GatewayWhat the 5-point questions actually ask for.
Appears in
5–6 · 7–8 · 9–10 · 11–124 levels
What it is
Every five-point problem has a short way in, but you do not always find it. The skill is deciding, quickly and deliberately, whether to keep hunting for the idea or to start the long reliable calculation - and then committing to that decision.
●●●5 points, difficulty 3 of 35 ptLevel 5–6about 135s
What is the value of (1 - 1/2) x (1 - 1/3) x (1 - 1/4) x ... x (1 - 1/10)?
Hints
Take them one at a time. The first gives nothing away.
1A nudge
Write each bracket as a single fraction.
2The strategy
Turn every bracket into one fraction before multiplying anything. Then write the whole product out in a line and look at what appears on the top and on the bottom.
3The full solution
The brackets are 1/2, 2/3, 3/4, ... , 9/10. Every top cancels the bottom before it, leaving 1 on top and 10 underneath.
Solution
The reliable way
Write each bracket as one fraction: 1 - 1/2 is 1/2, 1 - 1/3 is 2/3, 1 - 1/4 is 3/4, and so on up to 9/10. The product is (1 x 2 x 3 x ... x 9) over (2 x 3 x 4 x ... x 10). Everything cancels except the 1 on top and the 10 underneath, giving 1/10. Grinding it out multiplication by multiplication gives the same answer and takes ten times as long. The transferable idea: rewrite before you calculate - the shortcut only becomes visible once the terms are in a comparable form.
The elegant way
Each bracket is (k - 1)/k, so the product telescopes to 1/10 whatever the last denominator is.
Why this is on the test: It is the clearest case of a problem that is two minutes of grinding or fifteen seconds of noticing, and the decision between the two is the skill.
Where it goes wrong
Each of these is written the way a student would say it to themselves.
“I kept hunting for an elegant idea until the time ran out.”
“I started grinding immediately and missed a one-line observation.”
“I switched between the two approaches repeatedly and finished neither.”
Every problem in this skill
●○○3 points, difficulty 1 of 33 ptOne step from a skill you know.
Nothing at this tier yet for CFT-07.
●●○4 points, difficulty 2 of 34 ptTwo steps, or one step plus a trap.
Nothing at this tier yet for CFT-07.
●●●5 points, difficulty 3 of 35 ptNeeds an idea, not just a procedure.