A repeating unit

A row of 90 shapes is printed on a border strip. The shapes repeat in the same order all the way along: 4 circles, then 3 squares, then 4 circles, then 3 squares, and so on. How many squares are in the row? Cut the strip where the shapes begin again. Bracket 4 circles and 3 squares. One repeating unit is 7 shapes. Slide the bracket along to the next one. The same unit begins again. So how many whole units fit into the 90? Divide the 90 by the 7 shapes in one unit. 90 divided by 7 is 12 whole units with 6 shapes left over. Whole units and a leftover, never a decimal. Those 12 whole units use 84 shapes. The 6 left over have started a thirteenth unit. It needs 7 shapes and has only 6, so it is not a whole unit. Read those 6 off the start of the unit. Circle, circle, circle, circle, square, square. The 6 left over are 4 circles and 2 squares. Now count the squares. Each unit holds 3 squares, so 12 units hold 36. The leftover adds 2 more. 36 plus 2 is 38 squares.

Your turn. A garden border is twenty one point six metres long and 20 centimetres wide. It is paved from end to end with a pattern that repeats along its whole length: two square tiles of side 20 centimetres, then one rectangular tile measuring 40 centimetres by 20 centimetres. How many square tiles are used? Try it first. Pause if you like. First put the border into centimetres. Twenty one point six metres is two thousand, one hundred and sixty centimetres. One repeat is 20 plus 20 plus 40, which is 80 centimetres. Two thousand, one hundred and sixty divided by 80 is 27 whole repeats. Each repeat holds 2 square tiles. 27 times 2 is 54 square tiles.

One last pattern. A string of beads is made with 3 white beads between every 2 black beads. The string begins with a black bead and ends with a black bead, and there are 4 black beads in all. How many gaps are on the string, and how many on the same beads closed into a loop? The string goes black, white, white, white, black, and repeats. Each group of 3 white beads sits in a gap between 2 black beads. A string has 2 ends, so there is one fewer gap than black beads. 4 minus 1 is 3 gaps. Now close the string into a loop. The last gap closes back onto the first black bead. Going round, 4 black beads make 4 gaps. A string has two ends, and a loop has none. Going round the loop, every black bead is followed by a gap. A closed loop has the same number of gaps as black beads.