Plumber's Licensing Exam: Math For Plumbers Solved Exercises

Note: Answers may vary based on whether the IPC or UPC is applicable. Before you through the math solutions below, you will also find this math for plumbers guide.

1. What is the sum of 5/8 in. and 7/8 in.?
Answer: 11/2 in.
5/8 + 7/8
5 + 7/8 = 12/8 = 14/8 = 11/2
 

2. What is the sum of 3/4 in. and 1/8 in.?
Answer: 7/8 in.
3/4 + 1/8
6/8 + 1/8 = 7/8

3. What is 5/8 in. minus 5/8 in.?
Answer: 1/8 in.
3/4 – 5/8
6 – 5/8 = 1/8

4. What is the area of a hole drilled for a closet flange that has a radius of 21/2 in.?
Answer: 19.625 sq. in.
A = πr2
A = 3.14 × (2.5 × 2.5)
A = 19.625 sq. in.

5. The label is missing from a leaking water heater. What is the volume of the tank in gallons if the radius of the top is 9 in. and the height of the tank is 48 in.?
Answer: 52.85 gal.
V = πr2h/231
V = 3.14 × (9 × 9) × 48/231
V = 12208.32/231
V = 52.85 gal.

6. Two parallel 1/2-in. copper pipes must be connected on the end using two 90-degree elbows forming a U shape, which have a fitting allowance of 3/8 in., and a piece of 1/2-in. copper pipe. The two parallel pipes must be kept 8 in. apart, measured center to center. How long must the connecting piece of pipe be?
Answer: 71/4 in.
L = 8 – (3/8 + 3/8)
L = 8 – 6/8
L = 78/8 – 6/8
L = 72/8 = 71/4 in.

7. A pipe that is 85 ft. long has an inside diameter of 2 in. How many gallons of water will it take to fill it?
Answer: 13.86 gal.
First, convert 85 ft. to inches so that both of the factors are expressed using the same measurement.
85 × 12 = 1,020
V = πr2h/231
V = 3.14 × (1 × 1) × 1,020/231
V = 3,202.8/231
V = 13.86 gal.

8. A 1-in. steel gas pipe must be offset 1ft. 6in., using 45-degree elbows. How long must the connecting piece be to the nearest 1/16 in.? The fitting allowance for 1-in. steel pipe is 1 in.
Answer: 237/16 in.
Convert the feet to inches so one unit of measurement is used.
1 foot 6 inches = 18 inches
L = Offset × 1.414 – (fitting allowance × 2)
L = 18 × 1.414 – (1 × 2)
L = 25.452 – 2
L = 23.452
Convert .452 to sixteenths by multiplying by 16.
0.452 × 16 = 7.232
Round 7.232 to 7. That is the number of sixteenths we have (7/16). Put the 7/16 with the 23 in. and the result is 237/16 in.

9. What is 4 ft. minus 67/8 in.?
Answer: 411/8 in.
Convert the feet to inches so only one unit of measurement is used.
4 × 12 = 48
48 → 478/8
–67/8 → –67/8
411/8

10. To the nearest 1/16 in., how long must a piece of tape be to just go around a section of pipe insulation that has an outside diameter of 31/4 in?
Answer: 103/16 in.
C = πd
C = 3.14 × 31/4
Convert the inches to a decimal by dividing the numerator (top number) by the bottom number (denominator):
1 ÷ 4 = 0.25
C = 3.14 × 3.25
C = 10.205
Convert the decimal to sixteenths by multiplying by 16:
0.205 × 16 = 3.28
Round 3.28 to 3. This is the number of sixteenths we have: 3/16.
Put the 3/16 back with the 10: 103/16 in.

11. A pitch of 1/4-in. per foot must be maintained on a drainpipe that is being hung under floor joists. On the upstream end of the pipe, it measures 127/8 in. from the underside of the floor above to the centerline of the pipe. What will the measurement be from the underside of the floor to the centerline of the pipe 18 ft. downstream?
Answer: 173/8 in.
The pipe will drop 1/4 in. for every foot along the 18 feet.
It will drop 18 × 0.25, which equals 4.5 in.
The pipe is already 127/8 in. down.
Add the drop to the upstream distance:
D = 127/8 + 4.5
Convert 127/8 to a decimal.
7 ÷ 8 = 0.875
Solve for D:
D = 12.875 + 4.5
D = 17.375
Convert 0.375 to eighths by multiplying by 8: 8 × 0.375 = 3
3 is the number of eighths.
Put the 3/8 back with the 17.
D = 173/8 in.

12. Julio must offset a 11/2-in. copper pipe a distance of 12 in. using 45-degree elbows. How long must the diagonal pipe be if the fitting allowance for a 11/2-in. copper 45-degree elbow is 9/16 in.?
Answer: 157/8 in.
L = Offset × 1.414 – (fitting allowance × 2)
First convert the fraction to a decimal:
9 ÷ 16 = 0.5625
L = 12 × 1.414 – (0.5625 × 2)
L = 16.968 – 1.125
L = 15.843
Decide how precise the measurement must be and multiply 0.843 by the denominator. Use eighths for this problem:
0.843 × 8 = 6.744
Round 6.744 to 7. That’s 7 eighths (7/8).
L = 157/8 in.

13. What is 3 ft. 31/8 in. minus 2 ft. 3/4 in.?
Answer: 1 ft. 23/8 in.
2 ft. 3/4 in. minus 3 ft. 31/8 in.
Denominators must be the same to subtract fractions:
2 ft. 6/8 in. – 3 ft. 31/8 in.
391/8 → 389/8
–246/8 → –246/8
143/8 in.
143/8 in. or 1 ft. 23/8 in.

4. Antifreeze needs to be added to a heating system as a 50% antifreeze-water solution. How much straight antifreeze is needed to fill the two manifold pipes that are both 4 ft. 3 in. long and have a diameter of 2 in.?
Answer: 51/2 cups
V = πr2h/231 ÷ 2
V = 3.14 × 12 × 51/231 ÷ 2
V = 160.14/231 ÷ 2
V = 0.6932467 ÷ 2
V = 0.3466233 gal.
Round this off to 0.347 gal. Since this is a fractional part of 1 gallon, multiply by 16 to get the number of cups of antifreeze needed. There are 16 cups in a gallon:
0.347 × 16 = 5.552 cups.
Multiply 0.552 by 8 to get the number of ounces since there are 8 ounces in a cup:
0.552 × 8 = 4.416
Round to 4 ounces. Put that with the 5 cups and the answer is 5 cups, 4 ounces, or 51/2 cups.

15. What is the sum of 1 ft. 8 in. and 20 in.?
Answer: 3 ft. 4 in., 40 in.
Convert all to inches:
1 ft. 8 in. = 20 in.
Add 20 in. plus 20 in. = 40 in. or 3 ft. 4 in.

16. How many 1/8-in. are there in 4 in.?
Answer: 32
1 in. = 8/8
8/8 × 4 → 8/8 × 4/1 = 32/1

17. The volume of a section of pipeline is 182,822 cubic inches. How many gallons will it hold?
Answer: 791.44 gal.
There are 231 cubic inches in 1 gallon.
Divide 182,822 by 231
182,822/231 = 791.44 gal.

18. Evan must offset two parallel copper pipes 14 in. using 45-degree elbows. One of the pipes is 3/4 in. in diameter and the other is 1 in. in diameter. The fitting allowance for 3/4-in. pipe is 1/4 in. and for 1-in. pipe, it’s 5/16 in. The two pipes must remain 3 in. apart, center to center, throughout the offset. How long must each diagonal pipe be?
Answer: 195/16 in. and 193/16 in.
This requires two complete calculations. For the 3/4-in. pipe:
L = Offset × 1.414 – (fitting allowance × 2)
L = Offset × 1.414 – (0.25 × 2)
L = 14 × 1.414 – 0.5
L = 19.796 – 0.5
L = 19.296
Convert .296 to sixteenths by multiplying by 16:
0.296 × 16 = 4.736
Round 4.736 to 5. That’s the number of sixteenths we have. Put it back with the 19 in. to yield 195/16 in. or 1 ft. 75/16 in. for the 3/4-in. pipe.
For the 1-in. pipe:
L = Offset × 1.414 – (fitting allowance × 2)
L = Offset × 1.414 – (0.3125 × 2)
L = 14 × 1.414 – 0.625
L = 19.796 – 0.625
L = 19.171
0.171 × 16 = 2.736
Round to 3:
193/16 in. or 1 ft. 73/16 in. for the 1-in. pipe.
19. A 23/8-in. hole must be drilled through the center of a 2 × 6 stud. How much wood will be left on each side of the hole? Remember, a 2 × 6 is really 11/2 × 51/2 in.
Answer: 19/16 in.
First, find the center of the joist:
5.5 ÷ 2 = 2.75
If the hole is drilled in the center, half of the hole will be on either side of the centerline.
Divide the hole size by 2.
Convert 23/8 to decimal form by dividing 3 by 8, which is 2.375.
2.375 ÷ 2 = 1.1875
2.75 – 1.1875 = 1.5625 or 19/16 in.
The edge of the hole will be 19/16 in. from the edge of the 2 × 6.
 

20. A tank holds 6 gallons. How many cubic inches are in it?
Answer: 1,386 cu. in.
Since there are 231 cu. in. in a gallon, 231 × 6 gal. = 1,386 cu. in.
 

21. A sheet of insulation blanket must be cut to cover the sides of a tank with a diameter of 10 in. and a height of 22 in. What must the dimensions of the blanket be to cover only the sides of the tank?
Answer: 313/8 in. × 22 in.
C = πd
C = 3.14 × 10
C = 31.4
C = 313/8 in.
The insulation blanket must be 313/8 in. wide by 22 in. high.
22. 8 ft. 31/4 in. minus 1 ft. 41/2 in. equals what?
Answer: 6 ft. 103/4 in.
7 ft. 145/4 in.
– 1 ft. 42/4 in.
6 ft. 103/4 in.

23. Pat kicks over a bucket of antifreeze that is 12 in. in diameter and 19 in. tall. The level of the antifreeze was 10 in. down from the top before Pat kicked it over. How much antifreeze, in gallons, did Pat have to clean up?
Answer: 2.2 gal.
0.406 × 12 = 4.872
V = πr2h/231 ÷ 2
V = 3.14 × 62 × 9/231 ÷ 2
V = 3.14 × 36 × 9/231 ÷ 2
V = 4.4042 ÷ 2
V = 2.2021 rounded to 2.2 gal.

24. A 2-in. diameter pipe that is 3 ft. long has 1 ft. 6 in. added to its length. What is the volume of the lengthened pipe in gallons?
Answer: 0.73 gal.
V = πr2h/231 ÷ 2
V = 3.14 × 12 × (36 × 18)/231
V = 169.56/231
V = 0.73 gal.
 

25. Julio drilled a hole through a wall for a pipe to pass through. The outside diameter of the pipe is 2 1/4 in. There is 1/4 in. of clearance on all sides of the pipe to the edge of the hole. What is the circumference of the hole?
Answer: 85/8 in.
C = πd
C = 3.14 × (2.25 + 0.25 + 0.25)
C = 3.14 × 2.75
C = 8.635
Convert to inches with a fraction of an inch.
0.635 × 16 = 10.16 sixteenths.
Round to 10/16; reduce to 5/8.
Bring the 8 inches back to give 85/8 in.

26. A plumbing distribution system has 330 ft. of 1/2-in. copper pipe. What size bucket, in gallons, would you need to drain the whole system?
Answer: 3.36 gal.
V = πr2h/231
Convert 330 ft. to 3,960 in.
V = 3.14 × .252 × 3,960/231
V = 3.14 × 0.0625 × 3,960/231
V = 777.15/231
V = 3.36 gal.

27. The pitch of a large sewer is 1/8-in. per foot. What is the difference in elevation from the sewage treatment plant to the end of the sewer main, which is 1,575 ft. away?
Answer: 16 ft. 5 in.
Divide 1,575 ft. by 8 to get the number of inches vertically in pitch:
1,575/8 = 196.875 inches
Divide 196.875 inches by 12 to reduce the answer to feet and inches:
196.875/12 = 16.406 feet
Multiply 0.406 by 12 to get the number of inches after the 16 ft.:
0.406 × 12 = 4.875
Round to 5.
16 ft. 5 in.

28. What is the sum of 3/4 in., 1/8 in. and 5/16 in.?
Answer: 13/16 in.
All denominators must be the same:
12/16 + 2/16 + 5/16 = 19/16
19/16 reduces to 13/16 in.

29. Evan cut the 5-in. diameter hole for a water closet flange in the wrong place. He must cut a new hole 41/8 in., measured center to center, to the left of the original hole. Will the holes overlap?

Answer: Yes

30. How far is it from the edge of a 23/8-in. diameter hole to the edge of another 23/8-in. diameter hole drilled 8 in. apart, center to center?
Answer: 55/8 in.
L = 8 – 23/8
L = 78/8 – 23/8
L = 55/8 in.

31. How long must a piece of 1-in. copper pipe be to act as an offset using 90-degree elbows when the two pipes are 12 in. apart? The fitting allowance for a 1-in. copper 90-degree elbow is 3/4 in.
Answer: 101/2 in.
L = 12 – (2 × .75)
L = 12 – 1.5
L = 10.5 or 101/2 in.

32. What is the area in square inches of a manhole cover that is 2 ft. in diameter?
Answer: 452.16 sq. in.
A = πr2
Convert the 2 ft. to 24 in.
A = 3.14 × 122
A = 3.14 × 144
A = 452.16 sq. in.

33. If a 50-ft. length of a 24-in. diameter storm drain is running half full, how many gallons of water are in the pipe?
Answer: 587.22 gal.
V = πr2h/231 ÷ 2
Convert the 50 ft. to inches:
V = 3.14 × (122) × 600/231 ÷ 2
V = 3.14 × 271,296/231 ÷ 2
V = 1174.44 ÷ 2
V = 587.22 gal.

34. A hot water expansion tank holds 3 gallons and is 16 in. high. What is the diameter?
Answer: 77/16 in.
V = πr2h/231
3 = 3.14 × r2 × 16/231
3 = 50.24 × r2/231
231 × 3 = 50.24 × r2
693 = 50.24 × r2
693 ÷ 50.24 = r2
13.79 = r2
√13.79 = r
3.71 in. = r
D = r × 2
D = 7.42 Multiply 0.42 by 16 to convert to a fraction with 16 as the denominator:
0.42 × 16 = 6.72
Round to 7. That is the number of sixteenths.
Put 7/16 back with the 7, D = 77/16 in.

35. What is 35 in. minus 2 ft.?
Answer: 11 in.
Convert the feet to in.:
2 ft. = 24 in.
L = 35 – 24
L = 11 in.