Seahorse Series · Quiz 13 · Decimal Place Value

The Guide

Before you begin

Decimals look like a new country, and they are not. A decimal is a fraction whose denominator was agreed on in advance, so that nobody has to write it down: tenths, hundredths, thousandths, always. Everything about place value (Quiz 1, if you did it) continues to the right of the point without interruption. As in every quiz of this series: no tricks, no calculator, and whatever this shows is useful news.

The street continues

Going left, each place is ten times the one before. Going right, each place is one tenth of the one before, and the names take the ending -ths to say so. The point is not the center of the chart; the ones place is. The point is only a marker telling you where the wholes stop.

tens	ones		tenths	hundredths	thousandths
0	3	.	6	0	7

The chart is wider than your screen: slide it sideways with your finger, or turn your phone.

So 3.607 holds 3 wholes, 6 tenths, no hundredths, and 7 thousandths. As fractions: 610 + 71000, or all together, 3 and 6071000.

Reading them aloud, and a promise kept

Say the whole number, say and at the point, then say the decimal part as a whole number followed by the name of its last place. So 3.607 is three and six hundred seven thousandths, and 5.09 is five and nine hundredths.

Quiz 1 has been waiting twelve quizzes for this. Back then you were told never to write and inside a whole number, because and was reserved. This is what it was reserved for: and marks the decimal point, and nothing else. It is why a check reads "one thousand two hundred seventy-five and 00/100" — the word is doing its one real job.

Zeros, silent and loud

Zeros at the end of a decimal change nothing: 0.6, 0.60, and 0.600 are the same amount, because six tenths, sixty hundredths, and six hundred thousandths are the same size cut three ways. Zeros inside or in front change everything: 0.07 is a tenth the size of 0.7, since the 7 stands one place further right. The rule is Quiz 1's rule unchanged — a zero holds a place open — but it now has an exception at the tail, where there is no place left to hold.

Comparing: pad, then walk

Line up the points and compare place by place from the left, exactly as in Quiz 2. The trap is counting digits: 0.85 has more digits than 0.9 and is the smaller number. Pad the short one with trailing zeros (zeros at the end), which costs nothing, and the walk becomes easy: 0.90 against 0.85, and 9 tenths beats 8 tenths at the first place that differs.

Drawn to scale. More digits, less number: 0.85 has not yet reached 0.9.

Rounding, unchanged

The rounding rule from Quiz 2 still applies, only with smaller places. Find the place you are rounding to, look one door right, and go up on 5 or more. Rounding 3.847 to the nearest tenth: the tenths digit is 8, the hundredths digit is 4, so stay, giving 3.8. Everything to the right simply falls away; unlike whole numbers, no zeros are needed to fill the places that were dropped, because those places were never holding wholes.

Money is a decimal you already read

Dollars and cents are wholes and hundredths, which is why $0.75 and three quarters are the same thing said twice. Rounding to the nearest cent is rounding to the nearest hundredth, and every register you have ever used has done it.

Worked Examples

Three to study before you start

Example 1 · Reading and writing

4.25 is four and twenty-five hundredths, because the last digit stands in the hundredths place. As a fraction it is 4 and 25100, which simplifies to 4¼ — and that is exactly why $4.25 is four dollars and a quarter.

Example 2 · Padding to compare

Order 0.4, 0.38, 0.409 from least to greatest. Pad all three to the same length: 0.400, 0.380, 0.409. Now the walk is plain: 380, 400, 409. So the order is 0.38, 0.4, 0.409. Padding is free because trailing zeros change nothing, and it turns an unfamiliar comparison into Quiz 2's ordinary one.

Example 3 · Rounding to the cent

Round $8.436 to the nearest cent. Cents are hundredths, so look at the hundredths digit, 3, and then one door right at the thousandths digit, 6. Six is five or more, so round up: $8.44. Fractions of a cent turn up constantly in tax and interest, and this is how they are settled.

The Quiz · Ten Questions

Now you

Work without a calculator, on paper. Questions 4 and 7 are multiple choice: choose the one best answer.

Write 0.7 as a fraction.
In 3.607, which digit stands in the hundredths place?
Write 5.09 in words.
Which of these is the largest?
- A) 0.85
- B) 0.9
- C) 0.109
- D) 0.8999
True or false: 0.6 = 0.60. Give your reason in a few words.
Put these in order from least to greatest: 0.4 · 0.38 · 0.409 · 0.5

▶ VideoA video rest stop, for anyone who has kept going this far.

Take two minutes with an orange pygmy seahorse, barely the size of a fingernail.

Andre says 0.85 is greater than 0.9, because 85 is greater than 9. What went wrong?
- A) Nothing; he is correct.
- B) He compared the digits as whole numbers instead of by place: 0.9 is nine tenths, and 0.85 is only eight tenths and a bit.
- C) He should have counted the digits, and more digits means a larger number.
- D) The two are equal once you add a zero.
Round 3.847 to the nearest tenth.
Round $12.567 to the nearest cent.
Brian's caliper marks three drill bits at 0.375 inch, 0.5 inch, and 0.4375 inch. Put them in order from smallest to largest.

Answer Key

Check your work

Open the key: after you've finished all ten

1 7/10

The 7 stands in the tenths place, so it is seven tenths. The name of the place is the denominator, which is the whole idea of decimal notation: the bottom number was agreed in advance so it never has to be written.

2 0

Counting right from the point: 6 is tenths, 0 is hundredths, 7 is thousandths. That zero is doing Quiz 1's old job, holding a place open so the 7 stays in the thousandths. Remove it and 3.607 collapses into 3.67, a different number entirely.

3 Five and nine hundredths

The last digit stands in the hundredths place, so the whole decimal part is named in hundredths. The and marks the point, doing the job Quiz 1 reserved it for. "Five point zero nine" is fine in speech but is not how the GED words it, and "five and nine tenths" names the wrong place — that would be 5.9, ten times larger.

4 B: 0.9.

Pad them all to four places: 0.8500, 0.9000, 0.1090, 0.8999. Now the tenths decide almost everything: 9 beats 8 beats 1. D) 0.8999 is the trickiest wrong answer, being very close to 0.9 without reaching it, and having the most digits of the four. Length is not size.

5 True: six tenths and sixty hundredths are the same amount.

A trailing zero adds no value because there is no place left for it to hold open — it only re-cuts the same amount into smaller pieces, exactly as Quiz 8 re-cut 1/2 into 2/4. This is why padding to compare is always safe.

6 0.38, 0.4, 0.409, 0.5

Padded: 0.380, 0.400, 0.409, 0.500. The trap is 0.38 and 0.409, which look longest and largest but sit low; the tenths place settles both before the other digits get a vote — the same first-difference rule that decided whole numbers in Quiz 2.

7 B.

Reading the tail as a whole number is the signature decimal error, and it is the exact cousin of Andre's fraction error in Quiz 9, where he read 5/9 as larger than 5/6. Both times the digits were counted as though position did not matter. C) states the error as though it were a rule. D) confuses padding with equality: 0.90 equals 0.9, but neither equals 0.85.

8 3.8

Tenths digit 8, and one door right the hundredths digit is 4, so stay. Everything past the tenths falls away, and no zeros are needed to fill the vacated places because those places never held wholes. If you got 3.9, you may have rounded in stages, taking 47 up to 50 and then 3.85 up to 3.9, the same drift Quiz 2 warned about. One look, one decision.

9 $12.57

Cents are hundredths, so look at the 6 in the hundredths place and then at the 7 in the thousandths: five or more, so round up. Sales tax and interest produce these thousandths constantly, and the register rounds them exactly this way.

10 0.375, 0.4375, 0.5

Pad to four places: 0.3750, 0.4375, 0.5000. And these are old friends in new clothes: 0.375 is 3/8, 0.4375 is 7/16, and 0.5 is 1/2, the same wrench sizes Quiz 9 asked about in fractions. A toolbox speaks both languages, and the two orderings must of course agree.

Reading your results

Questions	The skill they test	If they gave trouble
1, 2	Naming the places	The place name is the denominator; zeros inside still hold places open.
3	Reading decimals aloud	Name the decimal part by its last place, and let and mark the point.
4, 6, 7, 10	Comparing and ordering	Pad with trailing zeros, then walk from the left; length is not size.
5	Trailing zeros	They re-cut without changing the amount, which is what makes padding safe.
8, 9	Rounding decimals	Quiz 2's landmarks, one look and one decision, never in stages.

Eight or more right: the places are yours — on to Quiz 14. 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. Write out a handful of prices with your teacher and say each one aloud in place-value words rather than "point," until the names of the places come without thinking. Then come back to these same ten.

The Company · An Interlude

Fellow travelers

The trio is back at the masthead: the leafy to the left, our kuda holding the center, and the opal pressing in from the right. Two of the masthead's seahorses, seen whole and out of formation.

An opalescent seahorse in coral pinks and iridescent blues against a blurred reef, an imagined creature rather than a photographed one — The right-hand supporter, in full and right side up. An artist's invention rather than a species — no ocean has issued this animal — which is entirely proper, since heraldic supporters were always imagined beasts.

A leafy seadragon drifting against a dark background, trailing ornate leaf-shaped appendages in violet and gold — And the left-hand supporter, who needs no inventing. Every one of those leaves is real, grown for the sole purpose of not being seen.

Prints only this gallery, full size. The main print button stays photograph-free to spare your ink.