Singapore School Mathematics Operating Manual · Chapter 25
Many Mathematics questions do not ask us to predict what will happen from known inputs. They ask us to work backward.
We observe a result and try to recover the hidden quantity that produced it. We know a final price and reconstruct the original price. We know the area and one side of a rectangle and recover the missing side. We know a graph and infer the equation. We know several measurements and solve for an unknown parameter. We know a sequence and try to infer the generating rule. We know the total and a relationship between parts and reconstruct the parts.
These are inverse problems. The forward problem goes from cause or parameter to observable result. The inverse problem goes from observable result back to cause, parameter or hidden state.
The inverse direction is often harder because information can be lost. Different hidden states can produce the same observable result. Measurement noise can be amplified. A model can be underdetermined. A neat numerical answer can appear even when the reconstruction is not unique.
This chapter develops a school-level operating discipline for inverse reasoning: identify the forward map, reconstruct candidates, test uniqueness, check stability, and return to the original conditions.
1. Forward and inverse questions are different jobs
Forward: a rectangle has length 8 cm and width 5 cm. Find area.
Inverse: a rectangle has area 40 cm² and width 5 cm. Find length.
The forward rule is A = lw. The inverse step is l = A/w, provided w ≠ 0.
Here the inverse is simple because one output and one known parameter determine the missing quantity uniquely.
2. Some inverse problems have more than one answer
Forward: square a real number.
Inverse: if x² = 25, recover x.
Both x = 5 and x = −5 produce the same forward output.
The forward map x ↦ x² is many-to-one over the real numbers.
This is why an inverse problem may need a branch set rather than one number.
3. Domain restrictions can restore uniqueness
If x² = 25 and x is known to be positive, then only x = 5 remains.
The extra condition removes the negative branch.
Inverse reconstruction therefore depends on both the equation and the admissible domain.
4. Percentage reversal is an inverse problem
A price after a 20% increase is $120. What was the original price?
Forward model: final = 1.2 × original.
Inverse: original = 120/1.2 = 100.
A common error is subtracting 20% of 120. That uses the wrong base because the increase was applied to the original quantity, not the final quantity.
5. Discount reversal is not ordinary addition
A $80 item after a 20% discount becomes $64.
If we observe $64 and reverse the discount, the original is 64/0.8 = 80.
Adding 20% of 64 gives only $76.80 because the inverse of multiplication by 0.8 is division by 0.8, not addition of 20%.
6. Algebraic equations are inverse machines
Suppose y = 3x + 7 and y = 31.
The forward map multiplies x by 3 and adds 7.
The inverse operation subtracts 7 and divides by 3, giving x = 8.
Solving equations often means reversing a chain of operations while preserving equivalence.
7. Reversibility matters
If every forward step is one-to-one over the active domain, inversion can be clean.
If a forward step collapses information—such as squaring, absolute value or rounding—the inverse problem may become ambiguous.
This connects directly to Reversible Steps, Lost Information and Extraneous Solutions.
8. Rounding creates an interval of hidden values
If a length is reported as 5.2 cm to the nearest 0.1 cm, the original value is not exactly known.
Under the usual rounding convention, the hidden exact value lies in 5.15 ≤ L < 5.25.
The inverse of rounding is therefore an interval, not one exact number.
9. A rounded total can make reverse decisions uncertain
Suppose a reported mass is 10.0 kg to the nearest 0.1 kg and a threshold is 10.04 kg.
The hidden exact value might be below or above the threshold.
The central reported value alone cannot reconstruct the true comparison.
This is why the Rounded Data chapter distinguishes guaranteed from possible conclusions.
10. Graph reading is often inverse reasoning
Given y = 2x + 3, forward evaluation sends x to y.
Given y = 11, finding x asks for the inverse image of 11.
On the graph, draw a horizontal line from y = 11 to the graph and then down to the x-axis.
The graphical process reverses output to input.
11. A horizontal-line test checks inverse uniqueness
If a horizontal line crosses a graph more than once, some output belongs to multiple inputs.
Such a function does not have a single-valued inverse over that full domain.
Restricting the domain may restore one-to-one behaviour.
12. Geometry often reconstructs hidden dimensions from invariants
A right triangle has hypotenuse 13 and one leg 5.
Using a² + b² = c², the hidden leg satisfies b² = 169 − 25 = 144, so b = 12 under the positive-length condition.
The Pythagorean relationship acts as the forward constraint linking the dimensions.
13. Similarity reconstructs inaccessible lengths
If two triangles are similar, corresponding side ratios are equal.
A measured small triangle can be used to infer an inaccessible larger length.
The reconstruction is only valid if the similarity assumption is justified.
This illustrates how inverse problems depend on model validity as well as algebra.
14. Simultaneous equations reconstruct several hidden quantities at once
Suppose two numbers have sum 20 and difference 4.
Let x + y = 20 and x − y = 4.
Solving gives x = 12, y = 8.
Each equation alone leaves infinitely many possibilities. Together they identify one pair.
Inverse reconstruction often needs enough independent constraints.
15. Underdetermined systems do not have unique reconstructions
If x + y = 20 is the only condition, many pairs work.
One observed total cannot identify both hidden components uniquely.
A solver should not invent a second condition.
The mathematically correct conclusion is that multiple reconstructions are possible.
16. Overdetermined data can help verify a reconstruction
Suppose x and y are reconstructed using two equations, and a third independent measurement is also available.
If the reconstructed values satisfy the third measurement, confidence increases.
If not, there may be measurement error, model mismatch or an algebraic mistake.
Redundant information can therefore become a verification layer rather than a waste.
17. Sequences invite inverse pattern reconstruction
Given 2, 5, 8, 11, … one natural rule is a_n = 3n − 1.
But a finite list of terms does not logically determine one unique infinite sequence rule.
Many more complicated formulas can match the same initial terms and diverge later.
School pattern questions usually imply a simple intended rule, but mathematically the reconstruction is model-dependent.
18. Pattern inference should distinguish evidence from theorem
Observing constant first differences strongly suggests a linear rule.
Once a linear model is assumed, two points determine the line uniquely.
The uniqueness comes from the chosen model family, not from the finite data alone.
19. Parameter fitting is inverse modelling
Suppose y = ax + b and two points are known: (1,5) and (3,11).
Then a + b = 5 and 3a + b = 11.
Subtracting gives 2a = 6, so a = 3 and b = 2.
The observed points reconstruct the hidden model parameters.
20. Noise changes exact inversion into estimation
If measured points are (1,5.1), (3,10.9), and (5,17.2), they may not lie exactly on one line.
Then “recover a and b” becomes a best-fit problem rather than exact inversion.
The solver must distinguish exact equations from noisy observations.
21. Small measurement errors can produce large inverse errors
Suppose we infer x from y = 1/x.
Then x = 1/y.
When y is very small, a tiny change in y can create a large change in x.
This is an early glimpse of conditioning: some inverse problems are sensitive even when the forward rule is simple.
22. Hidden causes may not be identifiable
If two different mechanisms produce exactly the same observable outputs under the available measurements, the data cannot distinguish them.
The problem is then not merely hard—it is non-identifiable under the current evidence.
More independent information is required.
23. Probability inverse questions require care
If an event occurred, we cannot automatically infer that it was likely.
Rare events can occur.
Observing an outcome and reconstructing the underlying probability model requires more information than one event.
Outcome-to-model inversion is fundamentally different from model-to-outcome probability calculation.
24. Statistics often separates parameter from estimate
A sample mean is observable.
The population mean is hidden.
Using the sample mean to estimate the population mean is an inverse inference from partial data.
The estimate carries uncertainty because many populations could generate similar samples.
25. Inverse problems can be local or global
A function may be invertible near one point but not over its entire domain.
x² is one-to-one on x ≥ 0 and on x ≤ 0 separately, but not on all real x.
Local domain restriction can therefore turn an ambiguous global inverse into a unique local one.
26. A complete inverse answer should state the reconstruction set
For x² = 16 over the reals, report x = −4 or 4.
For x² = 16 with x ≥ 0, report x = 4.
For a rounded 5.2 to nearest 0.1, report the interval 5.15 ≤ x < 5.25.
The answer form should reflect what the evidence actually determines.
27. Verification should run the reconstruction forward again
After reconstructing a hidden quantity, substitute it into the forward model.
If the original observable is not reproduced, the reconstruction is wrong or the model is inconsistent.
This forward-backward loop is one of the strongest general checks in Mathematics.
28. Multiple reconstructed candidates should all be checked
If an inverse step creates two candidates, do not verify only the preferred one.
Each candidate must be tested against the original domain and conditions.
The final set is the set that survives the full contract.
29. Reconstruction can fail because the data are inconsistent
Suppose x + y = 10 and x + y = 12 are both claimed exact.
No reconstruction exists.
The correct inverse conclusion is an empty feasible set, not an average of 11 unless an approximation model is explicitly introduced.
30. Reconstruction quality depends on the forward model
If the forward model is wrong, exact inversion of it can still be useless.
A travel-time reconstruction that assumes constant speed may not recover a real journey’s detailed speed history.
Inverse Mathematics inherits every assumption of the forward model.
31. A practical inverse-problem audit
Ask:
What is observable and what is hidden? What forward relationship links them? Is the forward map one-to-one? What domain restrictions are active? Are there enough independent conditions? Could rounding or measurement noise create an interval rather than a point? Is the reconstruction stable? Have I run the candidate forward to verify it?
32. Independent practice
1. A quantity is increased by 25% to become 150. Find the original.
2. Solve x² = 81 over the reals.
3. Solve x² = 81 given x > 0.
4. A rectangle has area 54 cm² and width 6 cm. Find length.
5. A number rounds to 7.3 to the nearest 0.1. State the interval of possible exact values.
6. Two numbers have sum 18 and difference 2. Find them.
7. Explain why x + y = 18 alone cannot recover x and y uniquely.
8. The line y = ax + b passes through (0,4) and (2,10). Find a and b.
9. Why does a horizontal-line test matter for inverse functions?
10. Give one example where inversion produces an interval rather than one number.
11. What is the best general verification step after reconstructing a hidden input?
12. Why can a finite sequence of observations fail to determine a unique infinite rule?
33. Worked answers
1. 1.25x = 150, so x = 120.
2. x = −9 or 9.
3. x = 9.
4. l = 54/6 = 9 cm.
5. 7.25 ≤ x < 7.35.
6. x + y = 18 and x − y = 2 give x = 10 and y = 8.
7. Infinitely many pairs have sum 18; one equation cannot generally identify two real unknowns uniquely.
8. b = 4. Then 10 = 2a + 4, so a = 3.
9. More than one intersection with a horizontal line means one output corresponds to multiple inputs, preventing a single-valued inverse on that domain.
10. Reversing a rounded value: 5.2 to nearest 0.1 corresponds to 5.15 ≤ x < 5.25.
11. Run the candidate through the original forward relationship and check every original condition.
12. Many different formulas can agree on finitely many terms and diverge later unless a model family is specified.
34. Continue through Batch 07
Use Symmetry, Redundancy and Case Reduction to avoid solving equivalent copies of the same case, Thresholds and Regime Changes when the governing behaviour changes across a critical value, and Conditioning and Ill-Posedness when reconstruction is unstable.
Return to the BTT Mathematics Hub for Batch 07.