Decimals & Place Value

What Is a Decimal?

A decimal is a way to write numbers that are not whole. The decimal point (the dot) separates the whole number part on the left from the fractional part on the right. For example, in the number 3.75, the 3 is the whole number and .75 means seventy-five hundredths — or three-quarters of a whole.

Decimals and fractions represent the same idea. 0.5 and 1/2 are identical in value. Understanding the connection between them makes both topics easier.

The Place Value Chart

Every digit in a decimal number has a specific place value. The chart below shows how the places are named. Notice that the places to the left of the decimal point are whole numbers, and the places to the right end in "-ths."

In the example row above, the number is 3,456.7892. Reading from left to right: 3 thousands, 4 hundreds, 5 tens, 6 ones, 7 tenths, 8 hundredths, 9 thousandths, 2 ten-thousandths.

  Place Value Visual for  47.356

  4   7  .  3   5   6
  |   |     |   |   |
  |   |     |   |   +-- thousandths (6/1000)
  |   |     |   +----- hundredths  (5/100)
  |   |     +-------- tenths       (3/10)
  |   +------------- ones          (7)
  +----------------- tens          (40)
    

Reading Decimals Aloud

There is a simple rule: read the whole number part normally, say "and" for the decimal point, then read the digits after the point as if they were a whole number, followed by the name of the last place value.

Adding and Subtracting Decimals

The one rule that makes decimal addition and subtraction work is: line up the decimal points. When the decimal points are vertically aligned, each digit is in the correct column and you can add or subtract normally.

   14.60    <-- write a zero to fill the empty place
 +  3.27
 -------
   17.87
      

   8.50    <-- add a zero placeholder
 - 2.73
 ------
   5.77
      

Multiplying Decimals

When multiplying decimals, you do NOT need to line up the decimal points. Instead, follow these three steps:

1. Ignore the decimal points and multiply the numbers as if they were whole numbers.
2. Count the total number of decimal places in both original numbers combined.
3. Place the decimal point that many places from the right in your answer.

  Step 1: 24 x 13 = 312

  Step 2: 2.4 has 1 decimal place
          1.3 has 1 decimal place
          Total = 2 decimal places

  Step 3: Place decimal 2 places from the right in 312
          Answer = 3.12
      

  Step 1: 5 x 3 = 15

  Step 2: 0.05 has 2 decimal places
          0.3  has 1 decimal place
          Total = 3 decimal places

  Step 3: 15 becomes 0.015
          (need 3 decimal places, so add a leading zero)
      

Dividing Decimals

Dividing decimals requires a slightly different approach depending on whether the divisor (the number you are dividing by) is a decimal.

Dividing a Decimal by a Whole Number

Set up the long division as normal. Place the decimal point in the quotient (your answer) directly above where it appears in the dividend.

     2.13
   ------
  3 ) 6.39
      6
      ---
       03
        3
       ---
        09
         9
        ---
         0
      

Dividing by a Decimal Divisor

When the divisor is a decimal, move its decimal point all the way to the right to make it a whole number. Then move the decimal point in the dividend the same number of places to the right. Now divide normally.

  Move decimal in 0.6 one place right  --> 6
  Move decimal in 4.8 one place right  --> 48

  Now solve: 48 / 6 = 8

  Answer: 8
      

  Move decimal in 0.12 two places right --> 12
  Move decimal in 1.44 two places right --> 144

  Now solve: 144 / 12 = 12

  Answer: 12
      

Converting Between Fractions and Decimals

Fraction to Decimal

Divide the numerator (top number) by the denominator (bottom number).

Decimal to Fraction

Write the decimal digits over the appropriate power of 10, then simplify.

Common Decimal-Fraction-Percent Equivalents

Rounding Decimals

Rounding reduces a decimal to fewer decimal places while keeping the value as close to the original as possible. Here are the steps:

1. Identify the place you are rounding to.
2. Look at the digit immediately to the right of that place.
3. If that digit is 5 or greater, round up (add 1 to the target place). If it is 4 or less, round down (keep the target digit the same).
4. Drop all digits to the right of the target place.

  7.3852
      ^
      Hundredths place = 8
      Next digit (thousandths) = 5

  Since 5 >= 5, round up: 8 becomes 9

  Answer: 7.39
      

  12.4749
     ^
     Tenths place = 4
     Next digit (hundredths) = 7

  Since 7 >= 5, round up: 4 becomes 5

  Answer: 12.5
      

Real-World Applications

Decimals appear everywhere outside the classroom:

• Money: $4.75 means 4 dollars and 75 hundredths of a dollar (75 cents).
• Measurements: A board 2.54 centimeters thick means a little over 2 and a half centimeters.
• Sports statistics: A baseball player batting .315 gets a hit about 31.5% of the time.
• Fuel efficiency: A car that gets 32.4 miles per gallon.
• Science: The mass of an atom or the pH of a solution.