Break it apart
Multiply a big number by splitting it into parts you already know, and learn the four shortcuts that cover most of the times table you never memorised: ×4, ×5, ×8 and ×9.
THE IDEA
Seven forty-sixes is seven forties plus seven sixes
46 is 40 + 6, so seven of them is seven 40s plus seven 6s. That is all the distributive law says: multiplying a sum is the same as multiplying each part and adding. Picture 7 rows of 46 dots, then draw one vertical line after column 40. You have two clean rectangles and neither dot moved.
7 × 46 = (7 × 40) + (7 × 6) = 280 + 42 = 322.
7 × 46
split: 46 = 40 + 6
big: 7 × 40 = 280
small: 7 × 6 = 42
join: 280 + 42 = 322322Do the big piece first, the way you learned in lesson 1. After one step you already know the answer is 'about 280 to 330', so a wild slip in the small piece cannot go unnoticed.
Check yourself
Omid orders 6 boxes of tea at 48 each. What does he pay?
- 298
- 288
- 280
- 308
Show the answer
288
Right. Either 6 × 40 = 240 and 6 × 8 = 48, giving 288, or 6 × 50 = 300 minus six 2s, which is 300 − 12 = 288.
SHORTCUT
×5 is half of ×10
5 is 10 ÷ 2, so five of something is ten of it, halved. Multiplying by 10 costs you nothing (add a zero) and halving is a move your head already does with prices and bills. Never multiply by 5 the long way again.
5 × 86: ten 86s is 860, and half of 860 is 430. Same for 5 × 74: 740, then 370.
SHORTCUT
×4 and ×8 are just doubling
4 is 2 × 2 and 8 is 2 × 2 × 2. So ×4 means double it twice and ×8 means double it three times. Doubling is the one operation almost everybody is fast at, and stacking it costs no memory: you only ever hold one number.
4 × 35: double 35 to 70, double again to 140. 8 × 45: 45 → 90 → 180 → 360.
Check yourself
Which shortcut would you reach for?
- 5 × 64
- 4 × 26
- 9 × 23
- 5 × 128
- 9 × 41
- 4 × 19
Show the answer
Halve ten times it: 5 × 64, 5 × 128
Double it twice: 4 × 26, 4 × 19
Ten times it, minus one copy: 9 × 23, 9 × 41
×9: buy ten, send one back
Nine of something is ten of it with one copy returned. So 9 × 27 is 270 minus one 27, which is 243. Where the analogy breaks: what you send back is a whole item, not a single unit. Returning 1 instead of 27 is the mistake almost everyone makes the first time, and it lands you 26 away from the answer.
Check yourself
Since 9 × 40 = 360, it follows that 9 × 38 = 358.
Show the answer
False
False. Dropping from 40 to 38 removes two nines, not two units: 360 − 18 = 342. Whenever you shrink the number being multiplied by 2, the product shrinks by 2 × the other factor.
| times | do this | example |
|---|---|---|
| ×4 | double twice | 4 × 35 → 70 → 140 |
| ×5 | ten times it, halved | 5 × 86 → 860 → 430 |
| ×8 | double three times | 8 × 45 → 90 → 180 → 360 |
| ×9 | ten times it, minus one copy | 9 × 27 → 270 − 27 = 243 |
| ×20 | double it, then add a zero | 20 × 35 → 70 → 700 |
Check yourself
Use the shortcut, then match each one to its answer
Show the answer
- 5 × 74 → 370
- 4 × 65 → 260
- 9 × 34 → 306
- 8 × 25 → 200
Make it a habit
- Split the longer number and leave the short one whole: 7 × 46 is seven lots of 40 and seven lots of 6.
- Big piece first. It doubles as your estimate and it catches a wrong answer early.
- Before you split, check whether the multiplier is 4, 5, 8, 9 or 20. Those five have a faster road.
- If a number is near a round one, compensation from lesson 2 works here too: 6 × 48 is 6 × 50 minus six 2s.
Check yourself
The hard one. A workshop runs 9 sessions and each one costs 47. Total?
- 423
- 430
- 469
- 413
Show the answer
423
Exactly. Ten sessions would be 470; take one 47 off and you get 423. Splitting works too: 360 + 63 = 423.
Lesson recap
- Split the big number into parts you know, multiply each part, add the results back.
- It works because multiplying a sum equals multiplying each piece: 7 × (40 + 6) = 280 + 42.
- ×5 is half of ×10, ×4 is double-double, ×8 is double three times, ×9 is ×10 minus one copy.
- On ×9 and every 'minus' shortcut, take away a whole copy, not a single unit.