Deep Sea Series · Quiz 17 · What Percent Means

The Guide

Before you begin

Percent is not a new kind of number. It is the third way of writing numbers you already have, and it is the one the world outside this page actually uses: on pay stubs, loan papers, rent increases, sale signs, lab results. Learning it is mostly learning that you have known it since Quiz 13.

If percents are new to you, start with Ratio, Proportion, and Percent, which introduces them from the beginning. Parts 9 to 11 of that page are about percent.

The word says what it is

Per cent means for each hundred. So 45% means forty-five for each hundred, which is 45100, which is forty-five hundredths, which is 0.45. Those are not four facts. They are one fact in four costumes, and the costume that ties them together is hundredths, which you have been working in since Quiz 13. Even the sign keeps the shape of it: two small zeros with a stroke between them.

Drawn to scale: exactly 45 of the 100 cells. This is the same square from Quiz 14 and Quiz 15, wearing its third label.

Moving between the three

There are only two moves, and you learned both in Quiz 15.

Percent to decimal: divide by 100. Every digit walks two doors right. 45% becomes 0.45, and 8% becomes 0.08, and the zero in the tenths place is holding a door open, exactly as it did before.

Decimal to percent: multiply by 100. Two doors left. 0.45 becomes 45, and we write 45%.

The point is a fence, not a traveler. Two doors left, and a decimal becomes a percent.

Percent to fraction: write it over 100 and simplify. 45% is 45100, which is 920. Fraction to percent: go through the decimal, using Quiz 16's dictionary. 35 is 0.6, and 0.6 is 60%.

The decimal is the hub. You do not need six separate conversions. Turn whatever you have into a decimal, then step out to whichever form you need. Everything in this topic passes through the middle.

Above a hundred, and below one

100% is the whole thing, because a hundred hundredths is one. So percents can go past 100 whenever an amount exceeds the whole it is measured against: 150% of a goal is one and a half times the goal, and 125% is 1.25, which is 54, which is one and a quarter.

They can also fall below 1, and this is where people get hurt. 0.5% is not 0.5. It is half of one percent, which is half of one hundredth, which is 0.005. The two differ by a factor of a hundred, and the difference between them on a loan or a fee is the difference between ten dollars and a thousand. Read the sign, then move two doors.

Worth knowing by heart

Percent	Decimal	Fraction	Percent	Decimal	Fraction
50%	0.5	12	20%	0.2	15
25%	0.25	14	10%	0.1	110
75%	0.75	34	1%	0.01	1100
100%	1	the whole	37.5%	0.375	38

That last row is the drill bit again, and it makes a point worth pausing on: a percent is allowed to have a decimal inside it. 37.5% is a perfectly ordinary number. Percent is a way of writing, not a restriction on what can be written.

The guard

Two checks catch nearly everything. First, a percent under 100 must give you less than what you started with; a percent over 100 must give you more. Second, 10% is easy to find by eye, since it is just one door right, so use it as a ruler. Is 4% of $18 about right at 72 cents? Ten percent of $18 is $1.80, and 4% is under half of that, so yes.

Worked Examples

Three to study before you start

Example 1 · One number, three costumes

Write 45% as a decimal and as a fraction in simplest form. For the decimal, move two doors right: 0.45. For the fraction, put it over a hundred, because that is what the word means: 45100. Both parts divide by 5, giving 920. Check it the other way round to be sure: 9 divided by 20 is 0.45. The three forms agree, and they always must, because they are the same amount.

Example 2 · A fraction into a percent

Write 38 as a percent. Go through the middle. Quiz 16 gave the decimal: 3 divided by 8 is 0.375. Now two doors left, which puts the 3 in the tens, the 7 in the ones and the 5 in the tenths: 37.5%. Slightly more than a third, which is what three eighths should be. Do not be unsettled by the point sitting inside the percent; percents carry decimals all the time, and every mortgage in the country is written that way.

Example 3 · Sales tax on a receipt

A purchase comes to $24.50 and the tax is 6%. First turn the percent into a decimal, since that is the only form you can multiply with: 6% is 0.06. Then 0.06 × 24.50 = $1.47, and the total is $25.97. Check it against the guard: 10% of $24.50 would be $2.45, and 6% should be a good deal less than that. It is. Notice that the percent had to become a decimal before any arithmetic could happen, which is true every single time.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Questions 4 and 7 are multiple choice: choose the one best answer. Give every fraction in simplest form.

Write 45% as (a) a decimal and (b) a fraction in simplest form.
Write 0.08 as a percent.
Write 35 as a percent.
Which of these is the largest?
- A) 0.7
- B) 23
- C) 68%
- D) 0.68
Write 125% as (a) a decimal and (b) a mixed number.
A home care worker earning $18.00 an hour is given a 4% raise. (a) How much is the raise, in dollars per hour? (b) What is the new rate?

An engraving of a jellyfish with a ribbed dome and long trailing tentacles

ReadingA reading rest stop, for the stubborn ones.

Three minutes on the giant sea spider, from the research institute that films them. Read to the end for what their cameras caught one doing to a pom-pom anemone.

The engraving above is figure 9 from Filhol's plate of 1885, Drymonema victoria, not a photograph from the article. It is here because jellyfish are what sea spiders eat, and because of what it weighs: see the gallery below.

A fee is 0.5% of a $2,000 balance. Andre works it out as $1,000. What went wrong?
- A) Nothing; $1,000 is correct.
- B) He read 0.5% as 0.5 and moved no doors. But 0.5% is half of one percent, which is 0.005, so the fee is $10.
- C) He should have divided by 2,000 instead of multiplying.
- D) He moved the point one door instead of two, which would give $100.
Write 0.5% as a decimal.
Without computing anything, explain how many times bigger half of a bill is than 0.5% of the same bill.
A jacket costs $80 and is marked 25% off. (a) How much comes off? (b) What do you pay?

Answer Key

Check your work

Open the key: after you've finished all ten

1 (a) 0.45 · (b) 920

Two doors right for the decimal. For the fraction, over 100 first, then divide both by 5. Leaving it as 45100 is the right number but not simplest form, which the question asked for.

2 8%

Two doors left. The 8 was in hundredths, and moving it twice puts it in the ones, so the answer is 8, not 0.8 and not 80. Eight hundredths is eight per hundred, which is what the answer says.

3 60%

Through the middle: 3 divided by 5 is 0.6, and two doors left gives 60. Sense check with a benchmark, since 35 is a little more than a half, and 60% is a little more than 50%.

4 A.

Put them all in one language and they line up at once: 0.7 is 70%, 23 is 66.7%, and C and D are both exactly 68%, since 0.68 is 68%. So 0.7 wins. Worth sitting with the fact that two of the four options were the same number in different clothes, which is the whole argument of this quiz.

5 (a) 1.25 · (b) 114

125 over 100 is more than a whole, which is allowed: it means one and a quarter times whatever it refers to. As a fraction, 125100 simplifies by 25 to 54, and four goes into five once with one quarter left over.

6 (a) $0.72 · (b) $18.72

4% is 0.04, and 0.04 × 18.00 is 0.72. Then add it back on, since a raise is added to the old rate and does not replace it. Guard check: 10% of $18 is $1.80, and 4% should be well under half of that. Over a 40-hour week that raise is $28.80, which is worth knowing how to work out before somebody tells you what it comes to.

7 B.

Andre's habit, from Quiz 9 onward, is reading size straight off the digits without asking what place they are in. Here 0.5% looks like a half and is a two-hundredth of one. The true fee is 0.005 × 2,000, which is $10. His answer is a hundred times too big. D) is worth a second look because the arithmetic in it is right, in that moving one door does give $100, but that is not what he did. Note how completely the guard settles this: a fee of less than one percent cannot be half the balance.

8 0.005

Start at 0.5 and move two doors right. The 5 goes from tenths to thousandths, and the two empty places behind it are held open by zeros. Writing 0.05 is one door only, and 0.5 is none.

9 A hundred times bigger. Half is 50%, and 50 divided by 0.5 is 100.

Any reasoning of that shape earns the mark. The useful version to carry around: moving a point two places is always a factor of a hundred, in either direction. That is the whole of question 7's damage, stated as a number.

10 (a) $20 · (b) $60

25% is a quarter, so the discount is a quarter of 80, which is 20, and you pay the other three quarters: 60. Doing it with benchmarks beats doing it with decimals here, and noticing that 25% off leaves 75% on will save you a step every time a sale sign is involved.

Reading your results

Questions	The skill they test	If they gave trouble
1, 2, 3	Moving between the three forms	Two doors right to leave a percent, two doors left to become one. Everything passes through the decimal.
4	Comparing across forms	Translate everything into one language first. You cannot compare until they match.
5	Percents above 100	A hundred hundredths is one whole. Anything more than that is more than the whole.
7, 8, 9	Percents below 1	The signature error of the topic. 0.5% is half of one percent, not a half. Read the sign, then move two doors.
6, 10	Percent in the world	Turn the percent into a decimal before multiplying, and check whether the question wants the part or what is left.

Eight or more right: the third alphabet is yours, and Quiz 18 is next. 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. Practice finding 10% of things by eye, since it is one door right and nothing else, then build the others from it: 5% is half of 10%, 20% is double, 1% is one more door. Most everyday percent work is that ruler used carefully. Then come back to these same ten.

The Company · An Interlude

An animal that is four percent itself

A wood engraving of a giant sea spider, eight jointed stilt legs converging on a body so small it is hard to pick out — Figure 20, a *Colossendeis* sea spider, at the masthead of this quiz. Look for the body. It is there, at the point where the legs meet, and it is almost nothing: in the largest living species the trunk measures two or three centimeters while the legs span up to seventy. Do the division and the body is about four percent of the animal.

That number has a consequence. There is no room in four percent of an animal for the organs an animal needs, so the sea spider keeps them somewhere else: its gut branches out into the legs, and so, in many species, do the gonads. It is not a body with legs attached. It is a set of legs with just enough body to join them, digesting its dinner halfway down a thigh.

Sea spiders are not spiders, and not insects either. They are their own class of arthropod, and some zoologists think they may be the last survivors of a lineage older than all the others. They walk the deep floor on those stilts, find a jellyfish or an anemone, and empty it through a tube. The one in this engraving was drawn from a specimen the Talisman hauled up in 1883.

A wood engraving of a smaller sea spider with thick jointed legs, resting among sediment — Figure 21, *Nymphon robustum*, a second sea spider from the same plate, stouter and shorter in the leg. Same class, same trade, different proportions, and the difference between the two is exactly the kind of thing percent was invented to say: not how big, but how much of the whole.

One last thing, for anyone who did the division. Three over seventy does not come out evenly. Seventy is 2 × 5 × 7, and Quiz 16 warned you what a seven in the denominator does: 0.0428571, the same six digits wheeling round forever. The sea spider is 4.2857142857 percent body, and it will go on being that for as long as you care to keep writing.

And the jellyfish at the rest stop

Figure 9 on the same plate is a jellyfish, and it carries the best percent in this whole document. Its oral arms, the frilled curtain hanging under the bell, make up about half the animal's weight. Fifty percent. The bell can reach a meter across and the whole creature twenty-five kilograms, and half of that mass is the feeding apparatus. Set it beside the sea spider and you have the two ends of the same idea: one animal that is four percent body, and one that is fifty percent mouth.

Its name on the plate is Drymonema victoria, and there is a small lesson in that too. Haeckel published the name in 1882, in his report on the deep-sea medusae dredged by the Challenger, and he took it for a deep-water animal. Filhol, printing three years later, followed the best authority of the day. Since then it has been folded into Drymonema dalmatinum, which turns out to live at the surface. So the plate is not mistaken. It is 1885, faithfully. Every document is, and knowing which year a thing was certain in is part of reading it.

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.

Prints only this gallery, full size. The main print button keeps the page ink-thrifty.