Ken has made a selection of square jam sandwiches for a party, which he arranges in a single layer on the table in a perfect rectangle. sarah has brought along the same number of square peanut butter sandwiches, which she puts out, also as a single layer, to form a border (one sandwich wide) around ken's rectangle. given that all of the sandwiches are the same size and they are all used, how many sandwiches could each person possibly have?

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  24 or 30

Step-by-step explanation:

If the rectangle dimensions are x by y, then there are xy sandwiches in the rectangle and (x+2)(y+2)-xy sandwiches in the border. If those are the same numbers, we require ...

  xy = (x+2)(y+2)-xy

  2xy = xy +2x +2y +4

  xy = 2(x +y +2)

There are no integer solutions for x=1 or 2. For x=3, we have ...

  3y = 2(3 +y +2)

  3y = 2y +10

  y = 10 . . . . . . . . 3·10 = 30 sandwiches will make a 3×10 rectangle

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For x=4, we have ...

  4y = 2(4 +y +2)

  4y = 2y +12

  y = 6 . . . . . . . 4·6 = 24 sandwiches will make a 4×6 rectangle

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There are no other integer solutions.

Ken and Sarah could each have 24 or 30 sandwiches.

c.   x=2.4cm

step-by-step explanation:

similar triangles are triangles which have the same shape but not the same size. this means their angle measures are equal but their side lengths are not instead they are proportional. to solve, we will set up a proportion and solve.

a proportion is two equal ratios set equal to each other. we form the ratios by finding corresponding sides (sides which match to each other on both triangles by their position). out ratios will be one side of the small triangle over the corresponding side on the big triangle as shown below:

.

to solve the proportion, we'll cross multiply by multiplying numerator with denominator across the equal sign.