Deep Sea Series · Quiz 14 · Adding and Subtracting Decimals

The Guide

Before you begin

This whole quiz obeys one rule, and it is not a new one. Line up the places. You obeyed it in Quiz 1 when you stacked ones under ones, and again in Quiz 3 when you carried and regrouped. In decimals it wears a disguise: lining up the places means lining up the points. Nearly every error in this topic is that one rule broken. As always: no tricks, no calculator, and whatever this shows is useful news.

Why the right edge stopped working

With whole numbers you never had to think about alignment, because the right-hand edge did the thinking for you. In 47 and 385 the ones sit at the right in both, so flushing the edges puts ones under ones by accident. Decimals break that accident. In 4.7 the right-hand digit is tenths; in 0.38 it is hundredths. Flush the edges and you have stacked tenths on hundredths, which is stacking dimes on pennies and calling them equal.

		4	.	7
+	0	.	3	8

Edges flush: the 7 tenths sit over 3 tenths and the 8 hundredths have nothing above them. Wrong.

	4	.	7	0
+	0	.	3	8
	5	.	0	8

Points aligned, tenths over tenths. The teal zero is the padding.

The point is not a digit, it is a fence. It marks where the wholes stop. When you line up two points you are not lining up a symbol, you are lining up two whole streets of places at once, so that every column is honest about what it holds.

The padding is now a necessity

Quiz 13 proved that trailing zeros are safe: 0.6 and 0.60 are the same amount, because six tenths and sixty hundredths are the same cut counted twice. There it was a convenience for comparing. Here it is a necessity. You cannot subtract 45 hundredths from a column that has no hundredths in it, so you write the hundredths in, at no cost, and then the subtraction is ordinary.

Every whole number has a point you cannot see

Where is the decimal point in 6? It is immediately after the 6, where it always was, silent because nothing follows it. Write it down and 6 becomes 6.00, and 6 minus 2.45 stops being strange:

	5		9	10
	6	.	0	0
−	2	.	4	5
	3	.	5	5

Quiz 3's regrouping, unchanged, now crossing the point.

That row of small teal numerals is exactly Quiz 3's trade. One whole is exchanged for ten tenths, one tenth for ten hundredths, and nothing is lost, because a fair trade is not a loan. The only new thing is that the trade now steps across the point without noticing it, which is the point's whole character: it is a fence, not a wall.

The odometer, with its tenths window

Quiz 2 watched an odometer roll 5,972 up to 6,000, and Quiz 3 called carrying the same machine run forward. Look at the odometer in an actual car and you will find it has been doing decimals all along: the last drum, usually a different color, counts tenths of a mile.

The gold drum is tenths. 12.9 + 0.1 = 13.0, and the carry crosses the point without pausing.

Tenths add to tenths, hundredths to hundredths

Here is the reason underneath the rule. Take one square as one whole. Cut it into ten columns and each column is a tenth; cut each column into ten cells and each cell is a hundredth. Now 0.4 is four columns and 0.38 is three columns and eight cells. They can only be added the way they are shaped.

Drawn to scale: 78 of the 100 cells are shaded. Columns are tenths, cells are hundredths, and they add in their own kind.

The guard is on duty, and this is its best shift

Before you compute, round each number to something easy and hold the answer in mind. For 4.7 + 0.38 the guard says about 5. Any answer of 0.85, or 8.5, or 85 is then dead on arrival, and you have caught a misalignment without checking a single column. Quiz 2 introduced the guard; decimal arithmetic is where it earns its wages, because misalignment produces answers that are wrong by factors of ten, and factors of ten are exactly what an estimate can see.

A working order. Estimate. Write the points under each other. Pad the short ones with zeros. Add or subtract as usual, regrouping when a column asks. Bring the point straight down. Then look at the guard, and see whether it is satisfied.

Worked Examples

Three to study before you start

Example 1 · Adding, with padding

Add 4.7 and 0.38. The guard first: roughly 5 plus nothing much, so about 5. Write the points under each other. The top number stops at tenths, so pad it to hundredths: 4.70. Now add columns from the right. Eight hundredths and no hundredths make 8. Three tenths and seven tenths make ten tenths, which is one whole and nothing left over, so write 0 and carry the whole. Four and nothing and the carried one make 5. The answer is 5.08, and the guard is satisfied.

Example 2 · Subtracting from a whole

You hand over $50 for a repair costing $37.68. The guard says about 12 dollars back. Fifty has an invisible point, so write $50.00 and line the points up. Now regroup, exactly as in Quiz 3: there are no hundredths to take 8 from, and no tenths to lend, so the trade travels along the row, one whole becoming ten tenths and one tenth becoming ten hundredths. Then the columns are ordinary. The change is $12.32.

	4	9		9	10
	5	0	.	0	0
−	3	7	.	6	8
	1	2	.	3	2

Example 3 · A ragged column

Add 3.5, 12, 0.75 and 8.4. This is the hard real case, because the four numbers are all different lengths and one of them is a whole number. Do not let the ragged right edge tempt you. Line up the four points, giving 12 its silent point, then pad every number out to hundredths: 3.50, 12.00, 0.75, 8.40. The guard, rounding as it goes: 4 and 12 and 1 and 8, so about 25. Adding down the columns gives 24.65, which the guard accepts.

		3	.	5	0
	1	2	.	0	0
		0	.	7	5
+		8	.	4	0
	2	4	.	6	5

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Questions 4 and 7 are multiple choice: choose the one best answer. Show your columns; that is where the marks and the mistakes both live.

Add: 3.6 + 2.8
Add: 12.45 + 3.7
Subtract: 9.2 − 4.65
Sam is working out 5 − 1.36. Which is the correct way to begin?
- A) Write it as 5.00 − 1.36, so the points line up and every column has a digit.
- B) Write the 5 above the 6, so the right-hand edges are flush.
- C) Write it as 05 − 1.36, adding a zero on the left.
- D) Round 1.36 to 1, then subtract.
Add: 3.5 + 12 + 0.75 + 8.4
Martha buys three things at $4.79, $12.50 and $0.99, and pays with a twenty-dollar bill. (a) What is the total? (b) What is her change?

▶ VideoA video rest stop, for the stubborn ones.

Take a few minutes at a hydrothermal vent, where whole communities live on chemicals instead of sunlight.

ReadingOr read instead of watching: the Smithsonian's guide to the deep sea. Find the part headed Open Ocean Zones. It names the five layers of the ocean and gives the depth where each one starts, which is the country this series is descending through.

The engraving above is Filhol's gulper eel from 1885, not a frame from the film.

Denise works out 4.7 + 0.38 and gets 0.85. What went wrong?
- A) Nothing; 0.85 is correct.
- B) She lined up the right-hand edges instead of the points, added 47 and 38 as though they were whole numbers, and then guessed where the point belonged.
- C) She forgot to carry the one.
- D) She should have subtracted, since 0.38 is smaller.
A hole must finish at 0.5 in. Brian's bit cuts 0.375 in. How much wider does the hole still need to be?
Without working it out exactly, explain how you can tell that 12.6 + 3.45 cannot possibly be 4.71.
Yuliana's time card reads 7.5 hours on Monday, 8 on Tuesday, 6.25 on Wednesday and 7.75 on Thursday. (a) How many hours so far? (b) How many more would bring her to 40?

Answer Key

Check your work

Open the key: after you've finished all ten

1 6.4

Six tenths and eight tenths make fourteen tenths, which is one whole and four tenths: write the 4 and carry the whole into the ones. Then 3 and 2 and the carried 1 make 6. This is Quiz 3's carrying with no change at all except the name of the column.

2 16.15

Pad 3.7 to 3.70 and line the points up. The hundredths column is 5 and 0. If you got 12.82, you added 45 and 37 with the edges flush, which is the error question 7 is about. The guard would have said about 16.

3 4.55

Pad to 9.20, then regroup: there are no hundredths to take 5 from, so one tenth is traded for ten hundredths, leaving 1 tenth and 10 hundredths. Then 10 − 5 is 5, and 11 tenths less 6 tenths is 5 tenths after the next trade. A common wrong answer is 5.45, which comes from taking the smaller digit from the larger in each column instead of trading: Denise's habit from Quiz 3, and the guard refuses it, since 9 less 5 is nowhere near 5.

4 A.

Five has a point that is simply not written, and writing it lets you pad to 5.00 so that every column has something in it. B) is the whole-number habit that decimals break, and it would put the ones digit above the hundredths. C) adds a zero on the left, where zeros do nothing; the missing places are on the right. D) rounds away 0.36 of the answer before starting, which is what estimates are for and what answers are not. Set up as in A, the subtraction comes to 3.64.

5 24.65

Padded to 3.50, 12.00, 0.75 and 8.40. The 12 is the trap: it looks short, so the eye wants to shove it right. Its point sits after the 2, and once that is written the column is ordinary. The guard, at 4 and 12 and 1 and 8, expects about 25.

6 (a) $18.28 · (b) $1.72

Add first: 4.79 and 12.50 and 0.99 give 18.28. Then $20 becomes $20.00 and the subtraction runs across two zeros, the odometer in reverse. Notice the second step could not begin until the twenty was given its point and its cents.

7 B.

Denise's habit is old and honest: with whole numbers the right-hand edge always aligned the places for her, so she never had to think about it. Here the right-hand digits are tenths in one number and hundredths in the other. Adding 47 and 38 gives 85, and the point then has nowhere principled to go. Lined up properly, 4.70 + 0.38 = 5.08. C) names a real step in the correct working, since seven tenths and three tenths do carry, but forgetting it would give 4.08, not 0.85. Note that the guard catches this before any diagnosis: 4.7 and a bit more cannot land below 1.

8 0.125 in

Pad to 0.500 and subtract: 0.500 − 0.375 = 0.125. These are the same tools Quiz 9 measured in fractions and Quiz 13 put in order, and the fractions agree: one half less three eighths is one eighth, and one eighth is 0.125. A toolbox speaks both languages, and the two answers must of course match.

9 Because 12.6 is already larger than 4.71, and adding 3.45 to it can only make it larger still. The guard says about 16, and the true answer is 16.05.

Any reasoning of that shape earns the mark. Where does 4.71 come from? From adding 126 and 345 as whole numbers, getting 471, and dropping a point in. That is the same misalignment as question 7 wearing different clothes, and an estimate made in three seconds refuses it without any column work at all.

10 (a) 29.5 hours · (b) 10.5 hours

Padded: 7.50, 8.00, 6.25, 7.75. The 8 needs its silent point exactly as the 12 did in question 5. Then 40 becomes 40.00 and the subtraction crosses two zeros. Half an hour written as 0.5 rather than 30 is worth dwelling on, since time cards, invoices and pay stubs all use decimal hours and none of them explain it.

Reading your results

Questions	The skill they test	If they gave trouble
1, 2	Aligning and carrying	Write the points under each other first, before a single digit. Ten tenths make one whole.
3, 8	Regrouping across the point	Quiz 3's trade, unchanged. Pad first, then trade; never take the smaller digit from the larger to avoid the trade.
4, 5, 10	The invisible point	Every whole number ends in a point you cannot see. Write it, then pad to the right.
6	Two steps in a real setting	Total first, then subtract. Money is hundredths, so both numbers want two decimal places.
7, 9	Diagnosing misalignment	The signature error of the topic. Estimate before you compute and it cannot survive.

Eight or more right: the rule is yours, and Quiz 15 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 a handful of old receipts and, for each one, write the numbers in a column with the points under each other before adding anything. Do that until the aligning happens before you think about it. Then come back to these same ten.

The Company · An Interlude

New company, and where it comes from

The seahorses have taken a bow after thirteen quizzes. From here the series descends, and the company is drawn from a single wood-engraved plate made in 1885 for Henri Filhol's La vie au fond des mers, an account of the French deep-sea voyages of the Travailleur and the Talisman. Twenty-nine creatures, each numbered, each named, though the fold does its best to hide one of them. The engravings were made from animals hauled up in a dredge from depths nobody had yet seen, by people drawing what had never been drawn.

A wood engraving of a humpback anglerfish, jaw hinged wide, a lure on a stalk arching forward over its head — Figure 28, *Melanocetus johnstonii*, the humpback anglerfish, at the masthead of this quiz. It lives in the midnight zone, roughly a thousand meters down and further, where no sunlight arrives at all. The stalk over its head carries a lure it can light. A creature that hangs a small bright thing out in front of itself and waits seemed the right animal for a quiz about a number sitting out past the point.

A wood engraving of a deep-sea grenadier with a swollen rounded head, a large eye, and a body tapering back to a whip-thin tail — Figure 27, *Macrurus globiceps*, known today as *Cetonurus globiceps*: the globehead grenadier, a rattail of the family Macrouridae, brown, growing to about half a meter, and living mostly between one and two thousand meters down. *Globiceps* means ball-headed. That swollen skull the engraver drew so carefully is not bulk but instrument: grenadiers carry wide mucous chambers in the head that house the lateral line, the sense by which a fish in total darkness feels what moves near it.

A correction, and how it was made. The first version of this page told you this fish had no recoverable name, because the binding fold had eaten it. That was wrong, and the error was ours. Enlarged five times, the fold gives up more than it seemed to: the surviving letters read …crurus, not the …cerurus we first transcribed, and the missing syllable is simply Ma. What looked like an extra letter was the facing page printing through the crease. Three letters came back by looking harder at the page, and the rest came from a record kept somewhere else entirely.

The record is worth the detour. This species was named by the zoologist Léon Vaillant in 1884, and the naming appeared inside a dispatch Filhol wrote for the popular weekly La Nature, reporting the Talisman voyage to a general readership. The formal first description of the globehead grenadier is therefore an article in a science magazine. Whatever we like to assume about the distance between work published for specialists and work published for everyone, it has not always been there.

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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