Check quantity bounds when terms share an input

Two quantities can each have a sensible range while their lowest values cannot occur together. Inspect the relationship before describing extreme totals. This manual exercise uses invented integers in an ordinary shared document; colleagues can review the reasoning in a virtual office without using actual project estimates or a Kiguri calculation feature.

Define the permitted inputs and expression

For this fictional fixture, x is an integer from three through seven. The second quantity is y = 10 minus x. The output is F = 2x plus y. These are stipulated rules, not observations of a team or a claim about how real quantities behave.

Both x and y individually range from three to seven. However, choosing x also determines y. Write that relationship beside the expression rather than storing two separate ranges with no explanation. A range describes permitted values here, not probabilities, confidence, or how often any value occurs.

Enumerate the joint assignments

There are five allowed input pairs. When x is three, y is seven and F is thirteen. For x four, y is six and F is fourteen. For x five, y is five and F is fifteen. For x six, y is four and F is sixteen. Finally, x seven gives y three and F seventeen.

Keep those assignments together in the review record. They show which values can occur simultaneously. Every pair satisfies x plus y equals ten. Another reader should be able to calculate each output from its actual pair, instead of combining whichever individual values produce an appealing result.

Compare independently combined extremes

If the relationship were ignored, someone might calculate a lower value of two times three plus three, giving nine. The upper calculation would be two times seven plus seven, giving twenty-one. Neither input pair satisfies the stipulated relationship: three and three sum to six, while seven and seven sum to fourteen.

The attainable output range in the enumerated model is thirteen through seventeen. This also follows by substituting y into the expression: F becomes x plus ten. Keep the joint-input explanation, not only the simplified formula, because it identifies why the independent endpoint combinations cannot occur.

Distinguish a loose enclosure from a false claim

Nine through twenty-one still contains every permitted output. It can therefore serve as a loose enclosure. Calling it an impossible or mathematically false bound would overstate the finding. The problem arises if someone calls nine and twenty-one attainable extremes or describes that enclosure as the tight range for this model.

Ask what the bound is meant to support. A deliberately broad enclosure may be sufficient for one check, while a claim about achievable endpoints needs feasible inputs. Do not narrow a real range merely because a smaller interval looks more useful; the relationship used to narrow it must be established.

Preserve the relationship in the handoff

Retain input domains, the shared constraint, the output expression, all five assignments, and the chosen bound description. Ask a colleague to explain why x three cannot be paired with y three here. If they treat the two ranges as independent, the record has lost its essential information.

The exercise establishes exact results for five invented assignments, not empirical dependence or a reliable delivery estimate. Its useful outcome is an inspectable quantity-bound argument that distinguishes permitted values, simultaneous values, attainable endpoints, and a valid but looser enclosure.