Two kinds and what is left

Wei Jie had a book of stamps. Some were used and the rest were unused. He gave away two thirds of the stamps in the book. One half of the stamps he gave away were used, and one quarter of the stamps left in the book were used. He gave away twenty one more used stamps than the number of used stamps left in the book. How many stamps were in the book at first? Try it first. Pause if you like. The total number of stamps is unknown. Use equal units for the book. The whole book has three equal parts. Two of those parts were given away. The part given away must split into two equal parts for one half. The part left must split into four equal parts for one quarter. Extend those small equal parts across the whole book. The book is twelve units. Two thirds times twelve units. That is eight units given away. Twelve units minus eight units leaves four units in the book. The lower row shows the used stamps in both places. Each small unit is the same size. One half times the eight units given away. That is four units of used stamps given away. One quarter times the four units left in the book. That is one unit of used stamps left. Compare the used stamps: four units given away and one unit left. Four units minus one unit is three units. This purple part is the difference. The difference is twenty one stamps. Three units are twenty one stamps. One unit is twenty one divided by three. Seven stamps. The book has twelve units. Twelve times seven. Eighty four stamps. There were eighty four stamps in the book at first. One half refers to the stamps given away. One quarter refers to the stamps left. These fractions refer to different parts of the book. Equal-sized units let us compare the two counts.