Operations with Decimals

First, the place-value map

Reading 3.142 from the decimal point outward, each digit has its own home. Knowing these names makes "line up the points" feel obvious instead of mysterious.

        3  .  1   4   2
        │     │   │   │
     ones     │   │   thousandths
         tenths   hundredths

• Tenths are the first place after the point. So 0.1 means one tenth.
• Hundredths are the second place. Cents live here, since 0.25 is twenty-five hundredths of a dollar.
• Adding zeros on the right after the decimal never changes the value. 0.4 and 0.40 are the exact same number, which is a trick we will lean on a lot.

Add and subtract: line up the decimal points

This is the whole rule for adding and subtracting decimals: stack the numbers so the decimal points sit in one straight vertical line. Fill any short gaps with zeros so every number has the same length, then add or subtract just like whole numbers and drop the decimal point straight down into the answer.

Worked example #1: 12.5 + 3.75 + 0.4

• Line up the points: stack the three numbers so every decimal point is in the same column.
• Pad with zeros: the longest number goes to hundredths, so write 12.5 as 12.50 and 0.4 as 0.40. Now every row is the same length.
• Add right to left: add each column and carry as usual, then drop the point straight down.

Worked example #2: 8.3 − 5.46

Subtraction works the same way, and the zero trick really earns its keep here. Write 8.3 as 8.30 so it has a hundredths digit to subtract from. Then borrow as normal.

Multiply: ignore the points, then count them

Multiplying decimals has a surprise: you do not line up the points at all. Instead, pretend the decimals are not there, multiply the numbers as plain whole numbers, and only at the very end count how many decimal places belong in the answer.

The rule for the count: add up the number of decimal places in both numbers you multiplied, and the answer gets that many decimal places.

Worked example #3: 3.2 × 1.5

• Drop the points: treat this as 32 × 15.
• Multiply normally: 32 × 15 = 480.
• Count the places: 3.2 has one decimal place and 1.5 has one decimal place, so the answer needs 1 + 1 = 2 decimal places. Place the point in 480 to make 4.80.
• Tidy up: a trailing zero after the decimal can be dropped, so 4.80 becomes 4.8.

Divide: shift the point to make a whole-number divisor

Dividing by a decimal feels awkward, so the trick is to get rid of the decimal in the divisor (the number you are dividing by). Slide its decimal point to the right until it is a whole number, then slide the point in the dividend (the number being divided) the same number of places. Moving both by the same amount keeps the answer unchanged, and now it is ordinary long division.

Worked example #4: 6.45 ÷ 1.5

• Fix the divisor: 1.5 has one decimal place. Move its point one spot right to get the whole number 15.
• Match the dividend: move 6.45 one spot right too, turning it into 64.5. The problem is now 64.5 ÷ 15.
• Bring the point up: place the decimal point in the answer directly above the point in 64.5, then divide as usual.
• Divide: 15 into 64 goes 4 times (4 × 15 = 60, leftover 4). Bring down the 5 to make 45. 15 into 45 goes 3 times exactly.

  6.45 ÷ 1.5   →   move both points 1 right   →   64.5 ÷ 15

       4 . 3
     ─────────
  15 │ 6 4 . 5
       6 0
       ───
         4 5
         4 5
         ───
           0

Habits that prevent silly mistakes

• For adding and subtracting, line up the points. For multiplying, do not. Mixing up those two rules is the most common decimal slip there is.
• Pad with zeros freely. Writing 0.4 as 0.40 never changes its value and keeps your columns straight.
• Estimate first. Round each number to something easy and do the rough math in your head. If your worked answer is far from the estimate, you misplaced the point.
• When dividing, move both points the same number of places. Move the divisor's point and the dividend's point together, or the answer will be off by a factor of ten.

Your turn: practice problems

Grab a pen and try each one before you peek. Remember the rule that matches the operation.

1. 4.7 + 12.38
2. 20.1 − 6.75
3. 0.6 × 0.04
4. 9.6 ÷ 0.8