Deep Sea Series · Quiz 16 · Converting Fractions and Decimals

The Guide

Before you begin

Three quizzes ago you were working in fractions. Two quizzes ago you moved to decimals. They have been the same numbers all along, in two different alphabets, and this quiz is the dictionary. It also settles something that may have nagged at you: why some fractions come out neat and others go on forever. That turns out to be predictable, from the denominator alone, before you divide anything.

The fraction bar was a division sign the whole time

Quiz 8 said a fraction is a number, a single amount you can point at on a ruler. True, and there is one more thing to notice: it is also an instruction. 34 means three divided by four, and always did. So to write any fraction as a decimal, you do the division. Top goes inside; bottom goes outside.

	0	.	7	5
4	3	.	0	0
	2	8
		2	0
		2	0
			0

Three divided by four. The teal zeros are borrowed from nowhere: 3 is 3.00, and the point has been there all along.

The remainder's fourth fate. Quiz 5 gave a remainder three futures: round up, drop it, or write it as a fraction. Here is the fourth. Instead of stopping at 3 ÷ 4 with a remainder of 3, put a point after the 3, bring down a zero, and keep going. Nothing is left over because nothing is abandoned.

Going back: say the decimal's name out loud

The other direction needs no division at all, because the decimal's name is already the fraction. Quiz 13 taught the place names for exactly this moment. Read 0.375 properly and you say three hundred seventy-five thousandths, and that sentence is the fraction: 3751000. Then simplify, as in Quiz 8, and you get 38.

The last place you say gives you the denominator. Tenths means over 10, hundredths over 100, thousandths over 1,000. That is the whole method.

Why some go on forever, and how to know in advance

To turn a fraction into a decimal you are really trying to rewrite it with a denominator that is a power of ten, because that is what a decimal is. Sometimes you can. 18 multiplied top and bottom by 125 becomes 1251000, which is 0.125. Sometimes you cannot, no matter what you multiply by, and the reason is Quiz 7's prime factors.

Ten is two times five, and that is all it is ever made of.

So a fraction can be rewritten over a power of ten only if its denominator is built out of twos and fives already. Eight is 2 × 2 × 2, so it fits. Twenty is 2 × 2 × 5, so it fits. But six is 2 × 3, and no power of ten will ever contain a three, so sixths can never be made to land on a power of ten. The division never finishes, and the digits repeat instead.

Fraction	Denominator	Decimal	Ends?
18	2 × 2 × 2	0.125	yes
320	2 × 2 × 5	0.15	yes
925	5 × 5	0.36	yes
16	2 × 3	0.1666…	no
27	7	0.2857…	no

The test, in one line. Put the fraction in simplest form, then break the denominator into primes. If you see only twos and fives, the decimal ends. If anything else appears, it repeats forever. You never have to divide to find out.

The bar, and what it means

A decimal that repeats forever needs a way of being written down in finite space. The mark used is a bar drawn over the digits that repeat. One third is 0.3, meaning 0.3333 going on without end. One sixth is 0.16: the 1 happens once, and only the 6 repeats. Two sevenths is 0.285714, where all six digits come round again in the same order, forever. When a question asks you to write a repeating decimal, put the bar over the repeating part and nothing else.

Worth knowing by heart

These come up constantly, on the test and off it. Learning them saves you the division every time.

Fraction	Decimal	Fraction	Decimal
12	0.5	15	0.2
14	0.25	18	0.125
34	0.75	110	0.1
38	0.375	13	0.3

The guard

Before dividing, ask roughly where the answer must land. Any fraction smaller than one becomes a decimal smaller than one, so 34 cannot possibly come out above 1. That single check catches the most common error in this topic, which is dividing the two numbers the wrong way round.

Worked Examples

Three to study before you start

Example 1 · Fraction to decimal

Write 58 as a decimal. The guard first: five eighths is a little more than a half, so the answer should sit just above 0.5. Now divide 5 by 8. Eight does not go into 5, so write 0, put the point, and treat the 5 as 5.000. Eight into 50 goes 6, carrying 2; eight into 20 goes 2, carrying 4; eight into 40 goes 5 exactly. The answer is 0.625, which is indeed just above a half. Notice it finished, and it had to: 8 is 2 × 2 × 2.

Example 2 · Decimal to fraction

Write 0.375 as a fraction in simplest form. Say its name: three hundred seventy-five thousandths. The name is the fraction, so start from 3751000. Now simplify. Both end in 5 or 0, so divide both by 5 to get 75200, again for 1540, and once more for 38. Three and eight share nothing, so that is simplest form. And it is a number you have seen before: it is the drill bit from Quiz 13.

Example 3 · Predicting before dividing

Which of 78 and 56 gives a decimal that ends? Break the denominators into primes. Eight is 2 × 2 × 2, all of them allowed, so seven eighths ends: it is 0.875. Six is 2 × 3, and the 3 is fatal, so five sixths cannot end: it is 0.83, the 3 going on for as long as you care to write. You knew both answers before doing either division, from the denominators alone.

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, and put a bar over any digits that repeat.

Write as a decimal: 34
Write as a fraction in simplest form: 0.6
Write as a decimal: 58
Which one of these gives a decimal that goes on forever?
- A) 38
- B) 720
- C) 56
- D) 925
Write as a fraction in simplest form: 0.375
A job needs a shim thicker than 0.31 in. The box holds shims marked 0.32 in and 516 in. Which is thicker, and by how much?

▶ VideoA video rest stop, for the stubborn ones.

Go down as far as the ocean goes, to the Mariana Trench, nearly eleven kilometers below the surface and deeper than Mount Everest is tall.

The engraving above is a sea cucumber from Filhol's plate of 1885, not a frame from the film. Sea cucumbers are still down there: the last expedition to the bottom of the trench found them on the floor.

Marcus writes 34 as 1.33. What went wrong?
- A) Nothing; 1.33 is correct.
- B) He divided the bottom by the top instead of the top by the bottom. Four divided by three is 1.33; three divided by four is 0.75.
- C) He should have added 3 and 4 first.
- D) He forgot to simplify the fraction before dividing.
Write as a decimal, using a bar for any digits that repeat: 16
Without dividing, say whether 740 gives a decimal that ends, and explain how you can tell.
A time sheet shows 634 hours on Monday and 712 hours on Tuesday. (a) Write each as a decimal. (b) What is the total?

Answer Key

Check your work

Open the key: after you've finished all ten

1 0.75

Three divided by four. Four will not go into 3, so the answer starts 0, then the point, then treat the 3 as 3.00. Four into 30 goes 7 with 2 left; four into 20 goes 5 exactly. It had to end, since 4 is 2 × 2.

2 35

Say the name: six tenths, which is 610. Both divide by 2, giving 35. Leaving it as six tenths is not wrong as a number, but the question asked for simplest form.

3 0.625

Worked in Example 1. If you got 1.6 you divided 8 by 5 instead of 5 by 8, which is question 7's error. The guard refuses it: five eighths is less than one whole.

4 C.

Factor the denominators. 8 is 2 × 2 × 2, 20 is 2 × 2 × 5, and 25 is 5 × 5, so all three of those end: 0.375, 0.35 and 0.36. But 6 is 2 × 3, and the 3 can never be part of a power of ten, so five sixths repeats: 0.83. No division was needed to pick it out.

5 38

Three hundred seventy-five thousandths is 3751000, which simplifies to 38. This is the drill bit that has followed you since Quiz 9, wearing its other name.

6 The 0.32 in shim, by 0.0075 in.

You cannot compare them until they speak the same language. 516 is 5 divided by 16, which is 0.3125. So 0.32 is the thicker of the two, and 0.32 − 0.3125 = 0.0075. Both clear the 0.31 minimum, but only just, and this is exactly the situation where a mental guess would have picked wrong: five sixteenths looks bigger written down, and is not.

7 B.

Marcus flipped a fraction in Quiz 12 and he has flipped a division here, which is the same habit: the two numbers went into the box in the wrong order. Four divided by three is 1.33, and three divided by four is 0.75. The guard catches it instantly, because three quarters of anything is less than the whole of it, so an answer above 1 cannot be right. D) is a distraction: three quarters is already in simplest form.

8 0.16

One divided by six gives 0.16666 and onward. Only the 6 repeats, so only the 6 takes the bar. Writing 0.16 would claim that 16 comes round again and again, which would be 0.161616, a different number.

9 Yes, it ends, because 40 breaks into 2 × 2 × 2 × 5, which is nothing but twos and fives.

Any answer of that shape earns the mark, and you were not asked for the value. For the record it is 0.175. The point of the question is that the denominator told you the answer before any dividing began.

10 (a) 6.75 and 7.5 · (b) 14.25 hours

Three quarters is 0.75 and a half is 0.5, both worth knowing by heart. Then add with the points lined up, padding 7.5 to 7.50, which is Quiz 14 again. Payroll systems keep hours as decimals while people say them as fractions, so this conversion happens every time somebody checks a pay stub against the hours they remember working.

Reading your results

Questions	The skill they test	If they gave trouble
1, 3, 8	Fraction to decimal	Divide top by bottom, not the other way. Put the point in and keep bringing down zeros.
2, 5	Decimal to fraction	Say the name out loud. The last place you say is the denominator. Then simplify.
4, 9	Predicting from the denominator	Simplify first, then factor the bottom. Only twos and fives means it ends.
6, 10	Converting to compare and to combine	Two numbers can only be compared or added in the same language. Translate first, then work.
7	Which number goes inside	The top number is the one being divided. Ask what the answer should be near before you start.

Eight or more right: both alphabets are yours, and Quiz 17 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. Learn the eight benchmark pairs in the table above, five minutes a day, until they come without thinking. Most of this topic is those eight facts and one division. Then come back to these same ten.

The Company · An Interlude

A star with five arms, and why that matters here

A wood engraving of a five-armed deep-sea star with a small central disc, spined arms, and rows of tube feet beneath — Figure 16, *Styracaster spinosus*, at the masthead of this quiz. Its family, the Porcellanasteridae, are called mud stars, and they live on the soft floor of the abyss and down into the trenches, some of them below six thousand meters. They eat by swallowing mud and taking what nourishment is in it, which is a living made from the thinnest possible margin.

Five arms is worth a moment. It means each arm is one fifth of the animal, and one fifth is 0.2 exactly: no remainder, nothing repeating, done in a single digit. That is not luck. Five is one of the two primes that build ten, which is the whole reason fifths convert so cleanly, as the Guide showed. Sea stars are not always five-armed, though, and the arithmetic notices. A seven-armed star would have arms of one seventh each, and one seventh is 0.142857, six digits wheeling round forever without ever landing. Same animal, same question, and the answer depends entirely on what the denominator is made of.

A wood engraving of a domed deep-sea star, its arms hidden beneath a smooth swollen membrane — Figure 17, *Hymenaster rex*, a cushion star from the same plate and the same depths. The arms are there, five of them, but a membrane stretches over the top and hides the shape entirely, so a five-pointed animal arrives looking like a dome. Which is more or less what this quiz is about: the same thing can be written two ways and be unrecognizable in the second one until you know how to read it.

These engravings come from a dredge, and a dredge does not descend in order. It goes down, drags, and comes up with whatever the floor was holding, from any depth the cable will reach. That is why this series wanders as it goes down rather than stepping neatly through the zones: Filhol's plate is a jumble of depths because the net was.

Both 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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