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  1. 1. Using Decimals Review of comparing, rounding, adding & subtracting, multiplying & dividing decimals created by Alane Tentoni (copyright 2007)
  2. 2. What is a decimal? <ul><li>A decimal is a dot that goes after the ones column. </li></ul><ul><li>It separates the whole numbers from the partial numbers. </li></ul>
  3. 3. About Decimals <ul><li>Decimals as we know them were first used by John Napier in the late 1500s in Scotland. </li></ul>
  4. 4. About Decimals <ul><li>In order to use decimals, you have to understand place value. </li></ul><ul><li>1 2 3 4 . 5 6 7 8 </li></ul>ones tens hundreds thousands To the left of the decimal, all the numbers are whole numbers. Each column is worth ten times the column to its right.
  5. 5. About Decimals <ul><li>To the right of the decimal, all the numbers are like fractions. Each column is still worth 10 of the column to the right. </li></ul><ul><ul><li>1 2 3 4 . 5 6 7 8 </li></ul></ul>Ten thousandths thousandths hundredths tenths
  6. 6. Reading Decimals <ul><li>Zeroes that come at the end of a decimal don’t add or take away any value. </li></ul><ul><li>.4 = .40 = .400  This is like saying “four tenths” = “four tenths and no hundredths” = “four tenths and no hundredths and no thousandths.” </li></ul>
  7. 7. Reading Decimals <ul><li>HOWEVER – Zeroes that come between the decimal and the other numbers are VERY important! </li></ul><ul><li>.4 is “four tenths” but .04 is “four hundredths.” Would you rather have four dimes or four cents? </li></ul>
  8. 8. Comparing Decimals <ul><li>To tell if one decimal is bigger than another, you have to compare the same column in both numbers. </li></ul><ul><li>The length of the number does NOT matter at all !!!! </li></ul>
  9. 9. Comparing Decimals <ul><li>Compare these two numbers: </li></ul><ul><li>Which is larger? </li></ul><ul><li>.6 or .599823 </li></ul><ul><li>All you need to do is look at the tenths column. 6 is more than 5, so .6 is more than .599823, even though .599823 has more digits! </li></ul>
  10. 10. Comparing Decimals <ul><li>Another comparison </li></ul><ul><li>Which is larger? .457 or .49? </li></ul><ul><li>The tenth columns are the same (both 4), but the hundredths columns are different. 9 is more than 5, so .49 is more than .457. </li></ul>
  11. 11. Rounding Decimals <ul><li>Rounding means cutting off unnecessary digits. </li></ul><ul><li>Why would you use fewer digits than you know? Sometimes it is more convenient to give an approximate answer. </li></ul>
  12. 12. Rounding Decimals <ul><li>First, decide how many decimal places you want in your answer. </li></ul><ul><li>Just throw away everything behind that place. . . </li></ul><ul><li>Except! You will have to decide whether to increase the last digit or leave it alone. </li></ul>
  13. 13. Rounding Decimals <ul><li>Let’s round .576 to the nearest hundredth. </li></ul><ul><li>.576 is somewhere between .57 and .58. Which one is it closer to? </li></ul><ul><li>To decide, simply look at the digit after the hundredths place. Is it 5 or more? If so, round up. If not, leave it the same. </li></ul>
  14. 14. Rounding Decimals <ul><li>In our case, 6 is more than 5, so .576 should be rounded up to .58. </li></ul><ul><li>What happens if you have a number like .398 to round to the nearest hundredth? (answer: .398 ~.40) </li></ul>
  15. 15. Rounding Decimals <ul><li>Be Careful!! Don’t just replace the “chopped off” numbers with zeroes! When you round, you are really reducing the number of digits behind the decimal! </li></ul>
  16. 16. Rounding Decimals <ul><li>Here are some numbers to round to the nearest hundredth. </li></ul><ul><li>1.3247  1.32 </li></ul><ul><li>0.987  0.99 </li></ul><ul><li>4.89721  4.90 </li></ul>Because we are rounding to the nearest hundredth, each of the numbers ends up with two digits behind the decimal. What if we had been rounding to the nearest tenth? (answer: Rounding to the nearest tenth leaves one decimal place. In the example: 1.3, 1.0, 4.9)
  17. 17. Adding & Subtracting Decimals <ul><li>When you add decimals, line the decimals up – one on top of the other. </li></ul><ul><li>You have to add the tenths to the tenths, the hundredths to the hundredths, and so on – just as when you add whole numbers, you add ones to ones and tens to tens. </li></ul>
  18. 18. Subtracting Decimals <ul><li>When you subtract, you may have to annex zeroes to the larger number so you can borrow. </li></ul><ul><li>Example: 35.7 – 20.94= ? </li></ul><ul><li>35.7 0 </li></ul><ul><li>- 20.94 </li></ul><ul><li>14.76 </li></ul>Annex a zero here so you can borrow.
  19. 19. Multiplying Decimals <ul><li>When you multiply decimals, you should set the problem up just as if you were multiplying whole numbers – longest number on top, shortest on bottom. </li></ul>
  20. 20. Multiplying Decimals <ul><li>After you multiply the numbers, you are ready to put your decimal in place. </li></ul><ul><li>Count the number of digits behind the decimal in both of the multiplied numbers. </li></ul><ul><li>Put that many total digits behind the decimal in your answer. </li></ul>
  21. 21. Multiplying Decimals <ul><li>Here’s an example: </li></ul><ul><li>1. 2  one digit here </li></ul><ul><li>x 3. 9  one digit here </li></ul><ul><li>108 </li></ul><ul><li>_ 36_ </li></ul><ul><li>4. 68  two digits here </li></ul>
  22. 22. Multiplying Decimals <ul><li>Another example – same numbers but with the decimals in different places. </li></ul><ul><li> 1. 2  one digit here </li></ul><ul><li>x . 39  two digits here </li></ul><ul><li>108 </li></ul><ul><li>_ 36_ </li></ul><ul><li>. 468  three digits here </li></ul>
  23. 23. WHOA! <ul><li>Hang on! Did that last problem say 1.2 x .39 = .468? </li></ul><ul><li>Question: How can you multiply 1.2 by something and get an answer less than 1.2? </li></ul><ul><li>Answer: Anytime you multiply by something less than 1, the answer is smaller than the number you started with. </li></ul>
  24. 24. Multiplying Decimals <ul><li>If the answer doesn’t have enough digits, you will have to put zeroes between the decimal and the first number. </li></ul><ul><li> . 12  two digits here </li></ul><ul><li>x . 39  two digits here </li></ul><ul><li>108 </li></ul><ul><li>_ 36 _ </li></ul><ul><li>. 0468  four digits here </li></ul>
  25. 25. Dividing Decimals <ul><li>Let’s name the parts of a division problem so we can talk about them. </li></ul><ul><ul><li>8 56 </li></ul></ul>7 dividend divisor quotient Notice that the 7 is over the 6, not the 5. The quotient goes over the LAST digit you are working with.
  26. 26. Dividing Decimals <ul><li>Dividing decimals is a lot like dividing whole numbers, but we need a way to get the decimals in the right place in the answer. </li></ul><ul><li>Before we start dividing decimals, let’s look at dividing some whole numbers. </li></ul>
  27. 27. Dividing Decimals <ul><li>42 ÷ 6 = 7 And 420 ÷ 60 = 7 </li></ul><ul><li>In the second equation, both 42 and 6 have been multiplied by ten. Because both numbers were multiplied by the same thing, the quotient did not change. </li></ul>
  28. 28. Dividing Decimals <ul><li>We can use that trick to divide numbers with decimals. </li></ul><ul><li>Because moving the decimal to the right is just like multiplying by ten, if we move the decimal the same number of places in both numbers, our quotient stays the same. </li></ul>
  29. 29. Dividing Decimals <ul><li>Here’s an example: .132 ÷ .12: </li></ul><ul><li>.12 .132 </li></ul>If these were whole numbers, you would say, “How many times will 12 go into 13?” But it’s harder to think of .12 and .13. If you could move the decimal of the divisor (.12) over 2 places, you would have a whole number. You can do that as long as you move the decimal of the dividend over 2 places as well.
  30. 30. Dividing Decimals <ul><li>So now our problem looks like this: </li></ul>NOTICE: The decimal moved straight up from the dividend to the quotient. Lining up the number in the quotient and the dividend is VERY important because if they are wrong, your decimal will be in the wrong place. 12. 13.2 1.1 -12 1 2 -1 2 0
  31. 31. ALWAYS Check! <ul><li>Now that we have an answer, we need to check our work. </li></ul><ul><li>Multiply the quotient by the divisor. You should get the dividend back. </li></ul>1.1 x.12 22 11 .132 1 digit 2 digits 3 digits
  32. 32. Hang on! <ul><li>How can we take two small numbers like .12 and .132 and divide them and get a bigger number? Doesn’t dividing always mean you get a smaller number? </li></ul>
  33. 33. Dividing Decimals <ul><li>Another way to look at .132 ÷ .12 is to say, “How many groups of .12 does it take to make .132?” </li></ul><ul><li>.12 + .012 = .132 </li></ul><ul><li>It takes one and a little more, so our answer of 1.1 looks reasonable. </li></ul>
  34. 34. Dividing Decimals <ul><li>Let’s try another example: </li></ul><ul><li>1.25 ÷ .4 </li></ul>First of all, let’s estimate how many .4’s it would take to make 1.25 .4 + .4 + .4 = 1.2 so it will take 3 groups of .4 plus a little more to make 1.25 .4 1.25
  35. 35. Dividing Decimals First, move the decimal in the divisor and the dividend. In this case, we have pulled down all our numbers, but we still have a remainder. DO NOT tack your remainder onto the end of your answer! 4. 12.5 3.1 -12 05 -4 1
  36. 36. Annexing Zeroes <ul><li>Remember that adding zeroes at the end of a number does not change its value. </li></ul><ul><li>12.5 = 12.50000 </li></ul><ul><li>If you need to keep dividing, just annex zeroes, pull down & keep dividing until you get a remainder of zero (or until you see a pattern.) </li></ul>
  37. 37. Annexing Zeroes- -8 20 -20 0 When you get a remainder of zero, you can stop pulling down zeroes. 4. 12.5 000 3.125 -12 05 -4 10
  38. 38. Check Your Work! The original problem was 1.25 ÷ .4. The quotient was 3.125 Check: 3.125 x .4 1.2500 3 digits 1 digit 4 digits Since 1.2500 = 1.25, our answer is correct.
  39. 39. Dividing Decimals <ul><li>Sometimes when we divide, the quotient of the two numbers makes a pattern that never stops! </li></ul><ul><li>This is called a “repeating decimal.” </li></ul><ul><li>The kind that does stop is called a “terminating decimal.” If you can work your problem to a remainder of zero, you have a terminating decimal. </li></ul>
  40. 40. Dividing Decimals <ul><li>Tip: </li></ul><ul><li>Divisors that have factors of all twos or fives will definitely terminate. (like 2, 4, 5, 8, 10. . .) </li></ul><ul><li>Everything else can repeat – it depends on the dividend. </li></ul>
  41. 41. Dividing Decimals <ul><li>Here is a repeating decimal. </li></ul>First, move the decimal. . Put the decimal on the quotient line. .3 5.56 3. 55.6
  42. 42. Repeating Decimals When you’ve pulled down all your numbers and you still have a remainder, you need to annex zeroes and keep going. 3. 55.6 18.5 -3 25 -24 16 -15 1
  43. 43. Repeating Decimals -9 10 From here on, no matter how many zeroes we pull down, we will always get 10 and the next number will always be 3. The 3 is repeating. 3. 55.6 000 18.533 -3 25 -24 16 -15 10
  44. 44. Repeating Decimals <ul><li>To show that a number repeats, place a bar over all the numbers that form the pattern. </li></ul><ul><li>In our example, only the 3 was repeating: </li></ul>18.53
  45. 45. Get the “point”? <ul><li>Decimals are a pretty convenient way to represent fractional values. </li></ul><ul><li>Decimal rules are not difficult, but even though you know the rules, you must practice them until they are second nature! </li></ul>