Working Backwards

Explanation

'I think of a number, multiply by 3, add 5, and get 26.' Working backwards means rewinding the movie: start from the END and undo each step with its OPPOSITE, in OPPOSITE order. Undo the +5 first: 26 − 5 = 21. Then undo the ×3: 21 ÷ 3 = 7. The number was 7 — and you can prove it by playing the movie forward: 7 × 3 + 5 = 26 ✓.

Your Turn

I double a number, subtract 4, and get 18. What was it? Check: 18 + 4 = 22, ÷ 2 = 11. Forward: 11×2−4 = 18 ✓.

Watch Out

Both flips matter: operations become opposites AND the order reverses. Undo the LAST step first.

Explanation

OC style, mixing directions: 800 coins, buy 6 potions at 95 and a mystery map, 110 left. Push FORWARD through what you know (potions: 570; after them: 230) and work BACKWARD around the mystery (the map dropped 230 down to 110, so it cost 120). Most OC problems are this hybrid — forward through the knowns, backward around the unknown. Full check: 800 − 570 − 120 = 110 ✓.

Your Turn

Ella had some money, earned $15, spent $22, and has $9. How much did she start with? Check: 9 + 22 − 15 = $16.

Watch Out

When rewinding a SPEND, you ADD it back. Money she lost must return in the rewind — directions flip!

Explanation

The classic backwards chain: Dad eats half the biscuits, Mia eats half of what's left, Leo eats 3, and 5 remain. Rewind: undo Leo (5 + 3 = 8), undo Mia — 'ate half' rewinds as DOUBLE (8 × 2 = 16), undo Dad (16 × 2 = 32). The jar held 32. The key insight: when someone ate half, what remained was the OTHER half, so rewinding doubles it.

Your Turn

A number is halved, then halved again, then 6 is added: result 16. Start? Check: 16 − 6 = 10, ×2 = 20, ×2 = 40.

Watch Out

'Ate half' undoes as ×2 only because exactly half remained. 'Ate a third' leaves two-thirds — its rewind is NOT ×3!

Explanation

Multiple choice hands you a gift: the answer is ONE of five numbers on the page. Stuck? Test the options forward! For ▲×4−7=29, try option C first (the middle): 9×4−7 = 29 ✓ — done. Middle-first is clever: if the middle is too small, only bigger options remain; too big, only smaller. You never test more than three. This weapon works on any 'find the number' question — it's slower than thinking, but it NEVER fails.

Your Turn

Options 4, 5, 6, 7, 8 for: "double it, add 3, get 15". Test the middle. Check: 6→15 ✓ first try!

Watch Out

Options are usually sorted smallest → largest — that ordering is what makes middle-first powerful.

Explanation

You now hold three roads to any answer. Know the start? Go FORWARD. Know only the end? Go BACKWARD with opposite operations in opposite order. Both feel messy? TEST THE OPTIONS, middle first — 45 seconds of testing beats three minutes of confusion. Strong OC students aren't faster at one method; they're quick to pick the right road for each question. Quiz and exercises — practise choosing!

Your Turn

Which road for: "Ben triples his sticker count then loses 5, ending with 22"? Check: backward — (22+5)÷3 = 9.

Watch Out

If your backward answer is ugly (like 8.5 stickers), a step got mis-rewound — switch roads and test options instead.