Deep Sea Series · Quiz 15 · Multiplying and Dividing Decimals

The Guide

Before you begin

Quiz 14 asked one thing of you everywhere: line up the points. This quiz suspends that rule, for multiplication, and for nothing else. A rule that gets suspended without a reason stops being a rule and becomes a mood, so the reason comes first. As always: no tricks, no calculator, and whatever this shows is useful news.

What the lining up was really for

Adding and subtracting combine like with like. Tenths join tenths, hundredths join hundredths, and a column that mixed them would be adding dimes to pennies. That is why the points had to sit under each other: the columns had to be honest about what they held.

Multiplying does not combine anything. It scales. When you work out 0.3 × 0.4 you are not joining three tenths to four tenths, you are taking three tenths of four tenths. Nothing is being stacked, so nothing needs to line up. The place the answer lands in is not inherited from the columns; it is worked out at the end. That is the whole of the suspension, and it is the reason it applies to multiplication alone.

Powers of ten: the digits move, the point stays

You will hear people say move the decimal point. It is a convenient untruth, of the same family as borrowing. The point marks where the wholes stop, and it does not wander. What moves is the digits. Multiply by ten and every digit walks one door to the left, into a place worth ten times more; divide by ten and every digit walks one door right.

The gold line never moves. The digits walk left, and a zero fills the ones place they vacate.

Use the shortcut, but know which one is true. On paper it is quicker to slide the point the other way, and it gives the same answer every time. Keep it. Just remember that the digits are what really change place, because that is what tells you a zero is needed when a place falls empty: 4.5 divided by 100 is 0.045, and the zero in the tenths place is holding a door open exactly as it did in Quiz 1.

The ocean in two units

A kilometer is a thousand meters, so moving between them is three doors in one step. This is the map this series is descending through, written both ways:

Zone	Meters	Kilometers
Sunlit	0–200	0–0.2
Twilight	200–1,000	0.2–1
Midnight	1,000–4,000	1–4
Abyss	4,000–6,000	4–6
Trenches	6,000–10,994	6–10.994

Multiplying two decimals: count the places

Take 0.4 × 0.3. Ignore the points and multiply as whole numbers: 4 × 3 is 12. The only question left is where those digits belong.

Drawn to scale. Four tenths across and three tenths down enclose twelve hundredths.

Tenths of tenths are hundredths. Written as fractions the reason is plain: 410 × 310 = 12100. The denominators multiplied, and ten times ten is a hundred, which is why one decimal place and one decimal place make two.

The rule, now that it is earned. Multiply as though the points were not there. Count the decimal places in both factors, add them, and give the product that many places. Nothing about the columns matters; only the count.

Notice what just happened. 0.12 is smaller than 0.4 and smaller than 0.3. Quiz 11 promised this: multiplying can shrink, once you multiply by less than one, because a part of a part is a smaller part.

Dividing by a decimal: slide them both

Nobody wants to divide by four tenths. So make the divisor a whole number by multiplying it by ten. But a division is a fraction, and you cannot change the bottom of a fraction without changing the top: Quiz 8 established that multiplying both by the same number leaves the value untouched. So 8.4 ÷ 0.4 becomes 84 ÷ 4, which is 21.

8	.	4	÷	0	.	4
8	4		÷		4

Both slide one door. The value does not change, only the look of it.

And 21 is larger than 8.4, which is Quiz 12's mirror arriving on time. It has to be larger: the question was how many four-tenths fit inside 8.4, and small things fit many times over.

The signature error of division. Sliding the divisor and forgetting the dividend. That turns 8.4 ÷ 0.4 into 8.4 ÷ 4, and hands back 2.1: exactly a tenth of the truth. Both numbers move, or neither does.

The guard, still standing

Estimation is what catches a misplaced point, and a misplaced point is the only serious danger in this topic. For 0.2 × 0.3 the guard says: a fifth of a third, so a small piece of a small piece, nowhere near a half. An answer of 0.6 dies on sight without any checking of digits. For 16.50 × 7.5 the guard says sixteen sevens is 112, so at least a hundred and twelve dollars. Round first, hold the number, then work.

Worked Examples

Three to study before you start

Example 1 · Both directions at once

Work out 4.7 × 100 and 4.7 ÷ 100. Multiplying by a hundred walks every digit two doors left: the 4 goes from ones to hundreds and the 7 from tenths to tens, giving 470, with a zero filling the ones place nobody is standing in. Dividing by a hundred walks them two doors right: the 4 lands in hundredths and the 7 in thousandths, giving 0.047. That zero in the tenths is not decoration. Leave it out and you have 0.47, which is ten times too big.

Example 2 · Counting the places

Work out 1.4 × 0.25. The guard first: 0.25 is a quarter, and a quarter of 1.4 is about a third of a unit. Now ignore the points and multiply 14 by 25, which is 350. Count the decimal places in the factors: one in 1.4, two in 0.25, so three in all. Put the point three places from the right of 350 and you get 0.350. The trailing zero is doing nothing at the tail, exactly as Quiz 13 showed, so write 0.35. The guard is satisfied.

		1	4
×		2	5
		7	0
	2	8	0
	3	5	0

No points anywhere in the working. The gray zero is Quiz 4's slide.

0

.

3

5

0

Then the point: three places from the right, and the tail zero falls away.

Example 3 · Dividing by a quarter

How many quarter-hours are there in 7.5 hours? That is 7.5 ÷ 0.25. Slide both numbers two doors left, so that the divisor becomes whole: 750 ÷ 25. That comes to 30. Check it against sense rather than against the page: seven and a half hours, four quarters in each hour, so 7.5 × 4, which is also 30. The answer is much larger than the number you started with, and it should be. You asked how many small things fit inside a larger one.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Questions 4 and 7 are multiple choice: choose the one best answer. Estimate before you compute; in this topic the estimate is the only thing that catches a misplaced point.

Work out: 3.7 × 10
Work out: 4.5 ÷ 100
Work out: 0.6 × 0.4
You are multiplying 4.7 × 0.38. What should you do about the decimal points?
- A) Line them up under each other, as you did when adding.
- B) Ignore them while multiplying, then count the decimal places in both factors and give the answer that many.
- C) Put the point in the answer wherever it sits in the longer factor.
- D) Slide both points right until both numbers are whole, then multiply.
Six steel spacers are stacked, each 0.375 in thick. How thick is the stack?
A home care worker is paid $16.50 an hour and works 7.5 hours. What does she earn for the shift?

An engraving of a halosaur, a long deep-sea fish tapering to a thin tail

ReadingA reading rest stop, for the stubborn ones.

Five minutes with the Smithsonian's guide to the deep sea. Find the part headed Finding Food, and the piece under it called Bioluminescence. It explains what the dragonfish on this page is doing with all those lights.

The engraving above is a halosaur from Filhol's plate of 1885, not a photograph from the article.

Brian works out 0.2 × 0.3 and gets 0.6. What went wrong?
- A) Nothing; 0.6 is correct.
- B) His digits are right but he gave the answer one decimal place instead of two. Tenths of tenths are hundredths, so it is 0.06.
- C) He should have lined up the points before multiplying.
- D) He multiplied when he should have added.
Work out: 8.4 ÷ 0.4
Dividing 6 by 0.5 gives 12, which is larger than the 6 you started with. Explain in a sentence why dividing has made the number bigger.
The same worker is paid $123.75 for a shift of 7.5 hours. What is her hourly rate?

Answer Key

Check your work

Open the key: after you've finished all ten

1 37

Every digit walks one door left. The 4 was in the ones and is now in the tens; the 7 was in the tenths and is now in the ones. Nothing is left to the right of the point, so there is nothing to write there.

2 0.045

Two doors right. The 4 lands in hundredths and the 5 in thousandths, and the tenths place is left empty, so a zero holds it open. Writing 0.45 is the common slip and it is ten times too large: that is only one door.

3 0.24

Six fours are 24, and there is one decimal place in each factor, so the answer gets two. If you wrote 2.4 you counted one place instead of two. The guard would have refused it: six tenths of four tenths is a piece of a piece, and cannot be larger than either.

4 B.

A) is Quiz 14's rule imported into the one place it does not apply, because multiplying scales rather than combines. C) has no reasoning behind it at all. D) is interesting, because it is a real technique in the wrong room: sliding both points is exactly right for division, where the two shifts cancel out. In a multiplication they do not cancel, they multiply, and the answer comes out a hundred times too big.

5 2.25 in

Six times 375 is 2,250, and 0.375 carries three decimal places while the whole number 6 carries none, so the answer gets three: 2.250, which is 2.25. Answering 22.50 means counting two places instead of three. Sense check: each spacer is a bit over a third of an inch, and six of them should come to a bit over two inches.

6 $123.75

1,650 times 75 is 123,750. Two decimal places in $16.50 and one in 7.5 make three, so the point goes three from the right. The guard had it at a minimum of $112 before any of that, from sixteen sevens.

7 B.

Brian's arithmetic is fine and his placement is not, which is his old habit from Quiz 4 in new country: there the missing slide put a partial product in the wrong column, and here a missing place puts the whole answer in the wrong one. Both are position errors, not number errors. His answer is ten times too big. C) is the tempting wrong answer, since it applies the rule that governed the whole of Quiz 14, and this is the single topic where that rule is off duty.

8 21

Slide both numbers one door so the divisor becomes whole: 84 ÷ 4. If you got 2.1 you slid the divisor and left the dividend where it was, which is the signature error of this operation. Both move, or neither does.

9 Because dividing by 0.5 asks how many halves fit inside 6, and halves are small, so a great many fit: twelve of them.

Any answer of that shape earns the mark. Dividing only makes a number smaller when you divide by something bigger than one. Below one it grows, which is Quiz 12's promise arriving in decimals, and it is the exact mirror of what multiplying did in question 3.

10 $16.50 an hour

Slide both one door: 1,237.5 ÷ 75, which is 16.5. This is question 6 run backwards, and it is worth noticing that it had to be: if 16.50 for 7.5 hours makes 123.75, then 123.75 shared over 7.5 hours must give 16.50 back. Every multiplication has a division standing behind it, which is the whole reason a pay stub can be checked.

Reading your results

Questions	The skill they test	If they gave trouble
1, 2	Powers of ten	Count doors, not points. When a place falls empty, a zero holds it open.
3, 5, 6	Counting decimal places	Multiply as whole numbers, then count the places in both factors and add. The columns do not matter here.
4, 7	Knowing which rule is on duty	Lining up points is for adding and subtracting only. Multiplying scales; it does not combine.
8, 10	Dividing by a decimal	Both numbers slide together, or the value changes. Make the divisor whole, and move the dividend the same distance.
9	Why the answers move the wrong way	Below one, multiplying shrinks and dividing grows. Ask how many of the small thing fit inside the big one.

Eight or more right: the exception is understood, not merely obeyed, and Quiz 16 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. Take any receipt with a price and a quantity on it and check the line total by hand, estimating first. Do that a dozen times and the place-counting will settle. Then come back to these same ten.

The Company · An Interlude

Two dragonfish, and a light meter

A wood engraving of a scaly dragonfish with a long fanged jaw, a barbel trailing from the chin, and two rows of light organs down the flank — Figure 29, *Stomias boa*, the scaly dragonfish, at the masthead of this quiz. It lives between about 200 and 1,500 meters, which is the twilight zone and the top of the midnight zone below it. It grows to roughly a foot long. Look at the row of dots the engraver put along its flank: those are light organs, and they are drawn accurately.

What this fish does with them is the reason it belongs in a quiz about scaling. It does not simply switch its lights on. It carries organs near the eyes that measure how much light is falling from above, and it adjusts its own glow to match, dimming and brightening so that its underside never shows as a silhouette to anything hunting from below. A fish with a light meter and a dimmer switch, a thousand meters down, in 1885 hauled up in a net by people who could only guess what the dots were for.

A wood engraving of a second, darker barbeled dragonfish with prominent teeth — Figure 23, *Eustomias obscurus*, a cousin from the same family and the same plate. Both hang a lit barbel below the chin and wait, which makes them relatives in trade as well as in taxonomy of Quiz 14's anglerfish and its lure on a stalk. Three quizzes, three animals, one idea: put out something small and bright, and let the arithmetic come to you.

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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