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UNDERSTANDING FRACTIONS AND THEIR DECIMAL EQUIVALENCIES Anyone working in the building trades, performing mathematical calculations must have an understanding of fractions and their decimal equivalencies, for two main reasons. First, because the architectural convention for dimensioning is in feet and inches. Inches represent a fractional portion of a foot. Secondly, most people working in the industry use calculators, which with a few exceptions only allow decimals to be entered and manipulated. Most calculators require feet and inches be converted to their decimal equivalency before any calculations can be carried out. You could simply avoid such calculators and use an HP 48G or a Construction Master which will allow the entry and manipulation of fractions and feet and inches, but you, deciding to attend a construction math class, are looking for a deeper understanding of the subject. Learning how to convert fractions to decimal and decimals to fractions lies in the heart of the craft you have chosen to pursue so lets take a closer look at fractions and decimals. The Anatomy of a Fraction 3 numerator 16 denominator The denominator indicates how many equal parts the whole (in the fraction above the whole is 1 inch) is divided into. For example 1 inch (the whole unit) can be divided into equal parts several ways. If it is divided into 2 equal parts they are called halves. If it is divided into 4 equal parts they are called quarters. The number expressed in the denominator tells how many parts are in a whole. In the example above the denominator is 16. This means the inch is divided into 16 equal parts. The numerator indicates how many of the units expressed in the denominator are present in the measurement taken. In the example above the numerator is 3. This means the measurement represents 3 of the 16 equal divisions that make up an inch. Complete the following: 1. 1 inch is divided into ____ sixteenths? Write that expression as a fraction __/__ 2. 1 foot is divided into ____ inches? Write that expression as a fraction __/__ 3. One half inch represents ________ 16 ths? Express that as a fraction __/__. Now reduce the fraction __/__ Converting Fractions to Decimals When adding, subtracting, multiplying and dividing on most calculators, fractions must be converted to decimals. To do this, simply divide the fractions' numerator by its denominator. For example 1/2 expressed as a decimal is .5. Would 4/8 and 8/16 also equal .5? Yes, because they can be reduce to 1/2 and the division of the numerator by the denominator equals .5. 10 Every fraction has a decimal equivalent, which can be obtained by dividing the numerator (the number above the line), by the denominator (the number below the line. i.e. 1/2 = 12 = .5 Convert the following fractions to decimals: 6. 3/8 = _______ 7. 3/16= _______ 8. 5/8 = _______ 9. 5/16= _______ 10. 15/16 = _______ Easy! Express a Decimal as a Fraction When a carpenter uses a calculator to obtain a measurement the answer is given in decimal form. Since the tape measure is in fractional form (sixteenths) the decimal must be converted so that it can be found on the tape measure. Before carrying out this operation decide what the denominator is going to be. Nine times out of ten a carpenter or cabinetmaker will want the denominator to be sixteen. Typically sixteenths are the smallest unit used by cabinetmakers and carpenters. Also as stated before, sixteenths are the basic unit used on most tape measures Once a fractional unit has been chosen (16ths, 8ths etc.), multiply the decimal by the fractional unit. Example 2-A: You want your fraction expressed in 16ths. So how would you convert .5 into a fraction? By deciding on 16ths as your fractional unit you already have part of your fraction ?/16. You know the denominator, because you chose it. What you don't know is the numerator. So, multiply the decimal (don't forget to include the decimal point when you multiply) by the denominator you chose. .5 16 = 8 Now you have your numerator. Put it all together and your fraction is 8/16. This fraction can of course be reduced to 1/2. Now try a few on your own: Express the following decimals as fractions. Use 16ths to express your answers. Reduce you answers if possible. 11. .0625 = _______" 12. .125 = _______" 13. .1875 = _______" 14. .25 = _______" 15. .3125 = _______" In the above problems everything worked out. When the decimals were multiplied by 16 the answers came out as a whole number. Things get a little sketchy when the answer comes out as a whole number followed by a decimal. Example: .3487 16 = 5.5792 Can the fraction be written as 5.5792/16? Of course not, but what does this number (5.5792) tell us? It tells us that the fraction is 5/16" plus a little over half of 1/16. What is half of 1/16? ______" If you answered 1/32 that is correct. If you weren’t sure how to find one half of one sixteenth, here’s how. Simply double the denominator and leave the numerator as is. In the example above .5792 may be eliminated by rounding the fraction up or down. In it’s final form the fraction could be written as 5/16 in which case .5792 is simply drop or round up to 6/16 or 3/8. Rounding up or down depends on the situation. Remember, as carpenters we work in sixteenths so there is no reason to calculate and answer more precise than sixteenths. 10 11 Try these: (round to the nearest 16th.) 16. .9173 ________ 17. .2843 ________ 18. .4958 ________ 19. .5824 ________ 20. .7478 ________ Using the Calculator to Add, Subtract, Multiply and Divide Fractions By now, converting fractions to decimals and visa versa should be no problem, but what about adding, subtracting, multiplying and dividing fractions? Armed with your trusty pocket calculator and the knowledge of converting fractions to decimals you have no worries. Believe it or not most carpenters and cabinetmakers memorize the decimal equivalents of the fractions they work with! If you haven't memorized them yet, don't worry, it just means one added step. In the example below you should have no problem adding the whole numbers. 1 + 3 + 5 = 9. In order to add the fractions, they must be converted into decimals. No mater what the fraction is simply divide the numerator by the denominator. Example: 1 5/8 + 3 3/16 + 5 7/16 = _____________ 1 5/8 = 1.625 3 3/16 = 3.1875 5 7/16 = 5.4375 Now add them up: 1.625 3.1875 + 5.437 10.25" Can you find 10.25 on your tape? Probably not, so to be able to find the fraction on your tape, convert the decimal into a fraction. How do you convert the decimal (.25) into a fraction? If you want your answer in 16ths, multiply the decimal (.25) by 16. What if you wanted your answer in 32nds? By what number would you multiply your decimal (.25)? ______ Example 2-C: .25 16 = 4 or 4/16 reduced to 1/4 .25 32 = 8 or 8/32 reduced to 1/4 Final answer = 10 1/4 Try these addition problems. Express your final answer in 16ths. 21. 15 3/16 + 32 9/16 + 54 13/16+ 23 3/8 = _____________ 22. 313/32 + 3 11/16 + 4 1/8 = _______________ 11 12 Decimal Feet vs. Feet & Inches Once again, architectural drawing are dimensioned in feet and inches so it is quit likely that you will need to convert feet and inches into decimal so you may enter them into your calculator and manipulate them. 1 foot = ______. So how hard can it be? Take a look at this problem: Express 12.7638 in feet and inches. First, look at what you already know. The question asks for feet and inches. You know part of the answer will be 12 feet. Its .7638' that must be expressed as inches. What part of a foot is .7638? How many inches are there in one foot? _______ Lets apply the same logic we used in our fraction problems to change .7638 into inches. For example .5 feet is 6 inches (.5 12 = 6). Couldn’t 6 inches be written as 6/12? By dividing the numerator by the denominator (6 divided by 12) you get .5. Apply the same principle to this problem. .7638 feet x 12 (why 12? Because there are 12 inches in one foot just like there are 16, 16ths in one inch.) = 9.1656 Answer: 12- 9.1656 Are you finished? You’re pretty close! All you have to do now is to convert the decimal inches into a fraction. You are already a pro at this! Remember that the only part of your answer you can’t read on your tape is the decimal. To get it into 16ths multiply by 16. .1656 16 = 2.6496 Are you finished now? Not yet! The answer as it stands is 12 9 2.6496/16 . Try finding that on your tape. Round the remaining decimal up or down. Since .6469 is greater than 1/2 of one sixteenth, round up to 3/16 (2.6496 rounds up to 3) Final answer: 12- 9 3/16 Try these. Express your answer in feet and inches with fractions of an inch in 16ths. 23. 13.5268 _______________ 24. 28.9736 _______________ 25. 2.3857 _______________ 26. .3974 _______________ 27. 150.34 _______________ 28. 12.692 _______________ 29. 4.1974 _______________ What if the measurement is in inches but you need to express it in feet and inches? Example: 246 11/16 Easy! Leave 11/16" as is and divide 246 by 12 (12 inches in one foot) = 20.5 Now .5 = How many inches? .5 12 = 6 Put it all together Final answer: 20' 6 11/16 12 13 Express the following problems as feet and inches 30. 123 5/16" = __________ 31. 16 9/16" = __________ 32. 143 1/2" = __________ 33. 3 3/16 = __________ 34. 7/16 = __________ 35. 15/16 = __________ Often times when converting decimals to fractions, students are confused whether to multiply the decimal by 12 (to convert to inches) or 16 (to convert to a fraction of an inch). Here’s a handy little rule. Look at the sign, after the decimal. Is it feet or inches? If it’s feet multiply by 12. If it’s inches multiply by 16. Example: Express 24.5634 as feet and inches 24 is a whole number so we can keep it, but .5634 must be converted to inches. It has a foot sign after it so, multiply by 12. 12 .5634 = 6.7608 6 is a whole number so we can use it. .7608 has an inch sign after it so multiply by 16 16 .7608 = 12.288 rounds to 12 Answer 24 6 12/16 reduced 24 6 3/4 Another point of confusion with students is whether to divide or multiply Here’s another handy rule: Fraction to Decimal - Divide Decimal to Fraction - Multiply If you get confused and are at a total loss, do a simple problem whose answer you already know. For example you should know that .5 feet = 6 inches. How do you make this happen? .5 divided by 12 = .04166 That can’t be right .5 multiplied by 12 = 6 Now substitute the decimal feet you really want to convert and multiply by 12. Use your calculator to solve the following problems: 36. You are building a landing off of an existing deck. The height of the deck is 72 5/8 off the ground. The deck and landing are constructed of 2 8 joists with 2 4 decking. The finished surface of the landing is to be 7 5/16 lower than the deck. The 2 8 landing will rest on 4 13 14 4 posts which are to be supported at their base by concrete piers projecting 4 above grade. First make a sketch and then answer the following questions. What is the elevation of the finished surface of the landing? _____________ To what length will the 4 4 posts be cut? ___________ What is the combined height of the pier, post and joist? ________________ How far below the finished surface of the deck will the top surface of the 2 8 joists rest? ____________ 37. A 2 4 partition with 1/2 drywall on each side will be added to divide a room in half. Assuming the room is square and measures 15-97/16 wide (finished surface to finished surface) answer the following questions. Assume the actual size of the 2 4 to be 11/2 31/2 What is the centerline dimension, measured from the interior wall surfaces of the room? Give answer in inches ______________ Convert to feet and inches ____________ After the wall is constructed and the room is divided in half, what will the width and length of each room measure? Give answer in feet and inches _____________ 38 You are asked to frame an interior wall 99 5/8 in height. Tou will use a single bottom plate and a double top plate. What length will you cut the studs? (Assume the plates are 1 1/2 in thickness) _______________ 39 Look at the foundation illustration on the following page and calculate the following. Dimension A _____________ Dimension B _____________ Footing Height ____________ Foundation Wall Height (not including the footing) __________ 40. You are asked to construct a deck with exterior staircase. The stair treads must be no more than 12 1/2 nor less than 11 in depth. You must use two by dimensional lumber. You can use more than one piece or more than one size to make the tread. Which size or combination of sizes would work? Answer ________________________________________________ Which would you choose? _______________________ TRICKS OF THE TRADE Converting Feet and Inches to Inches Here is an easy way to convert feet and inches into inches. Once you do this a few times you can amaze your friends and relatives. 14 15 Example: Convert 17’ 4 3/16” into inches You know that 17 feet can be converted to inches by multiplying 17 by 12, right? Another way to multiply 17 by 12 is to multiply 17 by 10, then 17 by 2 and add their answers together. 17 10 = 170 17 2 = +34 204 + 4 3/16 208 3/16” You would probably need pencil and paper to do the above computation, so here is the trick. Simply move the decimal point one place to the right including the whole inches. Moving the decimal point is the same as multiplying 17 by 10 and adding the inches that were already part of the problem. 17.4 becomes 174 Next, multiply 17 by 2 and add to 174 as below. 174 34 208 Now simply attach the fraction 208 3/16 The only hitch is when the inches are more than one digit such as 10 or 11. If it is 10 back it down to 9, add everything up and add one. If it is 11 back it down to 9, add everything up and add two. Examples: 21’ 10 5/16” 9’ 11 15/16 219 99 42 18 261 1 = 262 5/16 117 + 2 = 119 15/16 Try these: 12’ 7 3/16 ____________ 15’ 8 5/8____________ 17’ 11 7/8____________ 11’ 10 1/16___________ 32’ 4 9/16___________ 7’ 11 7/32 ____________ MORE TRICKS OF THE TRADE Halving fractions You can very easily find half of a fraction by simply doubling the denominator. Hence, half of 7/8” is 7/16”. Think about what we did. We doubled the number of equal units that one inch is broken into (from eighths to sixteenths) but left the number of units which were measured (seven) the same. 15 16 If you want to find half of a whole number and a fraction such as 6 5/16”, the answer would be 3 5/32” right? What if you needed half of 7 3/16”? Messy? You bet. Here is an easy way to solve this problem and further impress your friends and relatives. Using the example above, back 7 down to the next even number, which would be 6 and divide by two (62 = 3). Add the denominator and the numerator of the fraction together (3 + 16 = 19). This number becomes your new numerator. You denominator will be doubled as before (16 2 = 32). Answer: 3 23/32” Here is how it works. When seven was backed down to six, one inch was lost, then you divided 6 in half. When you added the numerator and the denominator you added the inch back in because the denominator was 16 (the number of units in one inch). Then you cut the fraction in half by doubling the denominator. Examples: Half of 17 7/8” is 8 15/16” Half of 21 15/16” is 10 31/32” Remember if the whole number is even, simply half the whole number and double the denominator of the fraction. Try these: 31 5/8___________ 161 7/8 __________ 55 13/16 ____________ 42 7/16___________ 203 5/16 __________ 62 11/16 ____________ 16

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