Tina is playing a computer game. She starts with 100 points, and she loses points based on the following rules:
• Each time a player passes a level, 8 points are lost.
• Each time a player catches a flower, 3 points are lost.
Suppose Tina catches 6 flowers per level, on average. Solve an inequality to determine the number of levels she must complete to have fewer than 20 points left.
Tina must complete at least
level(s).
Tina is playing a computer game. She starts with 100 points, and she loses points based on these rules:
• Each time a player passes a level, 8 points are lost.
• Each time a player catches a flower, 3 points are lost.
Part A
Suppose Tina catches 6 flowers per level. Which inequality determines the number of levels she must pass to have fewer than 20 points left?
Let's start solving the problem step-by-step.
Let L be the number of levels Tina needs to complete.
Each time Tina catches a flower, 3 points are lost. Therefore, the total number of points lost due to catching flowers is 6 flowers per level * 3 points per flower = 18 points per level.
In addition to that, Tina loses 8 points for passing each level.
So, the total number of points lost for L levels is (18 points per level + 8 points per level) * L levels = 26L points.
Tina starts with 100 points and loses 26L points, so the remaining number of points after completing L levels is given by:
100 points - 26L points < 20 points.
Simplifying the inequality:
-26L < 20 - 100
-26L < -80
To solve for L, divide both sides of the inequality by -26, remembering that when we divide an inequality by a negative number, the sign of the inequality is reversed:
L > (-80 / -26)
Simplifying further:
L > 80 / 26
L > 40/13
Since the number of levels must be a whole number, round up the result to the next integer:
L > 4.
Therefore, Tina must complete at least 5 levels to have fewer than 20 points left.
To solve the inequality, we need to express the number of points Tina has left in terms of the number of levels she completes.
Let's say the number of levels Tina completes is "x". Each time she passes a level, she loses 8 points, and for each level she catches 6 flowers, she loses 3 points.
So, the number of points Tina loses due to passing levels would be 8 multiplied by the number of levels (8x) and the number of points she loses due to catching flowers would be 3 multiplied by the number of levels (3x).
Therefore, the total number of points Tina loses would be 8x + 3x.
Since she starts with 100 points, the number of points she has left can be expressed as 100 - (8x + 3x).
To find the number of levels Tina must complete to have fewer than 20 points left, we need to solve the inequality:
100 - (8x + 3x) < 20
To simplify the inequality, let's combine like terms:
100 - 11x < 20
To isolate the variable, let's subtract 100 from both sides:
-11x < -80
Now, we need to solve for "x" by dividing both sides by -11. Remember that when dividing by a negative number, we must reverse the inequality sign:
x > (-80) / (-11)
Simplifying, we get:
x > 80/11
Since we're looking for a whole number of levels, we need to round up the result. Therefore, Tina must complete at least 8 levels to have fewer than 20 points left.