問56
頻出度★★☆
紙の大きさの規格は図のような相似形であり，例えばA判用紙では，A3の面積の半分がA4になるという関係にある。コピー機でA4サイズの原稿をA3サイズに拡大コピーするためには，どの拡大率（％）を選んだらよいか。ここで，拡大率は長い辺同士の比率を表す。
平成23年度 特別試験 ITパスポート試験 問56
テクノロジ系基礎理論 › 応用数学
This question's explanation and your answers so far.
解説
問 56
Incorrectア
At 122%, the length becomes only 1.22 times, so the area becomes 1.22² ≈ 1.49 times, which does not reach twice. This is the standard ratio used when enlarging from A4 to B4, not the ratio for A4 to A3.
- イ
正解 In the figure, the outer rectangle is divided in two by a vertical line, and the left half is A3; the right half is divided in two by a horizontal line, and the upper part is A4; the lower part is further divided by a vertical line, one side of which is A5 — a nested structure. In other words, halving an area the same size as A3 gives A4, and just as the question states, the area of A3 is twice the area of A4. A-series paper sizes are similar (proportional) shapes, so if the ratio of the long edges (that is, the enlargement ratio) is k, the ratio of the areas is k squared. To enlarge an A4 original to A3, the area only needs to be doubled, so k² = 2, that is, k = √2 ≈ 1.4142. Expressed as a percentage this is about 141.4%, so the enlargement ratio to select is 141%. Therefore the correct answer is イ.
Incorrectウ
At 150%, the area becomes 1.5² = 2.25 times, which exceeds the area of A3 (twice that of A4). The output becomes too large relative to the original and does not fit within the A3 frame.
Incorrectエ
200% comes from mistaking “doubling the area” for “doubling the length.” If the length is doubled, the area becomes 2² = 4 times, which is four times A4 — that is, the size of A2, the outer frame in the figure.
ここがポイント
- In similar figures, when the ratio of lengths is k, the ratio of areas is k². When you want to double the area, the multiplier for the lengths is not 2 but √2 ≈ 1.41.
- For A-series and B-series paper, the area ratio between adjacent sizes is 2 to 1, so enlarging by one step (A4 to A3, for example) uses a ratio of about 141%, and reducing by one step (A3 to A4, for example) uses 1/√2 ≈ about 71%. Keep these in mind as the standard ratios on a copy machine.
用語