UNDERSTANDING FRACTIONS AND THEIR DECIMAL EQUIVALENTS
Anyone performing mathematical calculations in the building trades must sooner or late convert fractions to decimals, and or decimals to fractions. The architectural convention for dimensioning is feet and inches, which is no problem when simply measuring with a tape measure. But, when dimensions must be added, subtracted, multiplied and divided, inches must be converted into decimal feet.
(Inches represent a fractional portion of a foot) With a few exceptions, most calculators only allow decimals to be entered and manipulated, therefore the conversion of feet and inches to decimal feet is required.
You could simply avoid such calculators and use a HP 48G or a Construction Master, which both allow the entry and manipulation of decimals, fractions and feet and inches but because you deciding to attend a construction math class, I am sure you want 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 let’s take a closer look at fractions and decimals.
The Anatomy of a Fraction
3
16
numerator
denominator example 3-1
The denominator indicates the number of equal parts the whole is divided into. For example, 1 inch (the whole unit) can be divided into equal parts several ways. If divided into 2 equal parts the denominator is 2, or if divided into 4 equal parts the denominator is 4.
In any case the number expressed in the denominator tells how many parts the whole is divided into. In example 3-1 the denominator is sixteen. This means the whole is divided into sixteen equal parts.
The numerator indicates how many of the units expressed in the denominator are present. In example
3-1 the fraction’s numerator is three, meaning there are three of sixteen equal divisions represented.
Complete the following :
1. How many sixteenths are present in one inch?_____
Express one inch as a fraction with sixteen as the denominator _______
2. How many inches are in one foot? ____ ?
Express one foot as a fraction with twelve as the denominator _______
3. How many 16 th ’s are present in one half-inch _______?
expression as a fraction __/__. Now reduce __/__
Express the following fractions as 16ths.
4.
1/8

= _______
5.
5/8

= _______
6.
3/4

= _______
7.
7/8

= _______
8.
3/8

= _______
19

Adding and Subtracting Fractions Without a Calculator
One of the easiest ways to add and subtract whole numbers and fractions is by simply using your tape measure and your thumbnail. For example, if you want to add 13 3/16

+ 4 3/8

+ 22 1/2

do the following:
Step 1: Add the whole numbers in your head, on paper or right on the tape. 13 + 4 + 22 = 39
Step 2: Extend the tape and add the fractions right on the tape. Count out the fractions using
your thumb nail as a guide.
I usually start at 10 inches so that I don’t have to deal with the first inch where the hook is in the way. Also, it will be easier if you start with the largest fraction first then the next and so on.
1/2

+ 3/8

+ 3/16

= 1 1/16

Step 3: Put it all together 39

+ 1 1/16

= 40 1/16

What about subtracting?
The steps are the same with one exception. Look at the following example:
23 5/16

– 14 7/8

Step 1: Subtract the whole numbers. 23 – 14 = 9
Step 2: Subtract the fractions.
Find 5/16 on your tape (again I usually start at 10 and then count back 7/8

). If you start at 10 and count forward 5/16


you end up at 9 7/16

. Because you are now one whole number less than your starting point (10) you must subtract 1 from your answer in step one. 9 – 1 = 8.
Step 3: Put it all together by tacking on the fraction 8 7/16

Her is another subtraction problem that is yet a little different.
46

– 22 7/16

= ?
Step 1: 46 – 22 = 24
Step 2 : Find 24 on your tape and subtract 7/16 by counting off.
24 – 7/16 = 23 9/16

Try these using your tape:
9.
12 5/8

+ 16 5/16

+ 32 9/16

= ________
10.
20 3/8

– 15 7/16

= ________
11.
18 7/8

+ 26 7/16

+ 9 13/16

= _________
12.
96 – 4 5/8

= _________
13.
12 7/16

+ 22 1/4

+ 16 1/16

= _________
14.
46 1/2

+ 22 13/16

- 37 9/16

= ____________
15.
25 5/16

- 7 15/16

= _________
20

16.
16 3/8

+ 12 7/16

+ 4 3/4

= ____________
17.
2 1/4

+ 7 5/8

- 4 1/2

= _____________
18.
22 1/2

+ 13 9/16

= ___________
19.
29 15/16

2 3/16

+ 32 7/8

-21 3/4

= _____________
20.
5/8

+ 14 1/4

3 1/2

= ____________
This method of adding and subtracting mixed numbers is a very visual way of carrying out mathematical calculations. It may seem crude, but when your calculator fails or you cannot remember your decimal equivalencies it works great and it’s a great way to get comfortable reading a tape measure.
When adding, subtracting, multiplying and dividing on most calculators, fractions must be converted to decimals . This is done by simply divide the numerator by the denominator. For example 1/2

expressed as a decimal is .5

.
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 = 1

2 = .5)
Convert the following fractions to decimals :
21. 3/8

= _______

22. 3/16

= _______

23. 5/8

= _______

24. 5/16

= _______

25. 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) decimal must be converted so the fraction may be found on the tape measure.
Before carrying out this operation the denominator must be determined. 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 3-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

.
21

Now try a few on your own: Express the following decimals as fractions. Use 16ths to express your answers. Reduce you answers if possible.
26. .0625

27. .125

= _______"
= _______"
28. .1875

29. .25

= _______"
= _______"
30. .3125

= _______"
In the above problems everything worked out. When the decimals were multiplied by 16 the answers came out as whole numbers. Things get sketchy when the answer comes out as a whole number followed by a decimal.
Example 3-B: Express .3487

in sixteenths .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 its 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.
Try these : (round to the nearest 16th.)
31. .9173

________
32. .2843

________
33. .4958

________
34. .5824

________
35. .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.
Most carpenters and cabinetmakers memorize the decimal equivalents of the fractions with which they work! If you haven't memorized them yet, don't worry, it just means one added step. In the example 3-C you should have no problem adding the whole numbers. 1 + 3 + 5 = 9. In order to add the fractions using the calculator, they must be converted into decimals. No matter what the fraction is simply divide the numerator by the denominator.
Example 3C: 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

22

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)? ______
.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.
36. 15 3/16

+ 32 9/16

+ 54 13/16

+ 23 3/8

= _____________
37. 31 3/32

+ 3 11/16 + 4 1/8

= _______________
Decimal Feet vs. Feet & Inches
Architectural drawing are dimensioned in feet and inches so it is quit likely that you will need to convert feet and inches into decimals so you may enter them into your calculator for manipulation.
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. It’s .7638' that must be expressed as inches.
What part of a foot is .7638
 ? How many inches are there in one foot? _______ Let’s 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? Dividing the numerator by the denominator (6 divided by 12) equals .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

One last detail! 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.
38. 13.5268

_______________
39. 28.9736

_______________
23

40. 2.3857

_______________
41. .3974

_______________
42. 150.34

_______________
43. 12.692

_______________
44. 4.1974

_______________
What if the measurement is in inches but you need to express it in feet and inches?
Example 3-D: 246 11/16

Easy! Leave 11/16" as is and divide 246

by 12 (12 inches in one foot) = 20.5

Now .5

equals how many inches? .5

12 = 6

Put it all together
Final answer: 20' 6 11/16

Express the following problems as feet and inches
45. 123 5/16" = __________
46. 16 9/16" = __________
47. 143 1/2" = __________
48. 13 3/16

= __________
49. 77/16

= __________
50. 29 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 3-E: 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

Here is a handy visual to help you remember:
Num.
Dec. Den.
24

The triangle in figure 3-1 has three compartments.
Figure 3-1
The fraction’s numerator is represented in the top compartment, the denominator and decimal equivalent are represented in the bottom compartments. You will know two of the three values. For example, in expressing
.6891

as a fraction of an inch you know decimal because it is given and you know the denominator because you are a carpenter and carpenters work in sixteenths. So, put what you know and what you need to know in the triangle see figure 3-2 . Place your finger over the value you want to know. If the remaining uncovered values are in the bottom compartments, then multiply. If the remaining uncovered values are in a top compartment and a bottom compartment, divide the bottom into the top as in figure 3-3.
.3125

16 = 5 Therefore 5/16

Num.
.3125 16
Figure 3-2
Again using the triangle, solve for the decimal equivalent of the fraction 7/16

. Place what you know and need to know into the triangle. See figure 3-3. Cover what you need to know with your finger. With the decimal covered you are left with the numerator and denominator. Treat them like a fraction and divide the top compartment by the bottom compartment.
7

16 = .4375

7
Dec. 16
Figure 3-3
In summary, if you are having trouble remembering whether to multiply or divide when deriving decimals from fractions and visa versa, draw the triangle on a piece of paper and use it as a tool.
Don’t hesitate to use it when taking a test.
Use your calculator to solve the following problems:
51.
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 surface. The 2

8
landing will rest on 4

4 posts, supported by concrete piers projecting 4

above grade. First
dimension the sketch below and then answer the following questions.
Note: The actual sizes:
2 x 8 is 1 1/2

x 7 1/4

2 x 4 = 1 1/2

x 3 1/2

2x4 decking
25

2x8 joist
2x8 joist
4x4 posts
pier pads
What is the measurement from the ground to 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? ________________
52.
A 2 X 4 wall 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

9 7/16

(finished surface to finish surface) and the wall studs are 1 ½ 
x 3 ½

, answer the following questions.
Measuring from finished surface to finished surface what is the distance to the centerline
of the room?
Answer ___________ express answer in feet and inches
After the wall is constructed and the room divided in half, what will the width and length of each room measure?
Answer______________ express answer in feet and inches
53.
You are asked to frame an interior wall 99 5/8

in height. You will use a single bottom plate and a double top plate. To what length will you cut the studs? (Assume the plates are 1 1/2

in thickness.)
Answer ______________ Express answer in inches
54 Use the illustration on page 27 to answer the questions below:
What is dimension A? __________ What is dimension B? __________
What is the foundation wall height not including the footing? _______
What is the footing height? _______
26

TRICKS OF THE TRADE
27

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.
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 products 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 imaginary decimal point between feet and inches one place to the right. Moving the decimal point one place to the right is the same as multiplying by 10 and notice that this simple procedure automatically adds in the inches that were already present.
17.4 becomes 174
Next, multiply 17 times 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 and complete the steps outlined above, adding one inch to your final answer.
If it is 11 back it down to 9, and complete the steps outlined above, adding two inches to your final answer.
Examples: 21’ 10 5/16” 9’ 11 15/16 
219

42
261

1 = 262 5/16

99

18
117 + 2 = 119 15/16

Try these:
12

7 3/16

____________
11

10 1/16

___________
15
32


8 5/8

4 9/16
____________ 17

___________ 7


11 7/8
11 7/32


____________
____________
MORE TRICKS OF THE TRADE
28

Dividing fractions in half
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 was done. The number of equal units in one inch was doubled but the actual number present was left the same.
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?
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
(6

2 = 3 ). Next, add the denominator and the numerator of the fraction together (3 + 16 = 19 ). This number becomes your new numerator. The denominator will be doubled as before (16

2 = 32 ). Answer: 3 19/32”
Here is how it works. When seven was backed down to six, one inch was lost. Adding the denominator to the numerator added one inch back in. 19/16

= 1 3/16

Finally, doubling the denominator cut the fraction in half.
Examples: What is half of 17 7/8

17 – 1 = 16

2 = 8, 7+ 8 = 15 , 8

2 = 16 8 15/16

What is half of 21 15/16

21 – 1 = 20, 15 + 16 = 31, 16

2 = 32 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

29