Module 5
Between, Inclusive, Exclusive.
範囲の境界 （Han'i no Kyōkai）
帰国子女・インターナショナルスクール生のための、デジタルSAT 数学の無料教材。独学で進められます。
When a question asks how many integers lie in a range, two decisions change the answer: whether the endpoints count, and the +1.
The counting rule. The number of integers from a to b inclusive is b − a + 1, not b − a.
From 12 to 40 inclusive: 40 − 12 + 1 = 29. Sanity-check on a tiny case: from 1 to 3 inclusive there are 3 numbers (1, 2, 3), and 3 − 1 + 1 = 3. ✓
- inclusive — both endpoints count （両端を含む）
- exclusive — neither endpoint counts （両端を含まない）
- between 3 and 11 — on the SAT, normally exclusive; the test says "inclusive" when it means it
日本人がハマる理由
日本語の「12から40まで」のまでは、端の数を含みます。だから特に指定がなくても両端を含めて読むのが普通で、inclusive はわざわざ書く必要のない指定に見えます。英語の between は逆で、指定がなければ両端を含みません。+1 の抜けは日本人に限った間違いではありませんが、この二つが重なるのが日本人の失点パターンです。両端を含むと思い込み、さらに +1 を忘れます。この二つの誤りは打ち消し合いません。
練習問題
難易度は出題者による目安です。College Board の判定ではありません。
How many integers are there from 12 to 40, inclusive?
Correct Answer: C
Explanation: 40 − 12 + 1 = 29.
- (A) 27: From
40 − 12 − 1— excluding both endpoints, which "inclusive" explicitly forbids. - (B) 28: From
40 − 12. The error this module is about — the +1 was forgotten. - (D) 30: Over-counts by one; usually from adding 1 twice or rounding.
Difficulty Justification
Rated Easy, but the +1 is easy to forget. Verify the rule on a three-number case before you trust it on a twenty-nine-number one.
The integer x is between 3 and 11, exclusive. How many possible values does x have?
Correct Answer: B
Explanation: Exclusive means 3 and 11 do not count, so x runs 4, 5, 6, 7, 8, 9, 10 — 7 values. (Formula: 10 − 4 + 1 = 7.)
- (A) 6: From
10 − 4— the +1 was dropped after correctly handling exclusivity. - (C) 8: From
11 − 3— treats the range as though one endpoint counted. - (D) 9: From
11 − 3 + 1— the inclusive count. The word "exclusive" was ignored.
Difficulty Justification
Rated Medium: two independent decisions (endpoints, +1) each have a wrong answer waiting. With a range this small, write the numbers out — it takes six seconds and cannot be wrong.
How many multiples of 5 are there from 10 to 60, inclusive?
Correct Answer: B
Explanation: (60 − 10) ÷ 5 + 1 = 10 + 1 = 11. They are 10, 15, 20, 25, 30, 35, 40, 45, 50, 55, 60.
- (A) 10: From
(60 − 10) ÷ 5— the +1 again. - (C) 12: From
60 ÷ 5, which counts multiples of 5 starting from 5 rather than from 10. - (D) 50: The size of the range, not a count of anything.
Difficulty Justification
Rated Medium. The +1 rule survives division by the step size — divide the span by the step, then add one.