A hard question on perimeter and area for P5–P6, with a step-by-step solution, diagrams and the answer. Common in the Hua Xia Cup's P4 to P6 papers; harder ones work out areas from the ratios between triangles' areas.
The question
A large rectangle is cut by two lines across and two lines down into 9 smaller rectangles, as shown (the lines are parallel to its sides).
Each number is the area of that small rectangle, in cm².
What is the area of the large rectangle, in cm²?
Show the hints (3)
💡Hint 1: Small rectangles in the same row have the same height. Their areas compare as what?
💡Hint 2: Of the four rectangles without numbers, three can be found in one step.
💡Hint 3: Find those three first, then use them to find the last one, and add up all nine.
Worked solution
Have a go first, then read the solution step by step.
1.The bottom-middle (21) and bottom-left (9) rectangles are in the same row, so they have the same height, and their widths compare as their areas: 21 : 9.
2.The top-middle and top-left (12) rectangles in another row have the same widths (they sit in the same two columns), so their areas compare the same way, 21 : 9: top-middle = 12 × 21 ÷ 9 = 28.
3.The middle-right (36) and centre (42) rectangles are in the same row, so they have the same height, and their widths compare as their areas: 36 : 42.
4.The top-right and top-middle (28) rectangles in another row have the same widths (they sit in the same two columns), so their areas compare the same way, 36 : 42: top-right = 28 × 36 ÷ 42 = 24.
5.The top-left (12) and top-middle (28) rectangles are in the same row, so they have the same height, and their widths compare as their areas: 12 : 28.
6.The middle-left and centre (42) rectangles in another row have the same widths (they sit in the same two columns), so their areas compare the same way, 12 : 28: middle-left = 42 × 12 ÷ 28 = 18.
7.The top-right (24) and top-left (12) rectangles are in the same row, so they have the same height, and their widths compare as their areas: 24 : 12.
8.The bottom-right and bottom-left (9) rectangles in another row have the same widths (they sit in the same two columns), so their areas compare the same way, 24 : 12: bottom-right = 9 × 24 ÷ 12 = 18.
⚠️Common mistake: thinking that if two rectangles in one row differ by some amount, the two below differ by the same amount. Rectangles in a row share a height, so their areas go in a ratio: multiply and divide.
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