KA-0125Spot the trick vs. decide to grind
5 points, difficulty 3 of 3Level 5–6about 135sWhat 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.