Learning Math: Number and Operations
Fractions and Decimals Part B: Decimals to Fractions (30 minutes)
In Part A of this session, you learned that the decimal representation for every rational number was either a terminating or a repeating decimal. You also learned how to find the decimal representation for any rational number. Is the converse of that statement true? That is, is every terminating or repeating decimal a rational number? The answer is yes. And any non-terminating, non-repeating decimal cannot be a rational number. So, for instance, is an irrational number, as is .
So how do we find the fractional representation of a decimal? See Note 2 below. If the decimal is terminating, it’s already a fraction; you just can’t see the denominator. For example, 0.25 means 25/100, which reduces to 1/4. However, if the fraction is repeating, the process isn’t quite so simple. To find the fractional representation for 0.232323…, for example, here’s what you need to do.
First, choose a letter to represent the fraction you are looking for; let’s say, F. This fraction, F, represents your repeating decimal; that is, F = 0.232323…. Now we need to think of a way to get rid of those repeating parts. To do this, multiply F by 10n, where n equals the size of the period. In this case, the period is two, so multiply F by 102, or 100. Finally, subtract F. The problem looks like this:
Since 99F = 23, F = 23/99.
This worked out nicely, didn’t it? But it does raise some questions:
• Why can we do this? We can do this because we subtracted equal quantities from both sides of an equation.
• How did we know to multiply by 100? The period of this decimal is two, so if we multiply by 102, the repeating part will “move over” two places and the repeating parts then “line up” under each other. In other words, if the period is p, we can multiply by 10p.
• What if the decimal doesn’t repeat right away? Then we need to modify the process. Let’s look at another decimal number, 0.45545454…. We know that F represents the repeating decimal number; that is, F = 0.45545454…. Once again, we need to think of a way to get rid of those repeating parts. To do this, we again find 100 times F (because the repeating part has a period of two) and then subtract F:
So, since 99F = 45.09, F = 45.09/99.
Notice that, unlike in the previous example, the first couple of digits didn’t “line up,” which resulted in having a terminating decimal number in the numerator. To simplify this fraction that contains a decimal point, multiply both top and bottom by 100, which gives us F = 4,509/9,900 = 501/1,100.
a. Find the fraction equivalent for 0.125.
b. Find the fraction equivalent for 0.125125125….
a. Find the fraction equivalent for 0.5436.
b. Find the fraction equivalent for 0.543654365436….
a. Find the fraction equivalent for 0.236.
b. Find the fraction equivalent for 0.2363636….
Look carefully at which digits repeat. This is not the same type of decimal as the ones used in Problems B1 and B2.
In this video segment, Donna and Tom convert into a fraction the type of decimal number where not all the digits repeat. Watch this segment after you’ve completed Problem B3.
Think about why, when using this method for converting decimals, the fraction’s denominator is always in the form of 9 multiplied by some power of 10.
You can find this segment on the session video approximately 18 minutes and 1 second after the Annenberg Media logo.
Find the fraction equivalent for 0.111111111….
In this session, you’ve learned that in our base ten system a decimal will terminate if all the factors of its denominator are factors of some power of 10. In other words, a decimal will terminate if the prime factorization of the denominator can be reduced to powers of the prime numbers 2 and 5.
Watch this video segment to find out how the people of ancient Babylonia used a base sixty number system, which allowed them to have more terminating decimals (simply because, in a base sixty system, 60 comprises more prime factors).
You can find this segment on the session video approximately 22 minutes and 22 seconds after the Annenberg Media logo.
Why would we want to change decimals to fractions or fractions to decimals? Sometimes computations are easier with decimals, and sometimes they’re easier with fractions. For example, it might be easier to multiply by 3/4 than 0.75. On the other hand, it may be easier to divide by 2 than to multiply by 0.5.
a. Since this is a terminating decimal, 0.125 = 125/1,000, which can be reduced to 1/8.
b. Use the method of multiplication. If F = 0.125125125…, then 1,000F = 125.125125125… Subtracting F from both sides gives you 999F = 125, so F = 125/999.
a. Since this is a terminating decimal, 0.5436 = 5,436/10,000 = 1,359/2,500.
b. F = 0.54365436…. Multiply by 10,000 to get 10,000F = 5,436.54365436…. Subtracting F from both sides gives you 9,999F = 5,436, so F = 5,436/9,999, which can be reduced to 604/1,111.
a. 0.236 = 236/1,000 = 59/250.
b. Note the difference; only the 36 is repeated! In this case, F = 0.2363636…. Multiply by 100 to get 100F = 23.6363636…. Subtracting F from both sides gives you 99F = 23.4, so F = 23.4/99, or 234/990, which can be reduced to 13/55.
Multiplying by 10 yields 10F = 1.111111…. Subtracting F from both sides gives you 9F = 1. Thus, F = 1/9. (It may seem counterintuitive at first that 0.11111111… = 1/9 since there are no 9s in the decimal!)
Session 1 What Is a Number System?
Understand the nature of the real number system, the elements and operations that make up the system, and some of the rules that govern the operations. Examine a finite number system that follows some (but not all) of the same rules, and then compare this system to the real number system. Use a number line to classify the numbers we use, and examine how the numbers and operations relate to one another.
Session 2 Number Sets, Infinity, and Zero
Continue examining the number line and the relationships among sets of numbers that make up the real number system. Explore which operations and properties hold true for each of the sets. Consider the magnitude of these infinite sets and discover that infinity comes in more than one size. Examine place value and the significance of zero in a place value system.
Session 3 Place Value
Look at place value systems based on numbers other than 10. Examine the base two numbers and learn uses for base two numbers in computers. Explore exponents and relate them to logarithms. Examine the use of scientific notation to represent numbers with very large or very small magnitude. Interpret whole numbers, common fractions, and decimals in base four.
Session 4 Meanings and Models for Operations
Examine the operations of addition, subtraction, multiplication, and division and their relationships to whole numbers. Work with area models for multiplication and division. Explore the use of two-color chips to model operations with positive and negative numbers.
Session 5 Divisibility Tests and Factors
Explore number theory topics. Analyze Alpha math problems and discuss how they help with the conceptual understanding of operations. Examine various divisibility tests to see how and why they work. Begin examining factors and multiples.
Session 6 Number Theory
Examine visual methods for finding least common multiples and greatest common factors, including Venn diagram models and area models. Explore prime numbers. Learn to locate prime numbers on a number grid and to determine whether very large numbers are prime.
Session 7 Fractions and Decimals
Extend your understanding of fractions and decimals. Examine terminating and non-terminating decimals. Explore ways to predict the number of decimal places in a terminating decimal and the period of a non-terminating decimal. Examine which fractions terminate and which repeat as decimals, and why all rational numbers must fall into one of these categories. Explore methods to convert decimals to fractions and vice versa. Use benchmarks and intuitive methods to order fractions.
Session 8 Rational Numbers and Proportional Reasoning
Begin examining rational numbers. Explore a model for computations with fractions. Analyze proportional reasoning and the difference between absolute and relative thinking. Explore ways to represent proportional relationships and the resulting operations with ratios. Examine how ratios can represent either part-part or part-whole comparisons, depending on how you define the unit, and explore how this affects their behavior in computations.
Session 9 Fractions, Percents, and Ratios
Continue exploring rational numbers, working with an area model for multiplication and division with fractions, and examining operations with decimals. Explore percents and the relationships among representations using fractions, decimals, and percents. Examine benchmarks for understanding percents, especially percents less than 10 and greater than 100. Consider ways to use an elastic model, an area model, and other models to discuss percents. Explore some ratios that occur in nature.
Session 10 Classroom Case Studies, K-2
Watch this program in the 10th session for K-2 teachers. Explore how the concepts developed in this course can be applied through case studies of K-2 teachers (former course participants) who have adapted their new knowledge to their classrooms.
Session 11 Classroom Case Studies, 3-5
Watch this program in the 10th session for grade 3-5 teachers. Explore how the concepts developed in this course can be applied through case studies of grade 3-5 teachers (former course participants) who have adapted their new knowledge to their classrooms.