Math 131

Spring 04

Fractions, Decimals, and Percents:

Connecting Representations

 

            This handout discusses moving between representations of rational numbers as decimals, fractions, and percents.  There are many ways to “convert” between equivalent representations.  Several different methods are discussed below; the discussion is intended to reinforce your understanding of these representations.  There is no need to memorize techniques without understanding!  Keep working on understanding, and the techniques will follow easily.

 

Decimals to Fractions:

 

            Recall that we defined decimals in terms of place values.  For example, the number 234.567 is equal to .  We also represented these place values in a chart:

 

            10,000’s     1000’s     100’s    10’s    1’s        ’s      ’s     ’s     ’s

 

                                                  2         3         4          5            6               7

 

 

 

 

We also used base-ten blocks to represent decimals.  If the large cube represents 1, then the flat represents , the rod represents , and the little cube represents .  Thus, the decimal .567 can be represented by 5 flats, 6 rods, and 7 little cubes.  We could also represent .567 with 567 little cubes, and thus this fraction is equal to , since each little cube represents .  Thus, the decimal 234.567 is equal to , as a mixed number. 

            Also note that we can find our final representation directly from the expanded notation above, instead of referring to the manipulatives: 

 

  

 

            Changing from decimal representations to fractional representations is computationally simple: find the appropriate power of 10, use it as a denominator, and use the portion after the decimal point as a numerator.  Here are some more examples; be sure you understand how they relate to place value:

 

Write 0.3 as a fraction:  The first place after the decimal place is the tenths place, and  .

 

Write .6 as a fraction: Similar to the above example, .

 

Write 7.9 as a fraction/mixed number:  .

 

Write 0.45 as a fraction:  The second place after the decimal place is the hundredths place.  We have .

 

Write 97.57 as a mixed number:  .

 

Write 0.0125 as a fraction: .

 

Write .22337 as a fraction:  .

 

            Note that all the examples of decimals in this section refer to terminating decimals.  It is also possible to find equivalent fractional representations of repeating decimals; see the exercises at the end of these notes.

 

Decimals to Percents:

 

            Recall that percent = per cent = per 100, and a percent is a fraction with denominator 100.  We can form an equivalent representation of any decimal as a fraction with denominator 1 (and numerator not an integer), i.e., , and , and .  To represent any of these fractions as equivalent fractions with denominator 100, we simply multiply by , which is equivalent to multiplying by 1, and hence does not change the value of the number.  Here are some examples:

 

Write .6 as a percent:   .  Note that we also can start with the fractional representation: .  Be careful!  The answer is not 6%, which is a mistake many students make when they don’t think about what they are doing.

 

Write 0.06 as a percentage: .

 

Write .45 as a percentage: .

 

Write .0125 as a percentage:  .

 

Write .004 as a percentage:  .

 

Write 1.7 as a percentage:  .

 

Write 22.03 as a percentage: .

 

            Note that in all cases, changing a decimal to a percent involves multiplying by 100, which is often colloquially referred to as “moving the decimal place two spaces to the right.”  It is, in fact, more accurate to think of keeping the decimal place in the same spot, and moving all the digits two spaces to the left (i.e. to a place value 100 times higher).

 

Percents to Fractions:

 

            As mentioned earlier, a percent is just a fraction with denominator 100.  Usually, changing percents to fractions involves first forming a fraction with denominator 100, and then possibly finding a simpler representation of the fraction.

 

Write 20% as a fraction:  Here we might just note that since , then  (five 20%’s make a whole), but we can also compute .

 

Write 8% as a fraction: .

 

Write 18.4% as a fraction: .

 

Write 340% as a fraction: .

 

Write .025% as a fraction: .

 

Percents to Decimals:

 

            Changing from percents to decimals is very similar to changing from decimals to percents  in this case we divide by 100 instead of multiplying by 100; hence, many people think of changing from percents to decimals as “moving the decimal point two places to the left,” which once again, can more accurately be described as keeping the decimal point stable, and moving the digits two places to the right (i.e. to a place value that’s 100 times smaller).

 

Write 43% as a decimal: .

 

Write 43.567% as a decimal: .

 

Write 2.3% as a decimal: .

 

Write .055% as a decimal: .

 

Write 150% as a decimal: .

 

Fractions to Decimals:

 

            If the denominator of a fraction is a factor of a power of 10, then we can find a decimal representation for the fraction by first finding an equivalent fraction whose denominator is a power of ten.  Here are some examples:

 

Write  as a decimal:  .

 

Write  as a decimal:  .

 

Write  as a decimal:  .

Write  as a decimal:  .

 

 

Write  as a decimal:  .

 

            If it is not easy to form an equivalent fraction with denominator a power of ten, we can simply use the model of a fraction as a division problem to find an equivalent decimal representation.  Often, it will be much easier to use a calculator to do the arithmetic.  Here it is important to realize that our representation might be a repeating decimal and that in many contexts we will need to round our answer (to how many places depends on the context):

 

 

Write  as a decimal:  .

 

Write  as a decimal:  .

 

Write  as a decimal:  , so  (note that the wavy equal sign means “is approximately equal to”).