Ms Newton's class took the same test as Mr. Euclid's students. Ms. Newton's class has 20 students, and the average(mean) test score was 90. The average of Mr. Euclids class was 80. If you combine both classes the average score on the test is 85. How many students does Mr. Euclid have in his Mathematics class?

If Newton had 20 students and Euclid had e students,

total points = 20*90 + e*80
total students = 20+e

total average = (20*90 + e*80)/(20+e) = 85

1800 + 80e = 1700 + 85e
100 = 5e
e = 20

Makes sense. 85 is halfway between 80 and 90, so there must have bee the same number of students in each class.

Luisa mows lawns during the summer. She charges $15 if she cuts the grass but charges $5 more if she also trims the grass. Last week she trimmed 5 more yards than she cut. If she made $415 last week, how many yards did she trim?

To find the number of students in Mr. Euclid's Mathematics class, we can use the concept of weighted averages.

Let's assume that Mr. Euclid's class has "x" students.

Since the average test score of Ms. Newton's class is 90, and there are 20 students in her class, the total test scores for her class would be 90 * 20 = 1800.

Now, let's calculate the total test scores for Mr. Euclid's class. We know that the average test score is 80, so the total test scores for his class would be 80 * x = 80x.

When we combine both classes, the average test score is 85. Since the total number of students is 20 (from Ms. Newton's class) + x (from Mr. Euclid's class) = 20 + x, the total test scores for both classes would be 85 * (20 + x) = 85(20 + x).

To find the total test scores for both classes, we can equate the expressions we found for Ms. Newton's class and Mr. Euclid's class:

1800 + 80x = 85(20 + x)

Now, let's solve this equation to find the value of "x" and determine the number of students in Mr. Euclid's class:

1800 + 80x = 1700 + 85x

Simplifying the equation, we get:

80x - 85x = 1700 - 1800
-5x = -100
x = -100 / (-5)
x = 20

Therefore, Mr. Euclid has 20 students in his Mathematics class.