Problem 19 · 2022 Math Kangaroo
Hard
Geometry & Measurement
proportionarea-decomposition
The large rectangle ABCD is made up of 7 congruent smaller rectangles (see diagram). What is the ratio AB⁄BC?
Show answer
Answer: D — 12⁄7
Show hints
Hint 1 of 2
The 7 small rectangles are all congruent; let one be \(a\) (long) by \(b\) (short) and read off how many fit across each row of the figure.
Still stuck? Show hint 2 →
Hint 2 of 2
Both rows span the same width AB, so equate the two row widths to get a relation between \(a\) and \(b\).
Show solution
Approach: equate the widths of the two rows of small rectangles
- Let each small rectangle have long side \(a\) and short side \(b\). One row places 3 rectangles standing up (width \(3b\)) beside others, and the other row lays 4 rectangles flat, but both rows span the full width \(AB\).
- Matching the row widths gives the relation \(4b = 3a\) (so \(a = \tfrac{4}{3}b\)), and the figure makes \(AB = 4b\) while \(BC = a + b\).
- Then \(\tfrac{AB}{BC} = \tfrac{4b}{a+b} = \tfrac{4b}{\frac{4}{3}b + b} = \tfrac{4}{7/3} = \tfrac{12}{7}\), so the answer is D.
Mark:
· log in to save