Questions such as “How many pieces will there be after cutting the tofu how many times?” are exactly problems in which a rectangular prism is cut repeatedly by flat surfaces.
Rules for Tofu-Cutting Problems
- If you make a cuts vertically, b cuts horizontally, and c cuts in the height direction, the number of pieces is (a+1) × (b+1) × (c+1)
- Remember the difference in “n cuts make n+1 pieces”
Example 1: The tofu was cut 3 times vertically, 2 times horizontally, and 1 time in the height direction. How many pieces will there be?
Solution: (3+1) × (2+1) × (1+1) = 4 × 3 × 2 = 24 pieces
Solution: (3+1) × (2+1) × (1+1) = 4 × 3 × 2 = 24 pieces
Example 2: A rectangular prism measuring 12 cm in length, 9 cm in width, and 6 cm in height is divided into 4 equal parts lengthwise, 3 equal parts widthwise, and 2 equal parts in height. What is the volume of one piece in cm³?
Solution:
① Total number of pieces = 4 × 3 × 2 = 24 pieces
② Total volume = 12 × 9 × 6 = 648 cm³
③ Volume of one piece = 648 ÷ 24 = 27 cm³
Solution:
① Total number of pieces = 4 × 3 × 2 = 24 pieces
② Total volume = 12 × 9 × 6 = 648 cm³
③ Volume of one piece = 648 ÷ 24 = 27 cm³
In problems about dividing into equal parts, you will get the same answer if you first find the size of one piece. For example, 3 cm × 3 cm × 3 cm = 27 cm³.