you are dividing by a number smaller than one, so the answer comes out larger than what you started with. Quiz 12 promised that and Quiz 15 showed you how.

The hard part is not the arithmetic

It is deciding which number is the whole. Nearly every wrong answer in this topic comes from measuring against the wrong thing, and the arithmetic afterwards is flawless, which is what makes it dangerous. Two reliable tells:

The word of points straight at the whole. In 15% of 60, the whole is 60. In what percent of 80 is 20, the whole is 80. Find the word of, and the number that follows it is what you are measuring against.

And where there is no of to follow, ask which number came first in time. A price before the sale, a rent before the increase, a paycheck before the deductions. The whole is almost always the original. A rent going from $1,200 to $1,320 has gone up by $120, and that $120 is measured against the $1,200 it started from, not against the $1,320 it ended at. Same rise, two different bases, two different percents, and only one of them is the answer to the question anybody asked.

The guard

Use 10% as a ruler, since it is one door and nothing else. Is 8.875% of $64 about $5.68? Ten percent of $64 is $6.40, and we want a bit less than that, so yes. And check direction before you check digits: finding a part gives you something smaller than the whole, finding the whole gives you something bigger than the part, and a rate under 100% means the part did not manage to be the whole thing.

Worked Examples

Three to study before you start

Example 1 · The part is missing

New York City sales tax is 8.875%. What is the tax on a $64 purchase, and what is the total? The whole is $64, because that is what the tax is a percent of. The rate is 8.875%, which becomes 0.08875 before anything can be multiplied. So the part is 0.08875 × 64 = $5.68, and the total is $69.68. Guard: 10% of $64 would be $6.40, and 8.875% should land a little under that. It does. That odd rate is real, by the way, and it is the number under every receipt in this city.

Example 2 · The whole is missing

A deposit of $340 is 20% of the full price. What is the full price? Here the part is $340 and the rate is 20%, and the whole is what you are after. Divide the part by the rate: 340 ÷ 0.20. Quiz 15's method applies, so slide both two doors and it becomes 34,000 ÷ 20, which is $1,700. Two checks. The answer is much larger than $340, as it must be, since $340 was only a fifth of it. And working forwards, 20% of 1,700 is 340, which closes the loop.

Example 3 · The rate is missing

You leave a $6 tip on a $40 meal. What percent is that? The whole is the $40 meal, since the tip is measured against it. Divide the part by the whole: 6 ÷ 40 = 0.15. Then two doors left, and the answer is 15%. Note the order, because reversing it is the commonest slip here: 40 ÷ 6 would give 6.67, and a tip is not 667% of the meal. If your rate comes out above 100%, ask whether you put the numbers in backwards.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Questions 4 and 7 are multiple choice: choose the one best answer. For each of the first three, name which piece is missing before you start working.

What is 15% of 60?
9 is what percent of 60?
9 is 15% of what number?
A rent goes up from $1,200 to $1,320. To find the percent increase, what should you divide the $120 rise by?
- A) $1,320, because that is what the rent is now.
- B) $1,200, because that is what the rise is measured against.
- C) $120, because that is the amount of the increase.
- D) $2,520, because that is both rents together.
A deposit of $340 is 20% of the full price. What is the full price?
Sales tax in New York City is 8.875%. On a purchase of $64: (a) how much is the tax? (b) what is the total?

▶ VideoA video rest stop, for the stubborn ones.

Take a few minutes at a hydrothermal vent, on the same abyssal floor as the sponge at the top of this page, where whole communities live on chemicals instead of sunlight.

The engraving above is figure 22 from Filhol's plate of 1885, a sea squirt, not a frame from the film. It is here as the sponge's opposite: it feeds by pumping water through itself and keeping the small percent that is food.

A $60 jacket is on sale for $45. Rosa says the discount is 33%. What went wrong?
- A) Nothing; 33% is correct.
- B) She divided the $15 discount by the sale price instead of the original price. The whole is $60, so the discount is 25%.
- C) She should have divided $45 by $60.
- D) She forgot to subtract before dividing.
A student answers 34 of 40 questions correctly. What percent is that?
Explain why finding the whole always gives you an answer larger than the part you started with, whenever the rate is under 100%.
A paycheck shows $58.50 withheld from gross pay of $650. (a) What percent was withheld? (b) At that same rate, what would be withheld from gross pay of $800?

Answer Key

Check your work

Open the key: after you've finished all ten

1 9

The part is missing, so multiply: 0.15 × 60. The word of is sitting right there in front of the 60, telling you which number is the whole.

2 15%

The rate is missing, so divide the part by the whole: 9 ÷ 60 = 0.15, then two doors left. Doing it the other way gives 6.67, which as a percent would be 667%, and nine cannot be six hundred percent of sixty.

3 60

The whole is missing, so divide the part by the rate: 9 ÷ 0.15 = 60. These first three questions are deliberately the same three numbers. One relationship, asked three ways, and the only thing that changed was which piece was blank.

4 B.

The rise is measured against what the rent was, not what it became: 120 ÷ 1,200 = 10%. A) uses the new rent and gives 9.09%, which is a real number and an answer to a question nobody asked. C) divides the rise by itself, which is always 100%. D) adds two things that were never meant to be added. Notice that A) is the dangerous one, because the arithmetic is faultless and only the base is wrong, so nothing about the working looks suspicious.

5 $1,700

340 ÷ 0.20, which is 34,000 ÷ 20 once both slide two doors. Multiplying instead would give $68, which fails the direction check immediately: a deposit is a piece of the price, so the price has to be bigger than the deposit.

6 (a) $5.68 · (b) $69.68

0.08875 × 64 = 5.68, and it comes out exact, which is a small mercy. Then add it on, since tax is charged in addition to the price. Guard: 10% would be $6.40. Worth remembering that this rate is the same on every receipt in the five boroughs, so 10% of the price minus a little is a lifelong shortcut.

7 B.

The discount is $15, and $15 divided by the original $60 is 25%. Rosa divided by $45 and got 33%, which measures the discount against what is left rather than against what it came off. This is her old habit from Quiz 10, where she added fractions across: she pairs numbers by where they sit rather than by what they are. C) would tell you what fraction of the original you still pay, which is 75%, a true and different fact. D) is wrong because she did subtract; subtracting was the one step she got right.

8 85%

34 ÷ 40 = 0.85. The whole is 40, since that is the number of questions there were, and the part is what was answered correctly. Benchmark check: 34 out of 40 is a bit more than four fifths, and four fifths is 80%.

9 Because you divide the part by the rate, and a rate under 100% is a number smaller than 1. Dividing by anything smaller than 1 makes a number bigger.

Any answer of that shape earns the mark. The plainer version, which is just as good: if the part is only a slice of the whole, the whole has to be bigger than the slice. This is Quiz 12's promise about dividing arriving in percent country, and it is also a free direction check on every question of this kind.

10 (a) 9% · (b) $72

First the rate is missing: 58.50 ÷ 650 = 0.09, which is 9%. Then the part is missing: 0.09 × 800 = 72. Two of the three questions in a single problem, which is what real pay stubs are like. It is worth being able to do this in both directions, because the only way to know whether a deduction is right is to work out the rate yourself and see whether it holds from one check to the next.

Reading your results

Questions	The skill they test	If they gave trouble
1, 6	Finding the part	Multiply the whole by the rate, and turn the rate into a decimal first.
3, 5	Finding the whole	Divide the part by the rate. The answer must come out bigger than the part.
2, 8	Finding the rate	Part divided by whole, in that order. If you get more than 100%, you probably reversed them.
4, 7	Choosing the whole	The real skill of this quiz. Follow the word of, or take the original, the before, the total.
9, 10	Direction, and both ways at once	Ask whether the answer should be bigger or smaller before you compute, and check it afterwards.

Eight or more right: all three questions are yours, and Quiz 19 puts them to work. 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. For a week, write the sentence at the top of the page before touching any percent problem: the part is the rate times the whole. Then label the numbers you were given, and the blank will tell you what to do. Most of the difficulty here is not knowing which number is which. Then come back to these same ten.

The Company · An Interlude

The sponge that stopped filtering

A wood engraving of a tall branching deep-sea sponge, a slender stem carrying tiers of feathered side branches — Figure 6, *Chondrocladia virgata*, at the masthead of this quiz. It was named in 1873 by Charles Wyville Thomson, the naturalist who ran the *Challenger* expedition, and it is the type species of its genus: the specimen everything else in *Chondrocladia* is compared against. Filhol printed it twelve years later without knowing what it did for a living, and nobody else knew either for another hundred and twenty-two years.

A sponge, by definition, is an animal that pumps water through itself. It draws a current in through pores, strains out bacteria and plankton, and pushes the water back out. The pumping machinery has a name, the aquiferous system, and the cells that drive it are called choanocytes, and those two things are how you know a sponge is a sponge. They are in the definition of the phylum.

In 1995 two French researchers showed that this family of sponges had abandoned both. Chondrocladia and its relatives catch small crustaceans on filaments studded with hook-shaped spicules, hold them like a burr holds wool, and then digest them where they stick, by sending cells to the spot. They gave up the aquiferous system and the choanocytes entirely. A sponge that does not filter, which by the rules of the phylum it belongs to is very close to a sponge that is not a sponge.

The reason is arithmetic, and it is this quiz's arithmetic. Filter feeding takes a very small percent of a very large volume. That is a fine bargain in shallow water thick with plankton. Down here the water is nearly empty, the percent shrinks toward nothing, and the pumping costs more energy than the catch returns. So the strategy inverted: instead of a tiny fraction of an enormous whole, take a large fraction of a rare and concentrated one. Same relationship, different blank filled in.

An engraving of a long-legged deep-sea shrimp with slender antennae — Figure 19, *Nematocarcinus gracilipes*, from the same plate: a small deep-sea crustacean, which is to say the item on the menu. What the sponge above gave up an entire body plan in order to catch.

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.

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