Squares and near-squares
Square any number ending in 5 in about two seconds, square numbers near 50 and near 100, and turn 19 × 21 into 20² − 1 without writing anything down.
ENDING IN 5
Front digit times the next one, then 25
Any number ending in 5 is 10a + 5. Square it and you get 100a² + 100a + 25, which is 100 × a × (a+1) + 25. So the last two digits are always 25, and everything in front is the front part times the number one above it.
35²: 3 × 4 = 12, then 25 → 1225. 65²: 6 × 7 = 42, then 25 → 4225. It also works past two digits: 105² is 10 × 11 = 110, then 25 → 11025.
35²
front part: 3
next one up: 4
3 × 4 = 12
tail: 25
answer: 1225
65²
6 × 7 = 42 → 42251225 and 4225The tail is always exactly 25, never 05 or 50. That is the one part you never have to think about.
Check yourself
A square courtyard is 85 steps on each side. How many square steps is it?
- 6,425
- 7,225
- 7,250
Show the answer
7,225
Yes. 8 × 9 = 72, then the tail 25, so 7,225.
NEAR-SQUARES
Two numbers either side of a round one
19 × 21 is 20 − 1 times 20 + 1. Multiply that out and the two middle terms cancel, leaving 20² − 1² = 399. The rule is (a − b)(a + b) = a² − b². Whenever two numbers sit the same distance either side of a friendly number, square the friendly one and subtract the square of the gap.
19 × 21 = 400 − 1 = 399. 18 × 22 = 400 − 4 = 396. 48 × 52 = 2500 − 4 = 2496.
Spotting the middle
- The problem: 48 × 52
Two numbers, four apart. The number halfway between them is 50, which you would happily square.
- Find the gap from the middle
48 is 2 below 50 and 52 is 2 above. So the gap is 2, and it has to be the same on both sides or the trick doesn't apply.
- Square the middle
50² = 2500.
- Subtract the gap squared
2500 − 2² = 2500 − 4 = 2496. Not 2500 − 2, and not 2500 − 8.
Check yourself
You know 30² = 900. So 29 × 31 = 899.
Show the answer
True
True. 29 and 31 sit one either side of 30, so the product is 30² − 1² = 899. Notice it is always less than the square: moving the same distance out on both sides always loses you a little area.
NEAR 50, NEAR 100
Squares that hang off a landmark
For a number near 50: start from 25, add the gap (or subtract it if you're below), and that's the hundreds. Then write the gap squared as the last two digits. For a number near 100: take the gap off the number itself for the front, and again the gap squared behind it.
47² → 25 − 3 = 22 in front, 3² = 09 behind → 2209. 53² → 25 + 3 = 28, then 09 → 2809. 97² → 97 − 3 = 94, then 09 → 9409.
Check yourself
A square tile pattern is 54 by 54. How many tiles?
- 2,516
- 2,904
- 2,816
- 2,916
Show the answer
2,916
Right. 54 is 4 above 50, so 25 + 4 = 29 in front and 4² = 16 behind: 2,916.
| problem | what you do | answer |
|---|---|---|
| 35² | 3 × 4 = 12, tail 25 | 1225 |
| 47² | 25 − 3 = 22, tail 3² = 09 | 2209 |
| 97² | 97 − 3 = 94, tail 3² = 09 | 9409 |
| 19 × 21 | 20² − 1² | 399 |
Check yourself
Pick the right family for each, then match it to its answer
Show the answer
- 45² → 2025
- 99² → 9801
- 18 × 22 → 396
- 25² → 625
Make it a habit
- Ends in 5? Front times the next one up, then 25. No exceptions, at any size.
- Two numbers the same distance either side of a round number? Square the middle, subtract the gap squared.
- Near 50 or near 100? Front part first, gap squared as a two-digit tail.
- All three come from the same algebra you met in lesson 4: multiply out the brackets and watch what cancels.
Check yourself
The hard one. A rectangle is 98 by 102. What's its area?
- 9,996
- 9,998
- 10,004
Show the answer
9,996
Exactly. Both sit 2 from 100, so it's 100² − 2² = 10,000 − 4 = 9,996. Very slightly less than the 100 × 100 square.
Lesson recap
- A number ending in 5: front part times the next number up, then a tail of 25. 35² = 1225.
- Two numbers either side of a round one: square the middle and subtract the gap squared. 19 × 21 = 399.
- Near 50: 25 plus or minus the gap in front, gap squared behind. 47² = 2209.
- Near 100: take the gap off the number for the front, gap squared behind, padded to two digits. 97² = 9409.