Work Backward after Inserting a Decimal Point
Lesson 15.4 · 添上小数点后倒推原数
A decimal point keeps the digits in the same order but changes their values. Use the amount lost, the point position, exact whole-number checks, and column subtraction to recover the original four-digit number.
See exactly what the decimal point changes
Every original number in this lesson is a whole number from 1000 to 9999. Imagine its decimal point at the right end first: 2004 means 2004.0. Inserting a point moves it left by 1, 2, 3, or 4 places without reordering the four original digits.
Four possible positions
Let N be a four-digit whole number. Placing the point before a digit gives:
The practice problem says the point is inserted before one of the digits. We will test every mathematically possible position rather than assume one.
The lost amount
If the point moves k places left, the new number is:
The absolute difference is:
Decimal-point explorer
Check the place-value model
Audit every point position in the practice problem
This worked example asks for a four-digit number whose value becomes 1,803.6 smaller after a decimal point is inserted.
Complete problem
A decimal point is inserted before one digit of a four-digit whole number. The absolute difference between the original number and the new decimal is 1,803.6. Find the original four-digit number.
Do not guess the point position
For each possible shift k, solve:
Keep a candidate only when it is a four-digit whole number.
What counts as a valid branch?
Exact position audit
| Shift | Point position | Candidate from the equation (≈ means rounded display) | Four-digit whole? | Verification |
|---|---|---|---|---|
| Run the audit to test every position before checking this mission. Exact fractions determine whether a candidate is whole; a rounded display does not. | ||||
Solve the practice problem with an equation
The position audit shows that the decimal point was placed before the final digit, so the new value is one tenth of the original.
Verification
The digits remain 2, 0, 0, 4 in the same order.
The next mission develops the worked example’s own digit-by-digit method.
Reconstruct the digits with the worked example’s column method
Align the decimal points. The new decimal is abc.d while the original is abcd.0.
Aligned subtraction
Borrowed top values, from thousands to tenths: 1 · 10 · 0 · 3 · 10. The hundreds receive 10 from the thousands; the tenths receive 10 from the ones.
Scroll the subtraction sideways on a small screen to keep each place-value heading aligned.
Let the original digits be a, b, c, d
Start at the tenths column and move left. Every borrow creates the next digit condition.
Tenths: borrow one whole, so 10−d=6. Therefore d=4.
Ones: after the borrow, (d−1)−c=3. With d=4, we get c=0.
Tens: the difference digit is 0, so c−b=0. With c=0, we get b=0.
Hundreds: borrow from the thousands place: 10+b−a=8. With b=0, we get a=2.
Final check: the thousands column is a−1=1, which agrees with a=2. The original is 2004.
Complete the worked example guided practice
A four-digit number becomes 2,003.4 smaller after a decimal point is inserted. First run the four-position search below: only shift 1 survives. The displayed equation then recovers that original number.
Test a candidate
Use a forward and reverse decimal-point laboratory
Generate a difference from a known original, then reverse the process by auditing every possible point position. For this mission’s fixed questions, run the reverse solver on 4277.79, regardless of any other examples you explore.
Forward generator
Reverse solver
Decide whether the answer is unique, multiple, or impossible
A difference alone does not always identify exactly one original number. A complete position audit tells us what the evidence proves.
Unique
Only one point position produces a four-digit whole number.
Multiple
Both 1100−110.0 and 1000−10.00 equal 990.
Impossible
No point position produces a valid four-digit whole number.
Complete case audit
1803.6
Not audited
990
Not audited
1803.5
Not audited
Verify a claim and transfer the method
A recovered number is not finished until the point is reinserted and the subtraction exactly reproduces the stated difference.
Verification checker
Transfer A
The point is inserted before the final digit, and the number becomes 3,888.9 smaller.
Transfer B
The point moves two places left, and the number becomes 4,752 smaller.
Decimal-point reverse-engineering workshop
Correct all eight answers to complete this workshop.
Worked explanation for question 1
Each move divides by 10; three moves divide by 1000.
Worked explanation for question 2
Keep digit order and divide by 100.
Worked explanation for question 3
3456 − 34.56 = 3421.44.
Worked explanation for question 4
9N = 11106, so N=1234.
Worked explanation for question 5
99N = 247500, so N=2500.
Worked explanation for question 6
1000 groups minus one group leaves 999 groups.
Worked explanation for question 7
2200−220=1980 and 2000−20=1980; shifts 3 and 4 give nonintegers.
Worked explanation for question 8
6789÷1000=6.789.
Objective exit ticket
Complete all ten missions, including a perfect exit score, to earn the certificate.
Worked explanation for question 1
9N=51003, so N=5667.
Worked explanation for question 2
99N=534600, so N=5400.
Worked explanation for question 3
2500−2.5=2497.5.
Worked explanation for question 4
Shift 1 gives 1000. Other shifts give values below 1000.
Worked explanation for question 5
Divide by 100 and retain all four digits.
Certificate of mathematical thinking
Decimal-Point Reverse Engineer
has completed Lesson 15.4 by locating decimal-point positions, recovering original numbers, and verifying every solution exactly.
Why is verification especially important here?
Explain why finding a whole-number candidate is not enough until the decimal point is reinserted in the claimed position.
About this lesson
The worked example uses digit-by-digit column subtraction to recover 2004. This lesson preserves that method in Mission 4 and adds a position audit, an equation route, a general forward/reverse solver, uniqueness classification, transfer problems, workshop, and exit ticket as instructional scaffolds.