Gaps & place value
The distance to 100 or 1,000, and moving the decimal point by tens.
Best in your head · no pen, no calculator why?
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the idea
A gap is the distance from one number up to another: 100 − 47 asks how far 47 is from 100. Count up in two jumps, first to the next round number, then on to 100. Each jump lands on a round number, so nothing needs carrying.
more of the idea
Place value is what a digit is worth because of where it sits: ones, tens, hundreds, and after the decimal point, tenths and hundredths. Multiplying by 10 moves every digit one place left, so the number is ten times bigger. Dividing by 10 moves every digit one place right. Name the direction first.
techniques
Bridge to the next ten
- Jump from the number up to the next ten.
- Jump from that ten up to 100.
- Add the two jumps. If the number already ends in 0, count up in tens.
100 − 68
- 68 up to 70 is 2.
- 70 up to 100 is 30.
- 2 + 30 = 32.
Answer: 32
Bridge to the next hundred
- Jump from the number up to the next hundred.
- Jump from that hundred up to 1,000.
- Add the two jumps.
1,000 − 735
- 735 up to 800 is 65.
- 800 up to 1,000 is 200.
- 65 + 200 = 265.
Answer: 265
Move the digits
- Name the direction: multiplying makes the number bigger, dividing makes it smaller.
- Move every digit one place for 10, and two places for 100.
- Fill empty places with zeros: 45 × 100 = 4,500, and 4.5 ÷ 100 = 0.045.
38 ÷ 100
- Dividing makes it smaller, and 100 means two places right.
- 38 becomes 0.38.
Answer: 0.38
watch out for
- Making each digit up to 10. In 100 − 47 the tens digit only goes up to 9, because the ones carry a ten. The gap is 53, not 63.
- Making the hundreds digit up to 10 when the number has tens or ones. They already carry a hundred, so 1,000 − 640 is 360, not 460.
- Moving the wrong number of places. ÷ 100 moves two places right, so 56 ÷ 100 is 0.56, not 5.6.
practice
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Gap to 100
worked example
100 − 99
- 99 → 100 is 1.
Answer: 1
Gap to 1,000
worked example
1,000 − 655
- 655 → 700 is 45, and 700 → 1,000 is 300, so the gap is 345.
Answer: 345
× and ÷ by 10 and 100
worked example
547 ÷ 100
- Dividing by 100 makes it 100 times smaller: every digit moves two places right, so 547 becomes 5.47.
Answer: 5.47