Arithmetic and Fractions

Question 1

Consider the positive integers from 1-100. If n is a number in that set, then for how many numbers n is it true that |n-30| > 20?

Answer 1

the distance between n and 30 is greater than 20

1-9 and 21-100 = 9points + 50 points = 59points

Question 2

At a certain school in December 2013, there were 6 teachers and 200 students. On January 2, 2014, 1 teacher and 35 students joined the school. No one left the school. compare A and B
A = teacher-student ratio in December 2013
B = The teacher-student ratio in late January, 2014

Answer 2

A = 6/200 = 3/100

if 1/35 is bigger than 3/100, then adding it to B will make it bigger. If it’s smaller, it’ll make B smaller

3/100 ?? 1/35
Cross multiply : 105 > 100
3/100 > 1/35
adding 1/35 to A makes it smaller so
A>B

Question 3

compare
A = 449/150
B = 20/7

Answer 3

notice that 3 = 21/7 = 450/150

A = 3 - 1/150
B = 3 - 1/7

Cleartly, 1/150 < 1/7

if we’re subtracting something smaller, the result has to be something bigger
A > B