01.03 Decimals & Percentages
- Type of Fraction expressed as Place Value in 10s (2/10, 14/100, etc.)
- Decimals are easier to perform basic operations with than fractions
- Decimals’ relative size is easier to conceptualize
- Learn to say decimals in a way that makes their amount clear
- Type of Fraction expressed as Place Value in 10s (2/10, 14/100, etc.)
- Solving Problems with Decimals – Points to remember
- Adding and Subtracting – Always line up decimal
- Multiplying – Multiply, then count digits in decimal places to complete problem
- Dividing – Divisor cannot be a decimal, so remember to convert it and the dividend
- Definition – A percentage is always an amount “Of one hundred” (e.g. 69% is 69/100)
- Percents are relative amount, not actual numbers.
- No solving problems with them
- Always convert them to decimals or fractions
- A percent can represent more than “total” (e.g. 150%)
- Relationship between Decimals, Fractions, and Percents is important to understand
- Perform conversions between all 3
- Memorize a set of “Go To” numbers for which you memorize the conversions for quicker mental math
Cornell Note-Taking System Instructions:
- Record: During the lecture, use the note-taking column to record the lecture using telegraphic sentences.
- Questions: As soon after class as possible, formulate questions based onthe notes in the right-hand column. Writing questions helps to clarifymeanings, reveal relationships, establish continuity, and strengthenmemory. Also, the writing of questions sets up a perfect stage for exam-studying later.
- Recite: Cover the note-taking column with a sheet of paper. Then, looking at the questions or cue-words in the question and cue column only, say aloud, in your own words, the answers to the questions, facts, or ideas indicated by the cue-words.
- Reflect: Reflect on the material by asking yourself questions, for example: “What’s the significance of these facts? What principle are they based on? How can I apply them? How do they fit in with what I already know? What’s beyond them?
- Review: Spend at least ten minutes every week reviewing all your previous notes. If you do, you’ll retain a great deal for current use, as well as, for the exam.
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Welcome to the course Decimals and Percents. When you understand the relationship between decimals and percents and can easily convert between them, you will begin using automatic “go to” numbers to sharpen your problem solving skills. We’ll work on all of this together.
Decimals are numbers, and they represent partial values similar to the way fractions do. Decimals are different from fractions because they are based on a unit of 10, just like whole numbers. To work with decimals, it’s important to know the names of the place values.
People find decimals easier to use when solving problems than fractions. I’d rather add 0.8 and 0.6 (=1.4) than 4/5 = 3/5, wouldn’t you? Also, it’s easier to judge the relative size decimals than fractions. There’s a lot to do to order 1/2, 2/3, and 4/5, right? But when you have .5, .66, and .8 – just add zeros on the ends so thaey all have the same number of place value spaces, and it’s so easy to see their size from smallest to largest.
So start using the names of the place values to say decimals correctly. Say the whole number first, then articulate AND for the decimal, and then say the decimal number followed by the place value (with -th at the end). Let’s practice a few. 104 and 3 tenths. 50 AND 24 hundred. See how quickly you can convert them to fractions this way?
Using the four basic operations to solve decimal problems is not a lot different than solving problems with whole numbers. When adding or subtracting with decimals, you have to line up the decimals so the place values are in the same column. Do that with 4.15 and 23.2. Then just add.
Multiplying with decimals is different only because you have to manage the decimal point at the end of the problem. First, align the numbers to the right, multiply as you would with whole numbers, but then you have to remember to deal with the decimal point. Simply count the decimal spaces to the right of the point in the problem, and count that many digits from right to left in the product. Let’s try it with 6.01 and .2 (=1.202). Big Note: Notice that the whole number got SMALLER when multiplying by a partial number.
As well, when dividing with decimals, do the math as you would with whole numbers. But here are some specific things to remember: there cannot be a decimal in the divisor. So if there is, you have to make that number a whole number (effectively that would be multiplying it by 100. Try it! Then, if you don’t do the same change to the dividend, you are changing the whole problem. SO then change dividends the same way. Now divide normally, and bring up the decimal point. KEY: Write neatly so you can see exactly where the decimal place goes.
Let’s take a look at when 0s have value, and when they don’t. This is an important concept to understand. If a 0 is between a number and the decimal point, it holds value and is key to the amount. If the 0 is to the far left or the far right, past the digits, it can be dropped, as it does not change the value. Look at these examples: 201.03 – Keep the 0s 02.34700 – Drop those 0s – no value.
The word “percent ” actually means per, or of, 100 (cent). That means that when you hear that “45% of teens in the local high school drive to school”. That number means 45 out of every 100. Percentages are relative numbers. 45% is not an absolute amount. If another HS reports that 34% of their students drive to school, you can see that more students relative to the total drive in the first report.
Key to all of this is to clearly see the relationships and the differences between decimals, fractions, and percents. I added fractions in here because we will work with them in our conversions later. Fractions show a part of 1 whole thing – like ¼ of a pizza. The 2 digits in a fraction together make 1 number, not 2. Decimals do the same thing, but the partial number is based on place value. Percents are easier to calculate problems than fractions.
Percents always represent relative amounts, always mean a part out of 100 (the “cent” in percent), and must always be converted to decimals or fractions to work with.
Learn how to convert fractions, decimals, and percents easily and automatically is super important. The math isn’t difficult, in fact it’s all math you already know. This chart gives you the instructions for all the conversions. To change from a fraction to a decimal, just divide the numerator by the denominator. To make a decimal a percent, move the decimal 2 spaces to the right and add a %. (Really, this is multiplying the decimal by 100.) Notice here that you cannot change a fraction to a percent without changing it to a decimal first.
Now the other direction. Change a % to a decimal by removing the % sign and adding a decimal 2 places from the right. Then, put that number, without the decimal, over the correct fractional part by saying it correctly – 75/100ths, and you have your fraction.
I suggest memorizing a specific list of go-to numbers because it’s easier to see the numbers in your head. The more mental math you perform the fewer mistakes you make, and the more confident you become with problem-solving. Also, having these numbers memorized helps you save time on standardized tests.
If you commit to understanding the relationships between percents and decimals, and learning how to automatically convert between without hesitation, and you commit to memorizing some basic “go to” numbers so you don’t have to convert, you will notice greater ease with solving problems involving decimals and percents. So go out there, solve math problems like a pro, and…