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Simplification of Denominate Numbers

Module by: Wade Ellis, Denny Burzynski. E-mail the authorsEdited By: Math Editors

Summary: This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to simplify denominate numbers. By the end of the module students should be able to convert an unsimplified unit of measure to a simplified unit of measure, be able to add and subtract denominate numbers and be able to multiply and divide a denominate number by a whole number.

Section Overview

  • Converting to Multiple Units
  • Adding and Subtracting Denominate Numbers
  • Multiplying a Denominate Number by a Whole Number
  • Dividing a Denominate Number by a Whole Number

Converting to Multiple Units

Denominate Numbers

Numbers that have units of measure associated with them are called denominate numbers. It is often convenient, or even necessary, to simplify a denominate number.

Simplified Denominate Number

A denominate number is simplified when the number of standard units of measure associated With it does not exceed the next higher type of unit.

The denominate number 55 min is simplified since it is smaller than the next higher type of unit, 1 hr. The denominate number 65 min is not simplified since it is not smaller than the next higher type of unit, 1 hr. The denominate number 65 min can be simplified to 1 hr 5 min. The denominate number 1 hr 5 min is simplified since the next higher type of unit is day, and 1 hr does not exceed 1 day.

Sample Set A

Example 1

Simplify 19 in.

Since 12  in.=1 ft12  in.=1 ft size 12{"12"" in" "." ="1 ft"} {}, and 19=12+719=12+7 size 12{"19"="12"+7} {},

19 in. = 12 in. + 7 in. = 1 ft + 7 in. = 1 ft 7 in. 19 in. = 12 in. + 7 in. = 1 ft + 7 in. = 1 ft 7 in.

Example 2

Simplify 4 gal 5 qt.

Since 4  qt=1 gal4  qt=1 gal size 12{4" qt"="1 gal"} {}, and 5=4+15=4+1 size 12{5=4+1} {},

4 gal 5 qt = 4 gal + 4 qt + 1qt = 4 gal + 1 gal + 1 qt = 5 gal + 1 qt = 5 gal 1 qt 4 gal 5 qt = 4 gal + 4 qt + 1qt = 4 gal + 1 gal + 1 qt = 5 gal + 1 qt = 5 gal 1 qt

Example 3

Simplify 2 hr 75 min.

Since 60 min =1 hr60 min =1 hr size 12{"60 min "="1 hr"} {}, and 75=60+1575=60+15 size 12{"75"="60"+"15"} {},

2 hr 75 min = 2 hr + 60 min + 15 min = 2 hr + 1 hr + 15 min = 3 hr + 15 min = 3 hr 15 min 2 hr 75 min = 2 hr + 60 min + 15 min = 2 hr + 1 hr + 15 min = 3 hr + 15 min = 3 hr 15 min

Example 4

Simplify 43 fl oz.

Since 8 fl oz=1 c8 fl oz=1 c size 12{"8 fl oz"="1 c"} {} (1 cup), and 43÷8=5 R343÷8=5 R3 size 12{"43" div 8=5" R3"} {},

43 fl oz = 40 fl oz + 3 fl oz = 5 ⋅ 8 fl oz + 3 fl oz = 5 ⋅ 1 c + 3 fl oz = 5 c + 3 fl oz 43 fl oz = 40 fl oz + 3 fl oz = 5 ⋅ 8 fl oz + 3 fl oz = 5 ⋅ 1 c + 3 fl oz = 5 c + 3 fl oz

But, 2 c=1 pt2 c=1 pt size 12{"2 c"="1 pt"} {} and 5÷2=2 R15÷2=2 R1 size 12{5 div 2=2" R1"} {}. So,

5 c + 3 fl oz = 2 ⋅ 2 c + 1 c + 3 fl oz = 2 ⋅ 1 pt + 1 c + 3 fl oz = 2 pt + 1 c + 3 fl oz 5 c + 3 fl oz = 2 ⋅ 2 c + 1 c + 3 fl oz = 2 ⋅ 1 pt + 1 c + 3 fl oz = 2 pt + 1 c + 3 fl oz

But, 2 pt=1 qt2 pt=1 qt size 12{"2 pt"="1 qt"} {}, so

2 pt + 1 c + 3 fl oz = 1 qt 1 c 3 fl oz 2 pt + 1 c + 3 fl oz = 1 qt 1 c 3 fl oz size 12{"1 pt "+" 1 c "+" 3 fl oz"=" 1 qt 1 c 3 fl oz"} {}

Practice Set A

Simplify each denominate number. Refer to the conversion tables given in (Reference), if necessary.

Exercise 1

18 in.

Solution

1 ft 6 in.

Exercise 2

8 gal 9 qt

Solution

10 gal 1 qt

Exercise 3

5 hr 80 min

Solution

6 hr 20 min

Exercise 4

8 wk 11 da

Solution

9 wk 4 da

Exercise 5

86 da

Solution

12 wk 2 da

Adding and Subtracting Denominate Numbers

Adding and Subtracting Denominate Numbers

Denominate numbers can be added or subtracted by:

  1. writing the numbers vertically so that the like units appear in the same column.
  2. adding or subtracting the number parts, carrying along the unit.
  3. simplifying the sum or difference.

Sample Set B

Example 5

Add 6 ft 8 in. to 2 ft 9 in.

6 ft  8 in. +2 ft  9 in.̲ 8 ft 17 in. Simplify this denominate number.6 ft  8 in. +2 ft  9 in.̲ 8 ft 17 in. Simplify this denominate number.alignr { stack { size 12{"6 ft 8 in" "." } {} # size 12{ {underline {+"2 ft 9 in" "." }} } {} # size 12{"8 ft 17 in" "." } {} } } {}

Since 12 in.= 1 ft12 in.= 1 ft size 12{"12 in" "." =" 1 ft"} {},

8 ft + 12 in. + 5 in. = 8 ft + 1 ft + 5 in. = 9 ft + 5 in. = 9 ft 5 in. 8 ft + 12 in. + 5 in. = 8 ft + 1 ft + 5 in. = 9 ft + 5 in. = 9 ft 5 in.

Example 6

Subtract 5 da 3 hr from 8 da 11 hr.

8 da 11 hr - 5 da   3 hr ̲ 3 da   8 hr 8 da 11 hr - 5 da   3 hr ̲ 3 da   8 hr alignr { stack { size 12{"8 da 11 hr"} {} # size 12{ {underline {"-5 da 3 hr"}} } {} # size 12{"3 da 8 hr"} {} } } {}

Example 7

Subtract 3 lb 14 oz from 5 lb 3 oz.

5 lb   3 oz - 3 lb 14 oz ̲ 5 lb   3 oz - 3 lb 14 oz ̲ alignr { stack { size 12{"5 lb 3 oz"} {} # size 12{ {underline {"-3 lb 14 oz"}} } {} } } {}

We cannot directly subtract 14 oz from 3 oz, so we must borrow 16 oz from the pounds.

5 lb 3 oz = 5 lb + 3 oz = 4 lb + 1 lb + 3 oz = 4 lb + 16 oz + 3 oz ( Since 1 lb = 16 oz. ) = 4 lb + 19 oz = 4 lb 19 oz 5 lb 3 oz = 5 lb + 3 oz = 4 lb + 1 lb + 3 oz = 4 lb + 16 oz + 3 oz ( Since 1 lb = 16 oz. ) = 4 lb + 19 oz = 4 lb 19 oz

4 lb 19 oz - 3 lb 14 oz ̲ 1 lb   5 oz 4 lb 19 oz - 3 lb 14 oz ̲ 1 lb   5 oz alignr { stack { size 12{"4 lb 19 oz"} {} # size 12{ {underline {"-3 lb 14 oz"}} } {} # size 12{"1 lb 5 oz"} {} } } {}

Example 8

Subtract 4 da 9 hr 21 min from 7 da 10 min.

7 da 0 hr 10 min - 4 da 9 hr 21 min̲ 7 da 0 hr 10 min - 4 da 9 hr 21 min̲ alignr { stack { size 12{"7 da 0 hr 10 min"} {} # size 12{ {underline {"-4 da 9 hr 21 min"}} } {} } } {} Borrow 1 da from the 7 da.

6 da 24 hr 10 min -4 da   9 hr 21 min̲6 da 24 hr 10 min -4 da   9 hr 21 min̲alignr { stack { size 12{"6 da 24 hr 10 min"} {} # size 12{ {underline {"-4 da 9 hr 21 min"}} } {} } } {} Borrow 1 hr from the 24 hr.

6 da 23 hr 70 min - 4 da   9 hr 21 min ̲ 2 da 14 hr 49 min 6 da 23 hr 70 min - 4 da   9 hr 21 min ̲ 2 da 14 hr 49 min alignr { stack { size 12{"6 da 23 hr 70 min"} {} # size 12{ {underline {"-4 da 9 hr 21 min"}} } {} # size 12{"2 da 14 hr 49 min"} {} } } {}

Practice Set B

Perform each operation. Simplify when possible.

Exercise 6

Add 4 gal 3 qt to 1 gal 2 qt.

Solution

6 gal 1 qt

Exercise 7

Add 9 hr 48 min to 4 hr 26 min.

Solution

14 hr 14 min

Exercise 8

Subtract 2 ft 5 in. from 8 ft 7 in.

Solution

6 ft 2in.

Exercise 9

Subtract 15 km 460 m from 27 km 800 m.

Solution

12 km 340 m

Exercise 10

Subtract 8 min 35 sec from 12 min 10 sec.

Solution

3 min 35 sec

Exercise 11

Add 4 yd 2 ft 7 in. to 9 yd 2 ft 8 in.

Solution

14 yd 2 ft 3 in

Exercise 12

Subtract 11 min 55 sec from 25 min 8 sec.

Solution

13 min 13 sec

Multiplying a Denominate Number by a Whole Number

Let's examine the repeated sum

4 ft 9 in. + 4 ft 9 in. + 4 ft 9 in. ︸ 3 times = 12 ft 27 in. 4 ft 9 in. + 4 ft 9 in. + 4 ft 9 in. ︸ 3 times =12 ft 27 in.

Recalling that multiplication is a description of repeated addition, by the distribu­tive property we have

3 ( 4 ft 9 in . ) = 3 ( 4 ft + 9 in. ) = 3 ⋅ 4 ft + 3 ⋅ 9 in. = 12 ft + 27 in. Now, 27 in. = 2 ft 3 in. = 12 ft + 2 ft + 3 in. = 14 ft + 3 in. = 14 ft 3 in. 3 ( 4 ft 9 in . ) = 3 ( 4 ft + 9 in. ) = 3 ⋅ 4 ft + 3 ⋅ 9 in. = 12 ft + 27 in. Now, 27 in. = 2 ft 3 in. = 12 ft + 2 ft + 3 in. = 14 ft + 3 in. = 14 ft 3 in.

From these observations, we can suggest the following rule.

Multiplying a Denominate Number by a Whole Number

To multiply a denominate number by a whole number, multiply the number part of each unit by the whole number and affix the unit to this product.

Sample Set C

Perform the following multiplications. Simplify if necessary.

Example 9

6⋅2 ft 4 in. = 6⋅2 ft+6⋅4 in. = 12 ft + 24 in. 6⋅2 ft 4 in. = 6⋅2 ft+6⋅4 in. = 12 ft + 24 in.

Since 3 ft= 1 yd3 ft= 1 yd size 12{"3 ft"=" 1 yd"} {} and 12  in.=1 ft12  in.=1 ft size 12{"12"" in" "." ="1 ft"} {},

12 ft + 24 in. = 4 yd + 2 ft = 4 yd 2 ft 12 ft + 24 in. = 4 yd + 2 ft = 4 yd 2 ft

Example 10

8⋅5 hr 21 min 55 sec = 8⋅5 hr + 8⋅21 min +8⋅ 55 sec = 40 hr +168 min + 440sec = 40 hr +168 min +7 min + 20 sec = 40 hr + 175 min +20 sec = 40 hr+ 2 hr + 55 min+20 sec = 42 hr + 55 min+ 20 sec = 24hr +18hr +55 min + 20 sec = 1 da + 18 hr + 55 min +20 sec = 1 da 18 hr 55 min 20 sec 8⋅5 hr 21 min 55 sec = 8⋅5 hr + 8⋅21 min +8⋅ 55 sec = 40 hr +168 min + 440sec = 40 hr +168 min +7 min + 20 sec = 40 hr + 175 min +20 sec = 40 hr+ 2 hr + 55 min+20 sec = 42 hr + 55 min+ 20 sec = 24hr +18hr +55 min + 20 sec = 1 da + 18 hr + 55 min +20 sec = 1 da 18 hr 55 min 20 sec

Practice Set C

Perform the following multiplications. Simplify.

Exercise 13

2⋅10 min2⋅10 min size 12{2 cdot left ("10 min" right )} {}

Solution

20 min

Exercise 14

5⋅3 qt5⋅3 qt size 12{5 cdot left ("3 qt" right )} {}

Solution

15 qt=3 gal 3 qt15 qt=3 gal 3 qt size 12{"15 qt"="3 gal 3 qt"} {}

Exercise 15

4⋅5 ft 8 in.4⋅5 ft 8 in. size 12{4 cdot left (5" ft 8 in" "." right )} {}

Solution

20 ft 32 in.= 7 yd 1 ft 8 in.20 ft 32 in.= 7 yd 1 ft 8 in. size 12{"20 ft 32 in" "." =" 7 yd 1 ft 8 in" "." } {}

Exercise 16

10⋅2 hr 15 min 40 sec10⋅2 hr 15 min 40 sec size 12{"10" cdot left (2" hr 15 min 40 sec" right )} {}

Solution

20 hr 150 min 400 sec=22 hr 36 min 40 sec20 hr 150 min 400 sec=22 hr 36 min 40 sec size 12{"20 hr 150 min 400 sec"="22 hr 36 min 40 sec"} {}

Dividing a Denominate Number by a Whole Number

Dividing a Denominate Number by a Whole Number

To divide a denominate number by a whole number, divide the number part of each unit by the whole number beginning with the largest unit. Affix the unit to this quotient. Carry any remainder to the next unit.

Sample Set D

Perform the following divisions. Simplify if necessary.

Example 11

12 min 40 sec÷412 min 40 sec÷4 size 12{ left ("12 min 40 sec" right ) div 4} {}

Long division. 12 min and 40 sec divided by 4. 4 goes into 12 minutes 3 times, making a quotient of 3 minutes with no remainder. 4 goes into 40 seconds 10 times, making a quotient of 10 seconds with no remainder. The total quotient is 3 min 10 sec.

Thus 12 min 40 sec÷4=3 min 10 sec12 min 40 sec÷4=3 min 10 sec size 12{ left ("12 min 40 sec" right ) div 4=3" min 10 sec"} {}

Example 12

5 yd 2 ft 9 in.÷35 yd 2 ft 9 in.÷3 size 12{ left ("5 yd 2 ft 9 in" "." right ) div 3} {}

Long division. 5 yd 2 ft 9 in divided by 3. 3 goes into 5 yards one time with a remainder of 2 yards. Bring down the 2 feet. 2 yards and 2 feet is eight feet. 3 goes into eight feet twice with a remainder of 2 feet. Bring down the 9 inches. 2 feet 9 in is equal to 22 inches. 3 goes into 33 inches exactly 11 times. The total quotient is 1 yd 2 ft 11 in. Convert to feet: 2 yd 2 ft = 8 ft . Convert to inches: 2 ft 9 in . = 33 in . Convert to feet: 2 yd 2 ft = 8 ft . Convert to inches: 2 ft 9 in . = 33 in .

Thus 5 yd 2 ft 9 in.÷3=1 yd 2 ft 11 in.5 yd 2 ft 9 in.÷3=1 yd 2 ft 11 in. size 12{ left ("5 yd 2 ft 9 in" "." right ) div 3=1" yd 2 ft 11 in" "." } {}

Practice Set D

Perform the following divisions. Simplify if necessary.

Exercise 17

18 hr 36 min÷918 hr 36 min÷9 size 12{ left ("18 hr 36 min" right ) div 9} {}

Solution

2 hr 4 min

Exercise 18

34 hr 8 min.÷834 hr 8 min.÷8 size 12{ left ("34 hr 8 min" "." right ) div 8} {}

Solution

4 hr 16 min

Exercise 19

13 yd 7 in.÷513 yd 7 in.÷5 size 12{ left ("13 yd 7 in" "." right ) div 5} {}

Solution

2 yd 1 ft 11 in

Exercise 20

47 gal 2 qt 1 pt÷347 gal 2 qt 1 pt÷3 size 12{ left ("47 gal 2 qt 1 pt" right ) div 3} {}

Solution

15 gal 3 qt 1 pt

Exercises

For the following 15 problems, simplify the denominate numbers­.

Exercise 21

16 in.

Solution

1 foot 4 inches

Exercise 22

19 ft

Exercise 23

85 min

Solution

1 hour 25 minutes

Exercise 24

90 min

Exercise 25

17 da

Solution

2 weeks 3 days

Exercise 26

25 oz

Exercise 27

240 oz

Solution

15 pounds

Exercise 28

3,500 lb

Exercise 29

26 qt

Solution

6 gallons 2 quarts

Exercise 30

300 sec

Exercise 31

135 oz

Solution

8 pounds 7 ounces

Exercise 32

14 tsp

Exercise 33

18 pt

Solution

2 gallons 1 quart

Exercise 34

3,500 m

Exercise 35

16,300 mL

Solution

16 liters 300 milliliters (or 1daL 6 L 3dL)

For the following 15 problems, perform the indicated opera­tions and simplify the answers if possible.

Exercise 36

Add 6 min 12 sec to 5 min 15 sec.

Exercise 37

Add 14 da 6 hr to 1 da 5 hr.

Solution

15 days 11 hours

Exercise 38

Add 9 gal 3 qt to 2 gal 3 qt.

Exercise 39

Add 16 lb 10 oz to 42 lb 15 oz.

Solution

59 pounds 9 ounces

Exercise 40

Subtract 3 gal 1 qt from 8 gal 3 qt.

Exercise 41

Subtract 3 ft 10 in. from 5 ft 8 in.

Solution

1 foot 10 inches

Exercise 42

Subtract 5 lb 9 oz from 12 lb 5 oz.

Exercise 43

Subtract 10 hr 10 min from 11 hr 28 min.

Solution

1 hour 18 minutes

Exercise 44

Add 3 fl oz 1 tbsp 2 tsp to 5 fl oz 1 tbsp 2 tsp.

Exercise 45

Add 4 da 7 hr 12 min to 1 da 8 hr 53 min.

Solution

5 days 16 hours 5 minutes

Exercise 46

Subtract 5 hr 21 sec from 11 hr 2 min 14 sec.

Exercise 47

Subtract 6 T 1,300 lb 10 oz from 8 T 400 lb 10 oz.

Solution

1 ton 1,100 pounds (or 1T 1,100 lb)

Exercise 48

Subtract 15 mi 10 in. from 27 mi 800 ft 7 in.

Exercise 49

Subtract 3 wk 5 da 50 min 12 sec from 5 wk 6 da 20 min 5 sec.

Solution

2 weeks 23 hours 29 minutes 53 seconds

Exercise 50

Subtract 3 gal 3 qt 1 pt 1 oz from 10 gal 2 qt 2 oz.

Exercises for Review

Exercise 51

((Reference)) Find the value: 582+3964582+3964 size 12{ left ( { {5} over {8} } right ) rSup { size 8{2} } + { {"39"} over {"64"} } } {}.

Solution

1

Exercise 52

((Reference)) Find the sum: 8+6358+635 size 12{8+6 { {3} over {5} } } {}.

Exercise 53

((Reference)) Convert 2.051112.05111 size 12{2 "." "05" { {1} over {"11"} } } {} to a fraction.

Solution

214275214275 size 12{2 { {"14"} over {"275"} } } {}

Exercise 54

((Reference)) An acid solution is composed of 3 parts acid to 7 parts water. How many parts of acid are there in a solution that contains 126 parts water?

Exercise 55

((Reference)) Convert 126 kg to grams.

Solution

126,000 g

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