A rounded price describes a range, not one exact execution.
With exactly $200 of assumed reinvested cash and a displayed price of $22.22, nine shares are compatible with our chosen rounding rule. A positive exact price that rounds half-up to $22.22 lies from $22.215 inclusive to $22.225 exclusive. The implied share range is greater than 8000/889 and at most 40000/4443. Nine falls inside it. This does not establish a broker's actual price or rounding policy.
The weighted-fill worksheet reconstructs an exact average from complete cash and share rows. This page asks the reverse question: how much information is lost when a price is shown with fewer digits? We assume exact cash and an exact candidate share count, excluding fees, taxes and separate share rounding.
Declare the display rule before inferring a range
Here, half-up means a positive value exactly halfway between two cent values rounds to the higher one. We deliberately choose cent precision for a mathematical test. It is not a statement about a platform's display settings, execution precision, truncation policy or account terms. Without a known rule, this particular interval cannot be applied to a statement.
Half a cent is $0.005. Subtract it from the displayed $22.22 to get $22.215, and add it to get $22.225. The lower endpoint is included because $22.215 rounds up to $22.22. The upper endpoint is excluded because $22.225 rounds up to $22.23 instead. Keeping those endpoint labels matters; replacing the range with a vague plus-or-minus tolerance loses the tie behavior.
For positive exact cash C and price P, shares Q = C/P. Increasing P decreases Q, so the endpoint order reverses when dividing. With C = $200, the lower quantity boundary is 200/22.225 = 8000/889 and is excluded. The upper boundary is 200/22.215 = 40000/4443 and is included. Exact fractions, rather than shortened decimal estimates, decide the boundary tests.
Check the candidate by going back to price
For a candidate of 9 shares, the reconstructed exact price is 200/9 = $22.222222… . It lies inside the price interval and rounds to $22.22 under the declared rule. Multiplying the displayed $22.22 by 9 gives $199.98, but the two-cent residual alone does not reject the candidate. The price digits discarded by the display explain that residual in this controlled example.
A candidate of 8.99 shares is below 8000/889 and therefore outside the interval. A candidate of 9.01 shares exceeds 40000/4443 and is also outside. “Outside” means incompatible with these exact cash, price-display and rounding assumptions together. It does not prove an account error: a different cash basis or an unknown rounding policy changes the question.
Test the two endpoints separately
| Case | Candidate shares | Compatible? |
|---|---|---|
| nine_shares | 9 | True |
| lower_excluded | 8000/889 | False |
| upper_included | 40000/4443 | True |
| below | 8.99 | False |
| above | 9.01 | False |
| positive_cash_zero_shares | 0 | False |
| zero_cash | 0 | True |
| zero_cash_positive_shares | 1 | False |
| missing_candidate | Unknown | unknown_candidate |
Download the candidate inputs · Download exact bounds and compatibility results
A candidate equal to 8000/889 shares reconstructs price $22.225 and must fail the selected $22.22 display test. A candidate equal to 40000/4443 reconstructs $22.215 and must pass. The test suite checks both by exact rational comparisons and independently by decimal half-up rounding. A boundary must not be accepted merely because its printed approximation appears close.
With positive $200 cash, a candidate of zero shares is outside the compatible range. With zero cash and a known positive displayed price, zero shares are compatible, while one share is not. Those are cash-to-quantity checks; zero cash does not reveal which price inside the display interval was used, nor does it prove an actual trade occurred.
Missing inputs do not become zero
A missing candidate quantity leaves the calculable interval intact but compatibility unknown. A missing cash amount or displayed price prevents construction of the interval. Negative cash, negative candidate shares or a nonpositive display increment is invalid in this nonnegative-purchase fixture. None of these failures returns a fake zero-share reconciliation.
The fixture requires a displayed price on its declared increment grid, so $22.222 with a $0.01 increment is rejected rather than silently rounded again. It also requires the entire underlying price interval to be positive. A displayed zero is outside that supported domain and is marked invalid, rather than generating division by zero or an invented finite upper share limit.
What if the statement has more precision?
Use its documented increment and tie convention instead of assuming cents. If cash or share count is also rounded, this one-uncertain-variable test is insufficient. Preserve the original statement fields, whether they are gross or net, and any documented rounding rules before drawing a conclusion.
The Schwab One Account Agreement discusses statement precision and partial-share calculations. That provides a reason to distinguish exact inputs from displayed fields; it does not establish the nearest-cent price-display rule chosen here. A public DRIP pricing question asks for the reinvestment price tied to a transaction. This worksheet is a way to test declared assumptions, not a substitute for that record.
Evidence and limits
Sources read October 27, 2026. Cash $200, display $22.22, increment $0.01, half-up ties and candidate quantities are explicit test inputs. Exact fractions and Decimal rounding independently verify results. The article adds an inverse interval and endpoint matrix to the prior weighted-price explanation. Comparison coverage is limited to freshly read bodies from the three fixed calculator sites; no global originality or broker-error claim is made.
When the cash field is rounded too
The exact-cash assumption above is essential. The two-rounded-input worksheet adds a cash interval, changes the outer boundary rules and saves feasible input pairs. It also distinguishes a displayed zero from known exact zero.
