Count the Digits Used to Number a Book
Lesson 13.2 — 数一数编页码用了多少个数字
Split page numbers into one-digit, two-digit, three-digit, and four-digit blocks. Count the page labels in each block, multiply by the width of each label, and add the subtotals.
Answers saved before this lesson update
Earlier answers — original questions
These answers belong to the earlier question wording. They are kept for reference; try the updated questions above and recheck your missions.
Decide what is being counted
A page count, a page number, and a digit symbol are different objects. A correct solution begins by naming the unit.
How many pages the book contains.
One complete label printed on a page.
One written symbol inside a page number.
Check the unit
Hint 1 — choose a starting point
Decide whether to count labels or symbols.
Hint 2 — take the next step
Count each visible position; do not add leading zeros.
Count an inclusive range correctly
When both endpoints are included, the number of integers from a through b is b − a + 1.
Inclusive-range laboratory
page labels in the selected range
Why the “+ 1”?
The subtraction 99 − 10 = 89 counts the jumps after page 10. Adding 1 also counts page 10 itself.
10 is item 1, 11 is item 2, …, 99 is item 90.
Count the labels
Hint 1 — choose a starting point
Try the small range 10–12 first.
Hint 2 — take the next step
Subtract to count gaps; add one to count both endpoints.
Build the complete digit-width blocks
Every page label inside one block has the same number of written digits.
| Page-number block | Number of labels | Digits per label | Digit subtotal |
|---|---|---|---|
| 1–9 | 9 | 1 | 9 × 1 = 9 |
| 10–99 | 90 | 2 | 90 × 2 = 180 |
| 100–999 | 900 | 3 | 900 × 3 = 2,700 |
| 1,000–9,999 | 9,000 | 4 | 9,000 × 4 = 36,000 |
Through page 99
The book uses the complete one-digit and two-digit blocks.
Through page 999
The complete three-digit block is included too.
Use the full blocks
Hint 1 — choose a starting point
Count labels in each complete block.
Hint 2 — take the next step
Multiply by width and add the subtotals.
Worked example: number pages 1 through 320
A 320-page storybook numbers every page from 1 through 320. How many individual digit symbols are printed?
Complete problem: A storybook has 320 pages. How many digit symbols are used to number all its pages?
The page numbers are printed normally, beginning with 1 and without leading zeros.
Rebuild the worked solution
Hint 1 — choose a starting point
Pages 1–99 already use 189 digits.
Hint 2 — take the next step
Count 100–320 inclusively, then multiply by three.
Guided practice: number pages 1 through 131
Now apply the same block method to a 131-page Grade 5 mathematics book.
Complete practice problem
A Grade 5 mathematics book has 131 pages. How many digit symbols are used in all of its page numbers?
Counting how many times a particular digit such as 1 appears is a different skill. That belongs to Lesson 13.3. Here we count every written digit symbol.
Complete the 131-page ledger
Hint 1 — choose a starting point
The last block starts at 100.
Hint 2 — take the next step
131 − 100 + 1 gives its label count.
Use a general page-digit counter
Choose a whole-number final page from 1 through 99,999. The counter automatically splits the numbering into the correct digit-width blocks.
| Range used | Labels | Width | Subtotal |
|---|
Read the counter at important endpoints
Hint 1 — choose a starting point
Predict the total before using the counter.
Hint 2 — take the next step
Adding a page adds exactly the width of its label.
Understand what happens at a boundary
Adding one new page adds exactly as many digit symbols as that page number contains.
8 digits total
page 8 adds 19 digits total
page 9 adds 111 digits total
page 10 adds 2187 digits total
page 98 adds 2189 digits total
page 99 adds 2192 digits total
page 100 adds 3From 9 to 10
The new page number 10 contains two digits.
From 99 to 100
The new page number 100 contains three digits.
Check the boundary totals
Hint 1 — choose a starting point
Compare the labels 9 and 10, then 99 and 100.
Hint 2 — take the next step
The new page, not the entire book, determines the added digits.
Detect and repair common errors
A reliable solution audits the range count, digit width, subtotal, and final sum.
Four-step audit
- Split at 9, 99, 999, …
- Count every used range inclusively.
- Multiply labels by digits per label.
- Add all subtotals and check the unit.
Unit check
A 320-page book has:
320 page numbers
but those labels use:
852 digit symbols
Judge each claim
Hint 1 — choose a starting point
Check the unit and both endpoints.
Hint 2 — take the next step
Recalculate a statement rather than guessing from its wording.
Independent workshop
Try all eight questions before opening the explanations. Correct all eight answers to complete this mission.
Hint
Split at 9, 99, and 999. Count the last block inclusively.
Worked explanation — open after trying
The complete lower-width blocks plus the final partial block give 87 digits. 9 + 39 × 2 = 87
Hint
Split at 9, 99, and 999. Count the last block inclusively.
Worked explanation — open after trying
The complete lower-width blocks plus the final partial block give 165 digits. 9 + 78 × 2 = 165
Hint
Split at 9, 99, and 999. Count the last block inclusively.
Worked explanation — open after trying
The complete lower-width blocks plus the final partial block give 207 digits. 189 + 6 × 3 = 207
Hint
Split at 9, 99, and 999. Count the last block inclusively.
Worked explanation — open after trying
The complete lower-width blocks plus the final partial block give 330 digits. 189 + 47 × 3 = 330
Hint
Split at 9, 99, and 999. Count the last block inclusively.
Worked explanation — open after trying
The complete lower-width blocks plus the final partial block give 516 digits. 189 + 109 × 3 = 516
Hint
Split at 9, 99, and 999. Count the last block inclusively.
Worked explanation — open after trying
The complete lower-width blocks plus the final partial block give 1113 digits. 189 + 308 × 3 = 1113
Hint
Split at 9, 99, and 999. Count the last block inclusively.
Worked explanation — open after trying
The complete lower-width blocks plus the final partial block give 2886 digits. 189 + 899 × 3 = 2886
Hint
Split at 9, 99, and 999. Count the last block inclusively.
Worked explanation — open after trying
The complete lower-width blocks plus the final partial block give 2941 digits. 2889 + 13 × 4 = 2941
Fresh exit ticket
Use the method on new problems. All five answers must be correct. Complete all ten missions to earn the certificate.
Hint
Count both endpoints.
Worked explanation — open after trying
172 − 100 + 1 = 73.
Hint
Multiply the label count by three.
Worked explanation — open after trying
73 × 3 = 219.
Hint
Include the shorter labels before page 100.
Worked explanation — open after trying
9 + 180 + 219 = 408.
Hint
Use the full total through 999, then count 1000–1005.
Worked explanation — open after trying
2889 + 6 × 4 = 2913.
Hint
Count symbols in the new label only.
Worked explanation — open after trying
The new label 1000 has four digits.
Lesson checkpoints completed
Page-Number Digit-Block Architect
This certifies that Learner can split page numbers into digit-width blocks, count inclusive ranges, and calculate the total digit symbols used to number a book.