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2.4 Arithmetic Operations with Fractions

Least or Lowest Common Denominator (LCD)

The Least Common Denominator (LCD) of a set of two or more fractions is the smallest whole number divisible by each denominator. It is the least common multiple (LCM) of the denominators of the fractions. Two methods of finding the LCM are explained in Chapter 1, Section 1.3.

In performing addition and subtraction of fractions, it is necessary to determine equivalent fractions with common denominators. The LCD is the best choice for a common denominator because it makes further simplification easier.

Example 2.4-a: Determining the Least Common Denominator

Determine the LCD of 49 and 715.

Solution

The LCD of the fractions 49 and 715 is the same as the LCM of the denominators 9 and 15.

Using one of the methods from Chapter 1, Section 1.3:

  • The largest number, 15, is not divisible by 9.
  • Multiples of 15 are: 15, 30, 45, …
  • 45 is divisible by 9.
  • Thus, the LCM of 9 and 15 is 45.

Therefore, the LCD of 49 and 715 is 45.

Comparing Fractions

Fractions can easily be compared when they have the same denominator. If they do not have the same denominator, first determine the LCD of the fractions and convert them into equivalent fractions with the LCD as the denominator.

Example 2.4-b: Comparing Fractions

Which of the fractions, 512 or 38, is greater?

Solution

Step 1: Since the fractions do not have the same denominator, we need to first determine the LCD of the fractions, which is the same as the LCM of the denominators. The LCM of 12 and 8 is 24.

Step 2: Convert each fraction to its equivalent fraction with 24 as the denominator.

To convert 512 to its equivalent fraction with 24 as the denominator, multiply the denominator by 2 to obtain the LCD of 24, and multiply the numerator by 2 as well to maintain an equivalent fraction.

512=5×212×2=1024

5 portions of 12 equal parts of a whole are equal to 10 portions of 24 equal parts of that whole.

Similarly, convert 38 to an equivalent fraction with 24 as the denominator:

38=3×38×3=924

3 portions of 8 equal parts of a whole is equal to 9 portions of 24 equal parts of that whole.

Step 3: Since the denominators are the same, we can compare the fractions’ numerators to identify the greater fraction.

Since 10 > 9, it implies: 1024 > 924.

Therefore, 512 > 38.

Addition of Fractions

The denominator of a fraction indicates the number of parts into which an item is divided. Therefore, adding fractions requires that every fraction’s denominators be the same. If the denominators are different, they must first be made the same by determining the LCD and changing each fraction to its equivalent fraction with the LCD as the denominator.

When the fractions have the same denominator, the numerators of each of the fractions may be added. The numerator of the resulting fraction is equal to this sum, and the denominator is equal to the common denominator of the fractions being added.

Express the final answer reduced to the lowest terms and as a mixed number, where applicable.

Example 2.4-c: Adding Fractions that Have the Same Denominator

Add 29 and 59.

Solution

29+59

The denominators of the fractions are the same. Adding the numerators and keeping
the common denominator,

=2+59

=79

 

2 over 9 (illustrated with 2 slices of the pie chart shaded) plus 5 over 9 (illustrated with 5 slices of the pie chart shaded) equals 7 over nine (illustrated with 7 slices of the pie chart shaded).

 

Therefore, the result from adding =29 and =59 is 79.

Example 2.4-d: Adding Fractions that have Different Denominators

Add 34 and 23.

Solution

34+23

LCM of 4 and 3 is 12 (i.e., LCD = 12). Determining the equivalent fractions with a denominator of 12,

=912+812

=9+812

Adding the numerators and keeping the common denominator,

=1712

=1512

Converting the improper fraction to a mixed number,

Therefore, the result from adding =34 and =23 is 1512.

Example 2.4-e: Adding a Mixed Number and a Proper Fraction

Add 356 and 49.

Solution

Method 1:

356 + 49

Converting the mixed number to an improper fraction,

=(3×6)+56+49=236+49

LCM of 6 and 9 is 18 (i.e., LCD = 18). Determining the equivalent fractions with a denominator of 18,

=6918+818

Adding the numerators and keeping the common denominator,

=7718

Converting the improper fraction to a mixed number,

=4518

Method 2:

356 + 49

Separating the whole number and the fractions,

=3+(56+49)

LCM of 6 and 9 is 18 (i.e., LCD = 18). Determining the equivalent fractions with a denominator of 18,

=3+(1518+818)

Adding the numerators and keeping the common denominator,

=3+2318

Converting the improper fraction to a mixed number,

=3+1518

=3+1+518

Adding the whole numbers and then the fraction,

=4+518

=4518

Therefore, the result from adding 356 and 49 is 4518.

Example 2.4-f: Adding Mixed Numbers

Add:

  1. 216 and 434
  2. 1523 and 335

Solution

  1. 216+434

    LCM of 6 and 4 is 12 (i.e., LCD = 12). Determining the equivalent mixed numbers with a denominator of 12,

    =2212+4912

    Separating the whole numbers and the fractions,

    =(2+4)+(212+912)

    Adding the whole numbers and the fractions,

    =61112

    Therefore, the result from adding 216 and 434 is 61112.

  2. 1523+335

    LCM of 3 and 5 is 15 (i.e., LCD = 15). Determining the equivalent mixed numbers with a denominator of 15,

    151015+3915

    Separating the whole numbers and the fractions,

    =(15+3)+(1015+915)

    Adding the whole numbers and the fractions,

    =18+1915

    Converting the improper fraction to a mixed number,

    =18+1415

    Adding the whole numbers and then the fraction,

    =19415

    Therefore, the result from adding 1523 and 335 is 19415.

Subtraction of Fractions

The process for the subtraction of fractions is the same as that of the addition of fractions. First, determine a common denominator and change each fraction to its equivalent fraction with the common denominator. When the fractions have the same denominator, the numerators of the fractions may be subtracted. The numerator of the resulting fraction is equal to this difference, and the denominator is equal to the common denominator of the fractions being subtracted.

Express the final answer reduced to the lowest terms and as a mixed number, where applicable.

Example 2.4-g: Subtracting Fractions that have the Same Denonimator

Subtract 38 from 78.

Solution

7838

The denominators of the fractions are the same. Subtracting the numerators and keeping the common denominator,

=738

=48=12

Reducing to the lowest terms,

=12

 

7 over 8 (illustrated with 7 slices of the 8 slice pie chart shaded) less 3 over 8 (3 slices of the 8 slice pie chart shaded a different colour) equals 4 over 8 (illustrated with 4 slices of the 8 slice pie chart shaded). 4 over 8 can be further reduced to 1 over 2.

Therefore, the result from subtracting 38 from 78 is 12.

Example 2.4-h: Subtracting Fractions that Have Different Denominators

Subtract 28 from 710.

Solution

71028

LCM of 8 and 10 is 40 (i.e., LCD = 40). Determining the equivalent fractions with a denominator of 40,

28401040

Subtracting the numerators and keeping the common denominator,

281040

1840=920

Reducing to the lowest terms,

920

Therefore, the result from subtracting 28 from 710 is 920.

Example 2.4-i: Subtracting Mixed Numbers

Subtract 723 from 1212.

Solution

Method 1:

1212723

Converting the mixed numbers to improper fractions,

=(12×2)+12(7×3)+23

=252233

LCM of 2 and 3 is 6 (i.e., LCD = 6). Determining the equivalent fractions with a denominator of 6,

=756466

Subtracting the numerators and keeping the common denominator,

=296

Converting the improper fraction to a mixed number,

=456

Method 2:

1212723

LCM of 2 and 3 is 6 (i.e., LCD = 6). Determining the equivalent mixed numbers with a denominator of 6,

=1236746

The fraction 46 is greater than 36. Therefore, we have to regroup the mixed number 1236 by borrowing 1 from 12:

1236=11+1+36=11+66+36=1196

=1196746

Subtracting the whole numbers and then the fractions,

=4(94)6

=456

Therefore, the result from subtracting 723 from 1212 is 456.

Multiplication of Fractions

First, convert any mixed number to its improper fraction to multiply two or more fractions. Then, multiply the numerators together to get the new numerator and multiply the denominators together to get the new denominator.

When multiplying fractions, you can reduce any numerator term with any denominator term. Reduce as much as possible before multiplying the numerators and the denominators together to keep the numbers as simple as possible.

Express the final answer reduced to the lowest terms and as a mixed number, where applicable.

Note: When multiplying mixed numbers, it is incorrect to multiply the whole number parts separately from the fractional parts to arrive at the answer.

Example 2.4-j: Multiplying Fractions

Multiply:

  1. 32×411
  2. 15×25
  3. 318×245

Solution

  1. 32×411=31×211

    Reducing the fractions,

    =31×211

    Multiplying the numerators together and denominators together,

    =611

    Therefore, the result of 32×411 is 611.

  2. 15×25=31×21

    Reducing the fractions,

    =31×21

    Multiplying the numerators together and denominators together,

    =61=6

    Therefore, the result of 15×25 is 6.

  3. 318×245

    Converting the mixed numbers to improper fractions,

    =(3×8)+18×(2×5)+45

    =258×145=54×71

    Reducing the fractions,

    =54×71

    Multiplying the numerators together and denominators together,

    =354

    Converting the improper fraction to a mixed number,

    834

    Therefore, the result of 318×245 is 834

Division of Fractions

When dividing fractions, as in multiplication, first convert any mixed number to its improper fraction. The division of fractions is done by multiplying the first fraction by the reciprocal of the second fraction. Then, follow the procedure used in multiplication to arrive at the final result.

Express the final answer reduced to the lowest terms and as a mixed number, where applicable.

Note:

Dividing by 2 is the same as multiplying by the reciprocal of 2, which is 12.

When multiplying or dividing mixed numbers, it is incorrect to multiply or divide the whole number parts separately from the fractional parts to arrive at the answer.

Example 2.4-k: Dividing Fractions

Divide 1516 by 920.

Solution

1516÷920

Multiplying 1516 by the reciprocal of 920, which is 209,

=1516×209=54×53

Reducing the fractions,

=54×53

Multiplying the numerators together and denominators together,

=2512

Converting the improper fraction to a mixed number,

=2112

Therefore, the result of 1516 divided by 920 is 2112.

Example 2.4-l: Dividing Mixed Numbers

Divide 3320 by 145.

Solution

3320÷145

Converting the mixed numbers to improper fractions,

=(3×20)+320÷(1×5)+45

=6320÷95

Multiplying 6320 by the reciprocal of 95, which is 59,

=6320×59=74×11

Reducing the fractions,

=74

Converting the improper fraction to a mixed number,

=134

Therefore, the result of 3320 divided by 145 is 134.

Complex Fractions

A complex fraction is a fraction in which one or more fractions are found in the numerator or denominator.

For example,

  • 1(58) is a complex fraction because it has a fraction in the denominator.
  • (23)6 is a complex fraction because it has a fraction in the numerator.
  • (25+14)3 is a complex fraction because it has two fractions in the numerator.
  • (56)(18) is a complex fraction because it has a fraction in both the numerator and the denominator.

A complex fraction can be simplified by dividing the numerator by the denominator and then, following the rule(s) for dividing fractions.

1(58)=1÷58=1×55=85

(23)6=23÷6=13×13=19

Example 2.4-m: Simplifying Complex Fractions

Add:

  1. (72)5
  2. 6(98)

Solution

  1. (72)5=72÷5=72×15=710
  2. 6(98)=6÷98=21×83 =513

Powers and Square Roots of Fractions

Powers of fractions are expressed the same way as whole numbers. When the base of a power is a fraction, it is written within brackets.

For example, (23)2 is read as “two-thirds squared”.

  • This means that 23 is used as a factor two times.
  • i.e., (23)2=23×23=49

A mixed number that is raised to a power is evaluated by first converting it into an improper fraction and then following the same procedure explained above.

For example, (123)4 is evaluated by first converting 123 into an improper fraction.

(123)4=(1(3)+23)4=(53)4=(53)(53)(53)(53)=62581 =75881

Example 2.4-n: Evaluating Powers of Fractions

Evaluate the following powers:

  1. (45)4
  2. (112)5

Solution

  1. (45)4

    Expanding by using 45 as a factor four times,

    =(45)(45)(45)(45)=256625

  2. (112)5

    Converting the mixed number into an improper fraction,

    =(1(2)+12)5=(32)5

    Expanding by using 32 as a factor five times,

    =(32)(32)(32)(32)(32)=24332

    Converting back to a mixed number,

    71932

Square roots of fractions are calculated the same way as square roots of whole numbers, but the numerators and denominators are evaluated separately.

For example, 916 is the same as 916=34.

Example 2.4-o: Evaluating Square Roots of Fractions

Evaluate the square root: 25144

Solution

25144

Determining the square root of the numerator and denominator separately,

25144=512

2.4 Exercises

Answers to the odd-numbered problems are available at the end of the textbook.

 

For Problems 1 to 8, identify the greater fraction in each pair.
  1. 25 or 38
  2. 43 or 65
  3. 1215 or 3545
  4. 54 or 76
  5. 87 or 1312
  6. 513 or 1639
  7. 89 or 3945
  8. 38 or 2548
For Problems 9 to 16, perform the addition, reduce to lowest terms, and express the answer as a mixed number, whenever possible.
  1. a. 58+78
    b. 712+34
  2. a. 59+79
    b. 710+920
  3. a. 43+56
    b. 1234+513
  4. a. 2312+13
    b. 1857+225
  5. a. 934+616
    b. 823+534
  6. a. 1114+523
    b. 7112+534
  7. a. 110+17100+391,000
    b. 35+710+915
  8. a. 310+47100+2411,000
    b. 23+34+58
For Problems 17 to 22, perform the subtraction, reduce to lowest terms, and express the answer as a mixed number, whenever possible.
  1. a. 2319
    b. 91235
  2. 1618
    b. 1920310
  3. a. 5338
    b. 1618112
  4. a. 17956
    b. 5231512
  5. a. 856539
    b. 9257310
  6. a. 8512436
    b. 558456
For Problems 23 and 24, perform the mixed additions and subtractions, reduce to lowest terms, and express the answer as a mixed number, whenever possible
  1. a. 3210081,000+325
    b. 58+131634
  2. a. 31041,000+5100
    b. 712+5623
For Problems 25 to 30, perform the multiplication, reduce to lowest terms, and express the answer as a mixed number, whenever possible.
  1. a. 165×54
    b. 3×79
  2. a. 125×253
    b. 69×1912
  3. a. 38×511
    b. 935×12996
  4. a. 45×239
    b. 1134×1174
  5. a. 938×1963
    b. 229×112
  6. a. 1527×1845
    b. 237×2545
For Problems 31 to 36, perform the division, reduce to the lowest terms, and express the answer as a mixed number whenever possible.
  1. a. 23÷49
    b. 38÷4
  2. a. 35÷34
    b. 17÷35
  3. a. 1015÷37
    b. 2312÷81316
  4. a. 812÷24
    b. 1014÷22748
  5. a. 515÷13
    b. 18÷445
  6. a. 514÷7
    b. 15÷313
For Problems 37 and 38, express the complex fractions as a proper fraction or a mixed number, where possible.
  1. a. 1(94)
    b. 434(38)
  2. a. 1(112)
    b. 849(112)
For Problems 39 to 44, evaluate the powers of the fractions.
  1. a. (35)2
    b. (67)2
  2. a. (34)2
    b. (29)2
  3. a. (34)3
    b. (53)4
  4. a. (27)3
    b. (65)4
  5. a. (113)2
    b. (312)3
  6. a. (214)2
    b. (123)3
For Problems 45 to 50, evaluate the square roots of the fractions.
  1. a. 19
    b. 149
  2. a. 116
    b. 110,000
  3. a. 425
    b. 8116
  4. a. 36100
    b. 14481
  5. a. 3116
    b. 614
  6. a. 11125
    b. 121100
For Problems 51 to 72, express your answers as a proper fraction or a mixed number, where appropriate:
  1. Peter spent 512 of his money on rent and 14 on food. What fraction of his money did he spend on rent and food?
  2. Alan walked 35 km to his friend’s house and from there, he walked another 34 km to his school. How far did Alan walk?
  3. Last night, Amy spent 316 hours on her math project and 2310 hours on her design project. How much time did she spend on both projects altogether?
  4. A bag contains 235 kg of red beans and 118 kg of green beans. What is the total weight of the bag?
  5. Thomas baked a 212 pound cake. He gave 158 pounds of it to his friend Yan. How much was left?
  6. Alexander bought 425 litres of milk and drank 123 litres of it. How much milk was left?
  7. Sarah had 34 kg of cheese. She used 27 kg of the cheese while baking. How many kilograms of cheese was left?
  8. Cassidy bought 58 litres of olive oil and used 13 litre of the oil while cooking. What quantity of olive oil was left?
  9. David spent 710 of his money on toys and 13 of the remainder on food. What fraction of his money was spent on food?
  10. Mary spent 25 of her money on a school bag. She then spent 13 of the remainder on shoes. What fraction of her money was spent on shoes?
  11. After selling 25 of its textbooks, a bookstore had 810 books left. How many textbooks were in the bookstore initially?
  12. Rose travelled 35 of her journey by car and the remaining 20 km by bus. How far did she travel by car?
  13. Cheng can walk 514 km in 112 hours. How many kilometres can he walk in 1 hour?
  14. 234 litres of juice weighs 423 kg. Determine the weight (in kilograms) of 1 litre of juice.
  15. A chain of length 78 metres is cut into pieces measuring 116 metres each. How many pieces are there?
  16. A cake that weighs 23 kg is cut into slices weighing 112 kg each. How many slices are there?
  17. A bottle of medicine contains 80 mg of medicine. Each dose of the medicine is 25 mg. How many doses are there in the bottle?
  18. A box of cereal contains 917 grams of cereal. How many bowls of cereal will there be if each serving is 3234 grams?
  19. Out of 320 bulbs, 120 of the bulbs are defective. How many of them are not defective?
  20. If 415 of the 1,800 students in a school enrolled for a mathematics course, how many students did not enroll for the course?
  21. The product of two numbers is 9. If one number is 334, what is the other number?
  22. If a wire that is 4234 cm long is cut into several 214 cm equal pieces, how many pieces would exist?

Unless otherwise indicated, this chapter is an adaptation of the eTextbook Foundations of Mathematics (3rd ed.) by Thambyrajah Kugathasan, published by Vretta-Lyryx Inc., with permission. Adaptations include supplementing existing material and reordering chapters.

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