A software programmer is writing test cases for a new application. The programmer must wri… — SAT

A software programmer is writing test cases for a new application. The programmer must write at least 15 test cases in total, consisting of $x$ unit tests and $y$ integration tests. Each unit test takes 10 minutes to write, and each integration test takes 30 minutes to write. If the programmer has at most 300 minutes to write all the test cases, what is the maximum number of integration tests the programmer can write?

Answer: 7

Answer

Step-by-Step Solution

Set up the system of inequalities representing the given constraints.

x + y \ge 15

and

10x + 30y \le 300

The total number of tests must be at least 15, and the total time taken by writing

x

unit tests (10 minutes each) and

y

integration tests (30 minutes each) cannot exceed 300 minutes.

Simplify the time inequality and combine it with the total test count constraint to isolate

y

x + 3y \le 30

. Substituting

x \ge 15 - y

into this inequality yields

(15 - y) + 3y \le 30 \implies 15 + 2y \le 30

Simplification and substitution help find the upper bound for the number of integration tests.

Solve for

y

and determine the maximum integer value.

2y \le 15 \implies y \le 7.5

. The largest integer satisfying this inequality is 7.

The number of integration tests must be a whole number, so we round down to the nearest integer.

Verify that a valid integer number of unit tests (

x

) exists when

y = 7

When

y = 7

, we get

x \ge 15 - 7 \implies x \ge 8

and

x + 3(7) \le 30 \implies x \le 9

. The integers

x = 8

and

x = 9

both satisfy the conditions.

We must confirm that the maximum value of

y

is achievable with an integer number of unit tests.

Key Concept

Systems of Linear Inequalities in Two Variables

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