Worked examples — Decimals — place value, reading and writing

Before starting, recall the one rule everything rests on (Base-10 Number System): each place is one-tenth of the place to its left. The first place after the decimal point (the little dot that marks "the ones place ends here") is tenths $=\frac{1}{10}=0.1$, then hundredths $=\frac{1}{100}=0.01$, then thousandths $=\frac{1}{1000}=0.001$.

Look at the picture: boxes to the left of the red dot get 10× bigger each step; boxes to the right get 10× smaller. That single ruler is the tool we use for every example.

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The scenario matrix

Here is every kind of case a decimal-place-value question can be. Each row is one "cell". The last column says which worked example covers it.

Cell	Scenario class	What makes it tricky	Example
A	Expand a "normal" decimal into place values	none — the baseline move	Ex 1
B	Internal zero in the fraction	zero shifts digits into smaller boxes	Ex 2
C	Leading zero only (value $<1$)	no whole part at all	Ex 3
D	Trailing zero — degenerate	extra digit adds nothing	Ex 4
E	Read a decimal aloud (both ways)	digit-by-digit vs fraction-name	Ex 5
F	Write a decimal from words	place-name fixes the last box	Ex 6
G	Limiting / long fraction (many places)	pattern must keep going, ten-thousandths+	Ex 7
H	Real-world word problem (money)	translate cents into place values	Ex 8
I	Exam twist — compare / trap	"more digits = bigger?" fallacy	Ex 9

We now clear every cell.

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Ex 1 — Cell A: expand a normal decimal

1. Split at the point. Left = 34, right = 256. Why this step? The point separates amounts $\ge 1$ from amounts $<1$; each side uses its own places.
2. Left side, ×10 going left. $3\to$ tens $=30$, $4\to$ ones $=4$. Why? Ordinary whole-number places (Place Value in Whole Numbers).
3. Right side, ÷10 going right. $2\to$ tenths $=2\times 0.1=0.2$ $5\to$ hundredths $=5\times 0.01=0.05$ $6\to$ thousandths $=6\times 0.001=0.006$ Why? Each step right is one-tenth of the step before — the base-10 rule extended.
4. Write the sum. $34.256=30+4+0.2+0.05+0.006$

Verify: add them back: $30+4+0.2+0.05+0.006=34.256$. ✔ Matches the original, so the places are right.

So the "6" was worth $0.006$ — did your forecast agree?

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Ex 2 — Cell B: an internal zero

1. Split. Whole $=5$, fraction $=036$. Why? Same first move — handle sides separately.
2. Assign right-side places, left to right. $0\to$ tenths $=0\times 0.1=0$ $3\to$ hundredths $=3\times 0.01=0.03$ $6\to$ thousandths $=6\times 0.001=0.006$ Why? The position decides the worth, not the digit. The first box after the point is always tenths, even when it holds a $0$.
3. Sum. $5.036=5+0+0.03+0.006=5.036$
4. What if we drop the zero? We'd get $5.36$, where the "3" jumps up to the tenths box: $5.36=5+0.3+0.06$. A totally different, ten-times-bigger fraction. Why this matters? An internal zero is a real place-holder — it pushes later digits into smaller boxes. It is not optional.

Verify: $5+0+0.03+0.006=5.036$ ✔ and $5+0.3+0.06=5.36\ne 5.036$ ✔ (they differ, proving the zero matters).

So the "3" is in hundredths, not tenths.

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Ex 3 — Cell C: value less than one (leading zero)

1. Split. Whole part $=0$, fraction $=408$. Why? The $0$ before the point tells us there is no whole thing — the whole number is zero. The leading zero is just polite, making the dot easy to spot.
2. Assign places. $4\to$ tenths $=0.4$; $0\to$ hundredths $=0$; $8\to$ thousandths $=0.008$.
3. Sum. $0.408=0.4+0+0.008=0.408$
4. Fraction name. The last digit sits in thousandths, so the whole fraction has denominator $1000$: $0.408=\frac{408}{1000}$ Why the last digit? The smallest place present sets the common denominator (Fractions).

Verify: $\frac{408}{1000}=0.408$ ✔.

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Ex 4 — Cell D: trailing zero (degenerate case)

1. Turn each into a fraction using its last place. $0.6=\frac{6}{10}$, $0.60=\frac{60}{100}$, $0.600=\frac{600}{1000}$. Why? The last digit's place gives the denominator (from Ex 3).
2. Reduce each to lowest terms. $\frac{60}{100}=\frac{6}{10}$ (divide top and bottom by 10); $\frac{600}{1000}=\frac{6}{10}$ (divide by 100). Why? Equal fractions represent the same amount; reducing reveals it.
3. Conclude. All three equal $\frac{6}{10}$, so $0.6=0.60=\mathrm{0.600.}$ Why a "degenerate" case? A trailing zero adds a box worth $0$ at the far right — it changes how the number looks, never its value.

Verify: $0.6=0.60=0.600$ as numbers ✔.

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Ex 5 — Cell E: read a decimal aloud (both ways)

1. Whole part first. $60\to$ "sixty". Why? Rule step 1 — read the whole part normally.
2. Say the point, then digits individually. $60.073\to$ "sixty point zero seven three". Why individually? Each digit lives in a different, shrinking box (tenths, hundredths, thousandths). Saying "seventy-three" would wrongly suggest they share one box.
3. Fraction-name version. The fraction part is $073$; its last digit is in thousandths, so: $0.073=\frac{73}{1000}$ Read: "sixty and seventy-three thousandths". Why "thousandths"? The last (smallest) place names the whole fraction's denominator.

Verify: $60+\frac{73}{1000}=60.073$ ✔.

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Ex 6 — Cell F: write a decimal from words

1. Read the place-name. "thousandths" $\Rightarrow$ the last digit lands in the 3rd place after the point. Why? The named fraction $\frac{\square }{1000}$ has denominator $1000$, which is the thousandths place.
2. Write the numerator's digits. "three hundred four" $=304$, which has three digits: $3,0,4$. Why? We place exactly these digits into tenths, hundredths, thousandths in order.
3. Fill the boxes. tenths $=3$, hundredths $=0$, thousandths $=4$ $\Rightarrow 0.304$. Why the internal $0$? Without it, "34" would slide left and "3" would fall into tenths, giving $0.34=\frac{34}{100}$ — a different number.
4. Answer: $0.304$.

Verify: $0.304=\frac{304}{1000}$ ✔.

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Ex 7 — Cell G: a long fraction (limiting behaviour)

1. Extend the ruler one box further. After thousandths ($\frac{1}{1000}$), divide by 10 again: the 4th place is ten-thousandths $=\frac{1}{10000}=0.0001$. Why? The pattern never stops — every step right is still ÷10. Nothing special happens at box three.

2. Assign places for $2.5008$. $5\to$ tenths $=0.5$; $0\to$ hundredths $=0$; $0\to$ thousandths $=0$; $8\to$ ten-thousandths $=0.0008$.
3. Sum. $2.5008=2+0.5+0+0+0.0008=2.5008$
4. Fraction name. Last digit in ten-thousandths $\Rightarrow$ denominator $10000$: $0.5008=\frac{5008}{10000}.$

Verify: $2+0.5+0.0008=2.5008$ ✔ and $\frac{5008}{10000}=0.5008$ ✔.

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Ex 8 — Cell H: real-world word problem (money)

1. Read each price by place. \3.07 = 3$dollars$+ 0$dimes$+ 7$cents$= 3 + 0 + 0.07$.$$3.70 = 3$dollars$+ 7$dimes$+ 0$cents$= 3 + 0.7 + 0$. Why? A dime is a tenth of a dollar (tenths box); a cent is a hundredth (hundredths box).
2. Compare left to right (Comparing and Ordering Decimals). Whole parts tie ($3=3$). Tenths: $0$ vs $7$ $\Rightarrow$ the ruler wins here. Why left-to-right? The leftmost differing place is worth the most, so it decides.
3. The ruler is dearer. Difference: $3.70-3.07=0.63=\$0.63.$
4. The two zeros. In $3.07$ the $0$ is internal — it holds the tenths box empty so "7" is truly $7$ cents. In $3.70$ the $0$ is trailing — pure decoration meaning "no loose cents". Why this matters? One zero changes the value, the other doesn't.

Verify: $3.70-3.07=0.63$ ✔ and $3.70>3.07$ ✔.

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Ex 9 — Cell I: exam twist (the classic trap)

1. Give every number the same number of places. Pad with trailing zeros (allowed — they change nothing, Ex 4): $0.5\to 0.500$, $0.45\to 0.450$, $0.409\to 0.409$. Why? Equal-length line-up lets us compare box-for-box fairly.
2. Compare tenths first. $0.500$ and the other $0.500$: tenths $=5$. $0.450$: tenths $=4$. $0.409$: tenths $=4$. Why tenths first? It is the biggest fractional box, so it outranks everything to its right.
3. Break the tie between $0.450$ and $0.409$ at the next box. Hundredths: $5$ vs $0$ $\Rightarrow 0.450>0.409$.
4. Order (smallest → largest): $0.409<0.45<0.5=\mathrm{0.5.}$
5. The trap. "$409>5$" treats the digits after the point as one big whole number. But they are not — each sits in a shrinking box. Tenths beat everything below, so $0.5$'s single "5" (worth $0.5$) crushes $0.409$'s "4" (worth $0.4$).

Verify: sorted list is $[0.409,0.45,0.5,0.5]$ ✔ and $0.409<0.5$ ✔.

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Recall drills

Recall In

$5.036$, what is the "3" worth, and why? $0.03$ (hundredths). The first box after the point is tenths and holds a $0$, so "3" is pushed one box right into hundredths.

Recall Why is

$0.60=0.6$ but $0.06\ne 0.6$? The zero in $0.60$ is trailing (worth nothing). The zero in $0.06$ is internal — it pushes "6" from tenths down to hundredths, dividing its value by 10.

Recall "Three hundred four thousandths" — how many decimal places and why?

Three. "Thousandths" fixes the last digit in the 3rd place, so $304$ fills tenths–hundredths–thousandths: $0.304$.

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Connections

• Base-10 Number System — the ÷10 ruler behind every example here
• Place Value in Whole Numbers — the left side of every split
• Fractions — each place-name is a power-of-ten denominator
• Comparing and Ordering Decimals — Ex 8 and Ex 9 use its left-to-right rule
• Rounding Decimals — next step once you can name every place

Case Map