Ablue paper clip is 1/6 inch wide . a silver paper clip is 3/8 inch wide, and a red paper clip is 1/3 inch wide. what color paper clip has the smallest widths

We have to make all the fractions have the same denominator in order to compare them, so we have to find the GCM(greatest common factor). The greatest common factor is 24, so

3 x 3 = 9    

8 x 3 = 24

9/24

1 x 4 = 4

6 x 4 = 24

4/24

1 x 8 = 8

3 x 8 = 24

8/24

Now we have the fractions 9/24, 8/24, and 4/24. We can compare the numerators and find the smallest number. The length of the blue paper clip which is 4/24 is the smallest width.

there are lots of ways.   pick's theorem tells us if the vertices are lattice points the area a is

a = b/2 + i - 1

where b is the number of boundary lattice points and i is the number of interior lattice points.

in the figure i count b=16, i=50 so

a = 8 + 50 - 1 = 57

let's try the shoelace formula.   the area is half the sum of the cross products of the sides.  

a(-2,6)   -2(2)-6(-5) = 26

b(-5,2)   -5(-3) - 2(-2) = 19

c(-2,-3) -2(-3) - (-3)(4) = 18

d(4,-3)   4(3) - (-3)(5) = 27

e(5,3)   5(3) - 3(1) = 12

f(1,3)     1(6) - 3(-2) = 12

a(-2,6)

each of the cross products is twice the area of the triangle with the associated side and one vertex the origin. so the area of abo=26/2=13.

area = (1/2) (26 + 19 +   18 + 27 + 12 + 12) = 57

that checks!

the volume is the product of the area of the base, 3x(3x+4), and the height, 3x-1

step-by-step explanation: