Thursday, May 21 — Day 8 (last day before Shavuot)

Today's new things: inverse + fraction division.

Both of these are short rules. The hard work was already done — once you have multiplication, division is essentially free.

Inverse — the easiest rule of the week

The inverse of a fraction is the fraction flipped.

• Inverse of 3/5 is 5/3.
• Inverse of 2/7 is 7/2.

For an integer, treat it as "n/1" and flip it:

• Inverse of 7 is 1/7.
• Inverse of 9 is 1/9.

That's it. There's no calculation. You don't even multiply anything.

Division — multiply by the inverse

Dividing by a fraction is the same as multiplying by its inverse.

a/b ÷ c/d = a/b × d/c

Worked example: 2/3 ÷ 4/5

1. Flip the second one: 4/5 → 5/4.
2. Now multiply: 2/3 × 5/4.
3. Top: 2 × 5 = 10. Bottom: 3 × 4 = 12. 10/12 = 5/6.

That's the whole technique. Three steps: flip the second one, multiply, reduce.

Today's deal

• Time. 9:00 → 12:30, 2×15-min breaks. 3 hours of work.
• Two new things, but easy ones. Inverse, then division.
• New on the side. Second half of row 3 — P, S, Cl, Ar.
• Everything else is review. Add, sub, mul, periodic rows 1-3, the last three wars, the rest of the evolution timeline.

Today's exam (22 questions)

• 2 add + 2 sub + 2 mul + 2 inverse (new) + 3 div (new)
• 4 periodic — P, S, Cl, Ar (one is the curveball: argon is Z=18 but the most common neutron count is 22, not 18)
• 4 wars — Vietnam, Gulf, Iraq + 1 review
• 3 evolution — first mammals, first hominids, Homo sapiens

Periodic — row 3 finishes

Argon is the trap. Atomic number 18, but its most common isotope has 22 neutrons (40Ar). Don't reflexively say "18 = 18 = 18" — that's the right answer for protons and electrons, but wrong for neutrons.

The last three wars

Vietnam runs ~1955–1975 (the start year is the answer). Gulf is the short one in 1990–91. Iraq is the 2003 one.

Evolution — the last three events

210 mya → first mammals (just before the dinosaurs took over). 7 mya → first hominids (split from chimps). 0.3 mya → Homo sapiens (300,000 years).

Practice before the exam

Do 10 division problems on /fractions before the exam. Then drill inverse out loud — pick 10 fractions and say their inverses without writing anything. Inverse should feel automatic by the time you sit down.

For periodic: write out all 18 elements with P / N / e from memory on a fresh piece of paper. If you miss any, that's where to focus for the next ten minutes.

Help links

• Dividing fractions
• Inverse
• Multiplying fractions
• Subtracting fractions
• Adding fractions
• Periodic table — rows 1-3
• The 15 wars
• Evolution timeline

And: what week 2 was for

Two weeks. Same content. Different shape. You came in cold on Tuesday May 12 and you're finishing today (Thursday May 21) able to do every single math operation, every periodic-table element from H to Ar, every war from 1775 to 2003, and the full evolution timeline from the Big Bang to last week.

That's not nothing. That's a lot. Whatever today's score is, the week's improvement is real.