Module 1
The Reversal Trap.
逆転の罠 （Gyakuten no Wana）
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。
This is the highest-value module in the guide. A small set of English phrases reverse the order of the numbers as you write them:
| English | Correct | The trap |
|---|---|---|
| 5 less than x | x − 5 | 5 − x ✗ |
| 5 fewer than x | x − 5 | 5 − x ✗ |
| 5 subtracted from x | x − 5 | 5 − x ✗ |
| x subtracted from 5 | 5 − x | x − 5 ✗ |
| x divided into 20 | 20 ÷ x | x ÷ 20 ✗ |
Reference — the rest of the code. Everything that does not reverse (the full table is Module 3):
is/equals → = · a number → x · sum / more than / increased by → + · product / times / twice / of → × · quotient / per / divided by → ÷
Note that "less than" written in words behaves the opposite way to "less than" written as a symbol. "5 less than x" is x − 5, but "x is less than 5" is x < 5. Same two words, different job, decided entirely by sentence position.
日本人がハマる理由
日本語では「x より 5 小さい数」と言います。基準になる x が先に来て、直後のよりがそれを基準だと示し、引く量の 5 は後です。英語はこれが逆で、量が先、基準が後に来ます。日本語の語順のまま訳すと、毎回 5 − x と書いてしまいます。しかも出てきた数はそれらしく見えることが多く、間違いを知らせてくれるものが何もありません。less than や subtracted from を見たら、後ろに出てくる数を実際に手で先に書いてください。
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。
Six less than twice a number is 10. What is the number?
Correct Answer: C
Explanation: "Six less than twice a number" → 2x − 6. So 2x − 6 = 10 → 2x = 16 → x = 8.
- (A) −2: The reversal trap. Writing
6 − 2x = 10givesx = −2. If your answer to a word problem comes out negative when the context has no reason to be negative, suspect this first. - (B) 2: From
(10 − 6) ÷ 2— the 6 was subtracted again instead of added back. - (D) 32: From
2 × (10 + 6)— multiplied where you should have divided.
Difficulty Justification
Rated Medium. The arithmetic is trivial; the difficulty is entirely in choice A, which is what the sentence says if you read it in Japanese word order.
When 4 is subtracted from a number, the result is 11. What is the number?
Correct Answer: C
Explanation: "4 is subtracted from a number" → x − 4. So x − 4 = 11 → x = 15.
- (A) −7: The reversal:
4 − x = 11. The word "from" is what tells you which number is being reduced. - (B) 7: From
11 − 4— subtracted where you needed to add to undo. - (D) 44: From
4 × 11; no multiplication appears anywhere in the sentence.
Difficulty Justification
Rated Easy because "from" is an explicit signpost. Learn to treat from as the marker that the following noun is the starting quantity.
A number is 9 less than 5 times another number n. Which expression represents the number?
Correct Answer: D
Explanation: "9 less than [5 times n]" — the whole quantity 5n is what gets reduced by 9. 5n − 9.
- (A) 9 − 5n: The reversal trap, in expression form. This is the answer the question is built to catch.
- (C) 5(n − 9): This subtracts 9 from n first, then multiplies. The sentence multiplies first — "5 times another number" is a complete quantity before "9 less than" acts on it.
- (B) 5(9 − n): Combines both errors: wrong order inside the parentheses and wrong grouping.
Difficulty Justification
Rated Medium. Expression-form questions are harder than value questions because there is no numerical answer to sanity-check. The only defence is the mechanical rule.