Learning Math: Number and Operations
Fractions, Percents, and Ratios Part B: Decimals and Percents (45 minutes)
In This Part: Percent as Proportion
The word “percent” means “out of 100.” See below. For example, 49% means 49 out of 100. Percents can also be expressed as fractions or decimals, since they too can be used to imply some part of a whole. So 49% can also be written as 49/100 or 0.49.
A percent implies a ratio: It is some part “per 100.” Ratios enable us to set up a relationship between two numbers. For example, in a water molecule, there is always one oxygen atom for every two hydrogen atoms, which means that the ratio of oxygen to hydrogen is 1:2. In a percent, the second number in the ratio is always 100. Such ratios always express a number of parts per 100 parts.
You can approach any kind of percent problem if you think of it as a proportion that equates two ratios: a data ratio and a percent ratio. In other words:
Since the percent whole is always 100, we can substitute 100 for “percent whole” in this formula:
Notice that there are three different unknowns in this equation. If you know any two of them, you can easily find the third.
For example, if you want to know how much 30% of $150 is, you’d write the proportion as follows:
From here, you can easily calculate the value you’re looking for, which in this case is $45.
a. You bought a new television set at a 20% discount and saved $80. What was the original price of the set?
b. How much did you pay for the set?
Jane bought a dress on a 25%-off sale for a total of $39. What was the original pre-sale price of the dress?
“Percent Part” for this problem should not be 25%. Why?
The bookstore reduced all items by 20% for the spring sale. After the sale, it increased the prices to 20% above the sale price. Were these prices the same as the original prices? Explain.
Try starting with an original price of $100. Note that when working with these types of percent problems, using 100 as a starting point can greatly simplify your calculations.
In This Part: Percents as Fractions and Decimals
As we’ve mentioned, percents can also be expressed as fractions and decimals. In this case, all three representations are used to indicate some part of a whole.
• What percent and decimal are represented by the fraction 1/8?
Using cross-multiplication (that is, multiplying both sides of the equation first by 100 and then by 8), we get 1 • 100 = 8x, so x = (1 • 100)8. One hundred divided by 8 is 12.5, so x = 12.5%, or 0.125 (i.e., 12.5/100).
• What fraction and decimal are represented by 35%?
You can use the same process as above, but in this case it is easier to remember the definition of percent. Thirty-five percent means 35 out of 100, which is the fraction 35/100 (which reduces to 7/20) and the decimal 0.35 (35 hundredths).
• What percent and fraction are represented by the decimal 1.8?
This decimal is 1 8/10, or 18/10:
Since the denominator of this fraction is 10, it’s easiest just to multiply both the top and bottom by 10, which gives us 180/100, or 180%.
What percent and decimal are represented by the fraction 1/200?
What fraction and decimal are represented by 0.2%?
What fraction and decimal are represented by 170%?
What fraction and percent are represented by the decimal 0.004?
Knowing some fraction, decimal, and percent equivalents allows you to estimate the answers for percent problems or conversions. Some critical values are shown in the following table:
The word percent comes from the Latin “per centum,” meaning “per 100.”
a. Set up the equation, knowing that the Data Part is $80 and the Percent Part is 20:
Here, Data Whole is the original price of the set, not the discounted price. The fractions can be made equal by multiplying the top and bottom of the right side of the equation by 4, which makes the original price $400. (You could also multiply 80 by 100 and then divide by 20.)
b. Since you saved $80 off the original price, the sale price was $320.
Again, we know the Data Part, but this time it represents the percentage after the discount, not the value of the discount (as it was in Problem B1). This means that the price we are given is 75% of the original price, not 25%.
You have several options at this point. You can multiply 39 by 100 and divide by 75. The original pre-sale price was $52.
No, the prices are not the same, because 20% of a sale price is less than 20% of the original price. For example, suppose that a set of books costs $100 before the sale. Reducing the items by 20% is a savings of $20, so the new price is $80. After the sale, the price is raised by 20%; 20% of $80 is $16, so the new price is $96.
Another way to think about this is that a 20% savings is equal to multiplying by 0.8, and a 20% price increase is equal to multiplying by 1.2. Doing both is equal to multiplying by (0.8 • 1.2) = 0.96, a 4% savings, or $96 for every $100 of the original.
This gives us 1 • 100 = 200 • x, so x = 1 • 100200, which is 0.5%, or 0.005.
This means 0.2 out of 100, or 2 out of 1,000, which is the fraction 2/1,000 (which reduces to 1/500) and the decimal 0.002.
This means 170 out of 100, which is the fraction 170/100 (which reduces to 17/10, or 1 7/10) and the decimal 1.7.
The fraction is 4/1,000, or 1/250; 1/250 is 0.4/100, so the percent is 0.4%.
Using the benchmark table, 25% of 12,000 is equivalent to 1/4 • 12,000 or 0.25 • 12,000, which equals 3,000.
Since 20% of the bridge has been built, 80% more remains to be completed. Using the benchmark fractions, this is equivalent to 4/5 • 80 = 320/5 = 64. Sixty-four meters must still be completed.
a. The shaded area is 68 out of 100; this represents 68%, 68/100 (which reduces to 17/25), and 0.68.
b. Thirty-nine percent is represented below:
To use the elastic model, use a meter stick. Expand your marked elastic so that 100% lines up with 80 centimeters. You should find that 40% of 80 is 32.
Then expand the elastic so that 100% lines up with 96 centimeters, and look for the percentage that lines up with 32 centimeters. You should find that 32 centimeters is exactly one-third along the elastic, or 33.33…%.
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.