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? Try it first. Pause if you like. The shapes start again after the 3 squares. So one unit is 4 circles and 3 squares, which is 7 shapes. One unit has 7 shapes and 3 squares. Two units have 14 shapes and 6 squares. Three units have 21 shapes and 9 squares. How many whole units are in 90 shapes? Divide 90 by the 7 shapes in one unit. 90 divided by 7 is 12, remainder 6. So there are 12 whole units, and 6 shapes are left over. 12 units use 12 times 7, which is 84 shapes. The 6 left over start a thirteenth unit, but it needs 7 shapes, so it is not whole. Read them from the start of the unit: 4 circles, then 2 squares. Each unit has 3 squares, so 12 units have 12 times 3, which is 36 squares. Add the 2 squares left over. 36 plus 2 is 38. So there are 38 squares in the row.

A string of beads has 3 white beads between every 2 black beads. It begins and ends with a black bead, and there are 4 black beads. How many gaps between black beads are on the string? The ends are then tied together to make a loop. How many gaps are on the loop? Try it first. Pause if you like. One unit is 1 black bead and 3 white beads. A gap is the space between 2 black beads. Count the gaps on the string: 1, 2, 3. The string has 2 ends, so it has 1 fewer gap than black beads. 4 minus 1 is 3. Now tie the two ends together to make a loop. The last black bead and the first black bead now have a gap between them too. On a loop, there are as many gaps as black beads. So the string has 3 gaps and the loop has 4 gaps. Divide by one repeating unit, then read any leftover from the start of the unit.