In a certain board game, a player receives 15 points for each quest completed, but loses 4… — SAT

In a certain board game, a player receives $15$ points for each quest completed, but loses $4$ points for each penalty card drawn. A player completes $12$ quests and draws $c$ penalty cards. If the player's total score is greater than $120$ points, what is the maximum possible value of $c$ ?

Cevap: 14

Cevap

The player earns a base score of

15 \times 12 = 180

points from quests, and loses

4

points for each of the

c

penalty cards, resulting in a total score of

180 - 4c

points. Because the score must be greater than

120

, we write the inequality

180 - 4c > 120

. Subtracting

180

from both sides yields

-4c > -60

. Dividing by

-4

and reversing the inequality sign gives

c < 15

. Since

c

must represent a whole number of cards, the maximum possible value is the largest integer less than

15

, which is

14

Adım Adım Çözüm

Set up the inequality representing the score constraint.

180 - 4c > 120

The player earns

15

points for each of the

12

quests (

15 \times 12 = 180

) and loses

4

points for each of the

c

penalty cards (

4c

), and this total must exceed

120

Isolate the variable term by subtracting

180

from both sides.

-4c > -60

To solve for

c

, we first subtract the constant term

180

from both sides of the inequality.

Divide both sides by

-4

and reverse the inequality sign.

c < 15

Dividing an inequality by a negative number requires reversing the direction of the inequality sign.

Determine the maximum integer value for

c

14

Since the number of penalty cards

c

must be a whole number and

c

must be strictly less than

15

, the largest possible value is

14

Anahtar Kavram

Solving linear inequalities in one variable involving multiplication or division by a negative number and interpreting the solution set in a discrete context.

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Adım Adım Çözüm

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