Adding a zero does not work
For whole numbers, multiplying by 10 looks like writing a 0 on the end: 45 × 10 = 450. Try that on 2.5 and you get 2.50, which is the same number as 2.5. A zero at the end, after the point, changes nothing, so it cannot be the rule.
The rule that always works is this: multiplying by 10 moves every digit one place to the left, into a column worth ten times as much. The decimal point stays where it is.
In 2.5, the 2 is 2 ones and the 5 is 5 tenths. Ten times 2 ones is 2 tens, and ten times 5 tenths is 5 ones. So the 2 moves to the tens column and the 5 moves to the ones column: 2.5 × 10 = 25.
× 1 moves no digit, so the value is unchanged
Turn 2.5 into 25
Choose × 10. The 2 and the 5 each move one column to the left, and the point stays where it is.
By 100
Multiplying by 100 is multiplying by 10 twice, so every digit moves two places to the left. For 0.7 × 100, the 7 moves from the tenths to the ones, then from the ones to the tens. The ones column is now empty, and a 0 fills it: 0.7 × 100 = 70.
× 1 moves no digit, so the value is unchanged
Turn 0.7 into 70
Choose × 100. The 7 moves two columns to the left, into the tens, and a 0 holds the ones place.
Dividing moves the digits right
Dividing by 10 undoes multiplying by 10, so the digits move the other way, one place to the right. In 45 ÷ 10, the 4 tens become 4 ones and the 5 ones become 5 tenths: 45 ÷ 10 = 4.5.
Dividing by 100 moves every digit two places to the right. In 3.2 ÷ 100, the 3 moves from the ones to the hundredths, and the 2 moves from the tenths to the thousandths. The ones and the tenths are now empty, so zeros fill them: 3.2 ÷ 100 = 0.032.
× 1 moves no digit, so the value is unchanged
Turn 3.2 into 0.032
Choose ÷ 100. The 3 and the 2 each move two columns to the right, and zeros fill the ones and the tenths.
By 1,000, and by 30
Count the zeros in 10, 100 or 1,000, and that is how many places each digit moves. Multiplying by 1,000 moves every digit three places left: in 2.5 × 1,000, the 2 goes to the thousands and the 5 to the hundreds, and zeros fill the tens and the ones, so 2.5 × 1,000 = 2,500. Dividing by 1,000 moves every digit three places right: 45 ÷ 1,000 = 0.045.
30 is 3 × 10, so multiplying by 30 is multiplying by 3 and then by 10. For 2.4 × 30, first work out 2.4 × 3. 2.4 is 24 tenths, and 24 tenths × 3 is 72 tenths, which is 7.2. Then multiply by 10, which moves every digit one place left: 7.2 × 10 = 72. So 2.4 × 30 = 72.
× 1 moves no digit, so the value is unchanged
Turn 2.5 into 2500
Choose × 1,000. The 2 and the 5 each move three columns to the left, and zeros fill the tens and the ones: 2,500.
Which way, and how far
Multiplying by 10, 100 or 1,000 makes a number bigger, so the digits move left. Dividing makes it smaller, so they move right. 2.5 × 10 is 25, not 0.25: 0.25 is what dividing by 10 gives.
Moving too far is the other mistake. 2.5 × 10 is not 250, because 250 is two moves, and that is 2.5 × 100.