Decimals — place value, reading and writing: Exercises

Figure 1 — the ruler of places. Below is a row of boxes, one per place. The two purple boxes on the left are whole-number places (tens, ones); the coral dot is the decimal point; the three green boxes on the right are the fractional places (tenths, hundredths, thousandths). Whenever a solution says "look at the tenths box in Figure 1," it means the first green box to the right of the coral dot. Keep glancing back at this ruler — every solution below points to one of its boxes.

Figure — Decimals — place value, reading and writing

Level 1 · Recognition

Goal: name a place, or state what a single place is worth. No arithmetic yet.

L1.1 In the number $6.379$ , which place does the digit $7$ sit in, and what is that digit worth?

Recall Solution L1.1

Step 1 — Split at the point. Left part 6 (whole), right part 379 (fractional). Why? The dot marks the end of the ones place; places to its right shrink by $\div 10$ each step. Step 2 — Walk right from the point (in Figure 1, step through the green boxes one at a time):

1st place right = tenths → digit $3$
2nd place right = hundredths → digit $7$
3rd place right = thousandths → digit $9$ Step 3 — Value of that digit. The hundredths place is worth $\frac{1}{100} = 0.01$ , so the $7$ is worth $7 \times 0.01 = 0.07$ . Answer: $7$ is in the hundredths place; it is worth $0.07$ .

L1.2 Name the place of every digit after the point in $0.508$ .

Recall Solution L1.2

Walk right from the point, one box at a time:

$5$ → tenths ( $\frac{1}{10}$ )
$0$ → hundredths ( $\frac{1}{100}$ )
$8$ → thousandths ( $\frac{1}{1000}$ ) The internal zero is not decoration — it holds the hundredths box empty so that $8$ is genuinely in the thousandths place. Remove it and $8$ would slide into hundredths, changing the number.

L1.3 True or false: the 2nd place to the right of the point is worth $\frac{1}{20}$ .

Recall Solution L1.3

False. The pattern is $\div 10$ each step, not $\div 2$ then something else. 1st place $= \frac{1}{10}$ , 2nd place $= \frac{1}{10} \div 10 = \frac{1}{100}$ . So the 2nd place is hundredths $= \frac{1}{100}$ , not $\frac{1}{20}$ .

Level 2 · Application

Goal: read decimals aloud and write them from words.

L2.1 Read $40.062$ aloud in the "digit-by-digit" style.

Recall Solution L2.1

Step 1 — Whole part: $40$ → "forty". Step 2 — Say "point". Step 3 — Each fractional digit individually, left to right: $0, 6, 2$ → "zero six two". Answer: "forty point zero six two". Why individually? Grouping them as "sixty-two" would suggest they share one place. They do not — $0$ is in tenths, $6$ in hundredths, $2$ in thousandths.

L2.2 Write "nine and forty hundredths" as a decimal.

Recall Solution L2.2

Step 1 — Whole part: "nine" → $9$ , write 9. Step 2 — The word "hundredths" fixes the last digit's place: the last digit must land in the 2nd place (hundredths). Step 3 — "forty" $= 40$ . Place these two digits so the final digit sits in hundredths: $4$ in tenths, $0$ in hundredths. Answer: $9.40$ . Check: $\frac{40}{100} = 0.40$ , and $9 + 0.40 = 9.40$ . ✔ (This equals $9.4$ ; the trailing zero is honest but optional.)

L2.3 Write "three hundred five thousandths" as a decimal.

Recall Solution L2.3

Step 1 — Whole part: there is none, so we start 0. Step 2 — "thousandths" fixes the last digit in the 3rd place. So we need exactly 3 fractional boxes. Step 3 — "three hundred five" $= 305$ , placed so the last digit is in thousandths:

$3$ → tenths
$0$ → hundredths (internal zero: it must stay to keep $5$ out of hundredths)
$5$ → thousandths Answer: $0.305$ , i.e. $\frac{305}{1000}$ .

L2.4 Give the formal (fraction-name) reading of $0.017$ .

Recall Solution L2.4

The last digit ( $7$ ) sits in the thousandths place, so the whole fractional part is measured in thousandths. As a number, $017 = 17$ . Answer: "seventeen thousandths" $= \frac{17}{1000}$ .

Level 3 · Analysis

Goal: compare decimals correctly and expose why the naive method fails.

L3.1 Which is larger, $0.7$ or $0.68$ ?

Recall Solution L3.1

Step 1 — Equalise the number of places by appending a trailing zero (which never changes value): $0.7 = 0.70$ . Step 2 — Compare from the left, tenths first (in Figure 2, follow the coral arrow down the tenths column):

tenths: $7$ vs $6$ → $7$ wins immediately. Answer: $0.7 > 0.68$ . Once the tenths differ, no later digit can overturn it — a whole tenth ( $0.1$ ) is bigger than any amount of hundredths ( $< 0.1$ ).

Figure 2 — comparing $0.7$ and $0.68$ . The two numbers are padded to the same length and stacked. The coral arrow points down the tenths column, the first place where they differ; that column alone decides the winner.

L3.2 Order from smallest to largest: $0.5, 0.45, 0.405, 0.54$ .

Recall Solution L3.2

Step 1 — Pad to 3 places: $0.500, 0.450, 0.405, 0.540$ . Step 2 — Compare tenths first:

$0.405$ has tenths $4$ (smallest so far).
$0.450$ has tenths $4$ too — tie with $0.405$ , so go to hundredths: $0.405$ has $0$ , $0.450$ has $5$ → $0.405 < 0.450$ .
$0.500$ and $0.540$ have tenths $5$ (larger). Between them, hundredths: $0$ vs $4$ → $0.500 < 0.540$ . Answer (smallest → largest): $0.405 < 0.45 < 0.5 < 0.54$ .

L3.3 Is $0.9$ equal to, greater than, or less than $0.90$ ? Justify with fractions.

Recall Solution L3.3

$0.9 = \frac{9}{10}$ and $0.90 = \frac{90}{100} = \frac{9}{10}$ . Same value. Answer: equal. The trailing zero adds an empty hundredths box; empty boxes contribute nothing.

Level 4 · Synthesis

Goal: rebuild a decimal from scattered place-value clues.

L4.1 A number has: $8$ in the ones place, $0$ in the tenths place, $3$ in the hundredths place, and $6$ in the thousandths place. Write the number and give its sum-of-places expansion.

Recall Solution L4.1

Step 1 — Place each digit in its box (left of the point: ones; right: tenths, hundredths, thousandths): $8.036$ Step 2 — Expansion using each place's worth: $8 \times 1 + 0 \times \frac{1}{10} + 3 \times \frac{1}{100} + 6 \times \frac{1}{1000} = 8 + 0 + 0.03 + 0.006$ Answer: $8.036$ , expanding to $8 + 0.03 + 0.006$ . Why the tenths zero matters: it holds the $3$ in hundredths. Drop it and you'd get $8.36$ — a hundred-fold error on those digits.

L4.2 Build the decimal that is $\frac{2}{10} + \frac{7}{1000}$ (note: no hundredths term). Then read it aloud digit-by-digit and formally.

Recall Solution L4.2

Step 1 — Match each fraction to its place:

$\frac{2}{10}$ → tenths → digit $2$
hundredths → nothing given → digit $0$
$\frac{7}{1000}$ → thousandths → digit $7$ Step 2 — Assemble: $0.207$ . Step 3 — Read digit-by-digit: "zero point two zero seven". Step 4 — Formal reading: last digit is thousandths, and $207$ as a number → "two hundred seven thousandths" $= \frac{207}{1000}$ . Answer: $0.207$ .

L4.3 "Five and three tenths" plus "two hundredths" — combine into a single decimal.

Recall Solution L4.3

Step 1 — First number: five and three tenths $= 5.3 = 5.30$ . Step 2 — Second amount: two hundredths $= 0.02$ . Step 3 — Add, aligning the points (tenths under tenths, hundredths under hundredths): $5.30 + 0.02 = 5.32$ Answer: $5.32$ — "five point three two".

Level 5 · Mastery

Goal: everything at once — read, write, compare, expand, spot traps.

L5.1 Consider $N = 100 + 0.4 + 0.008.$ (a) Write $N$ as a single decimal. (b) Read it formally. (c) State the value of the digit in the hundredths place.

Recall Solution L5.1

(a) Assemble by boxes:

hundreds: $1$ ; tens: $0$ ; ones: $0$
tenths: $4$ ; hundredths: none given → $0$ ; thousandths: $8$ $N = 100.408$ (b) Formal reading: whole part "one hundred", then fractional part — last digit is thousandths and $408$ as a number → "one hundred and four hundred eight thousandths". (c) Hundredths digit: it is $0$ , worth $0 \times 0.01 = 0$ . (That empty box is exactly what keeps $8$ in thousandths.) Answer: $N = 100.408$ .

L5.2 Two students measure a ribbon. Aya writes $1.4 m$ ; Ben writes $1.40 m$ ; Cara writes $1.04 m$ . Who agrees, and who is different? Order all three.

Recall Solution L5.2

Step 1 — Pad to 2 places: $1.40, 1.40, 1.04$ . Step 2 — Aya vs Ben: $1.4 = 1.40$ (trailing zero, no change) → they agree. Step 3 — Cara has tenths $0$ , hundredths $4$ : that is $1.04$ , an internal zero which does matter. Step 4 — Compare tenths: Aya/Ben have $4$ , Cara has $0$ → Cara is smallest. Answer: Aya $=$ Ben $= 1.40 >$ Cara $= 1.04$ . Order: $1.04 < 1.40 = 1.4$ .

L5.3 Write the number that is one thousandth larger than $0.6$ . Then explain, using place values, why the answer is not $0.7$ .

Recall Solution L5.3

Step 1 — Line up places. "One thousandth" lives in the 3rd box: $0.001$ . Step 2 — Write $0.6$ with three boxes: $0.600$ . Step 3 — Add in the thousandths box: $0.600 + 0.001 = 0.601$ . Answer: $0.601$ . Why not $0.7$ ? $0.7$ is one tenth larger than $0.6$ — that is $0.100$ , a hundred times bigger than a thousandth. The size of the step depends entirely on which box you add into.

L5.4 Fill the blanks so the statement is true, then verify by expansion: $\underline{7} . \underline{0} 5 \underline{2} = 7 + 0.05 + 0.002.$

Recall Solution L5.4

Step 1 — Read the target expansion box by box:

$7$ → ones place → whole digit $7$
$0.05 = \frac{5}{100}$ → hundredths → digit $5$ (matches the given $5$ )
$0.002 = \frac{2}{1000}$ → thousandths → digit $2$
tenths box has no term → digit $0$ Step 2 — Fill: ones $= 7$ , tenths $= 0$ , hundredths $= 5$ , thousandths $= 2$ → $7.052$ . Step 3 — Verify: $7 + 0.05 + 0.002 = 7.052$ ✔. Answer: $7.052$ .

Rapid-fire recall

Recall The digit

$7$ in $6.379$ is worth… $0.07$ (hundredths place).

Recall Write "three hundred five thousandths"

$0.305$ .

Recall Which is larger,

$0.7$ or $0.68$ ? $0.7$ (pad to $0.70$ ; tenths $7 > 6$ ).

Recall Assemble

$\frac{2}{10} + \frac{7}{1000}$ $0.207$ (placeholder $0$ in hundredths).

Recall One thousandth larger than

$0.6$ $0.601$ , not $0.7$ .

Connections

Decimals — place value, reading and writing — the parent this drill practises
Place Value in Whole Numbers — the left-of-point boxes reused here
Comparing and Ordering Decimals — L3's pad-then-compare method
Rounding Decimals — L5's "which box" discipline
Fractions — every place is a power-of-ten fraction
Base-10 Number System — the $\div 10$ -per-step master rule