Budgets, bounds and integer decisions · GMAT Quant

An inequality describes a range rather than one value. A business constraint may add an integer requirement, turning an interval endpoint into a rounding decision.

A reliable method

Solve the inequality while preserving the direction, reversing it when multiplying or dividing by a negative number. Apply integer conditions only after the interval is clear.

The common trap

A strict bound excludes the endpoint. Rounding to the nearest integer can exceed a budget; maximum feasible counts require rounding down when the bound is nonintegral.

Check the conclusion

Test the proposed greatest integer and the next larger integer in the original constraint. For counts, list the first and last permitted values before using an endpoint formula.

Contrast the cases

A $100 budget with a $12 fixed fee and $7 per report allows at most twelve reports: the remaining $88 permits 88/7 reports, which must be rounded down. Thirteen reports cost $103 and fail the original constraint. In −3x>−12, division by −3 reverses direction to x<4. A strict endpoint is excluded; with integer x, the greatest allowed value is three.

Transfer and explain

Transfer: count integers satisfying |x−5|≤2. The interval is 3 through 7 inclusive, giving five values. If the symbol changes to <, only 4,5,6 survive. Explain the difference using endpoint inclusion.

WORKED EXAMPLE

A team has a budget of $139. A service charges a fixed $12 plus $7 per report. What is the greatest whole number of reports the team can buy?

The report budget is 139−12 = 127.
Divide by 7 and take the greatest integer that does not exceed the result.
The maximum is 18 reports; 19 would cost $145.

Answer: 18

Compare the other answer choices

Charges the fixed fee separately for every report.

Leaves room for one more permitted report, so it is not the greatest count.

Rounds a noninteger affordable count up past the spending limit.

127

Returns the remaining money rather than report count.

Try it, then explain it.

Complete a short set without looking at the example. After each answer, explain which operation preserved the relationship and why the result makes sense.

Practice this skill

Sources & authorship

Original Slope explanations and examples. References support the concepts; they are not the source of copied questions. Independent educator review is pending.

GMAT exam content Siyavula Grade 10 mathematics foundation reference