doing math

courses › Mental Arithmetic

Gaps & place value

level 2 course

The distance to 100 or 1,000, and moving the decimal point by tens.

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the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

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.

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

Finding the gap from a two-digit number up to 100.

  1. Jump from the number up to the next ten.
  2. Jump from that ten up to 100.
  3. Add the two jumps. If the number already ends in 0, count up in tens.
worked example

Example: 100 − 68

  1. 68 up to 70 is 2.
  2. 70 up to 100 is 30.
  3. 2 + 30 = 32.

Answer: 32

Bridge to the next hundred

Finding the gap from a three-digit number up to 1,000.

  1. Jump from the number up to the next hundred.
  2. Jump from that hundred up to 1,000.
  3. Add the two jumps.
worked example

Example: 1,000 − 735

  1. 735 up to 800 is 65.
  2. 800 up to 1,000 is 200.
  3. 65 + 200 = 265.

Answer: 265

Move the digits

Multiplying or dividing by 10 or 100.

  1. Name the direction: multiplying makes the number bigger, dividing makes it smaller.
  2. Move every digit one place for 10, and two places for 100.
  3. Fill empty places with zeros: 45 × 100 = 4,500, and 4.5 ÷ 100 = 0.045.
worked example

Example: 38 ÷ 100

  1. Dividing makes it smaller, and 100 means two places right.
  2. 38 becomes 0.38.

Answer: 0.38

Tips by skill

  • TipGap to 100: Jump up to the next ten, then on to 100. Add the two jumps.
  • TipGap to 1,000: Jump up to the next hundred, then on to 1,000. Add the two jumps.
  • Tip× and ÷ by 10 and 100: Decide first: bigger or smaller? Then move every digit one place for 10, two places for 100.

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.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

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  3. 7 days
  4. 14 days
  5. 30 days
  6. 60 days
  7. mastered · every 90 days

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