Business Mathematics: Review of Arithmetic - Fractions

Fractions are used to represent quantities which cannot be represented as whole numbers. A fraction is written in the form alb. a is called the numerator and b is denominator of a fraction can never be 0.
A quantity can be written as a fraction many different ways. For example, 1/2, 2/4, and 3/6 all represent the same value. The essential feature of fractions, which will be used extensively, is the following “fundamental principle”:
The value of a fraction is unchanged if its numerator and denominator are multiplied or divided by the same nonzero number.

Example 15
Express the fraction 2/5 three equivalent ways. If we multiply the numerator and denominator by 2, 3, and 7 the fraction becomes, respectively, 4/10, 6/15, and 14/35. Any nonzero numbers may be used as multipliers.

Example 16
Simplify the appearance of 21/35 without changing its value. Since the numerator and denominator are both divisible by 7, the fraction can be reduced to 3/5.
A fraction is reduced to lowest terms if its numerator and denominator cannot be divided by the same whole number.

Example 17
Reduce 154/182 to lowest terms. Since both numerator and denominator are even, they can both be divided by 2. Applying the fundamental principle gives 77/91. Both 77 and 91 are divided by 7 to yield 11/13. Since the numerator and denominator cannot be divided by the same whole number, the fraction is reduced to lowest terms.
As a check to see if two fractions have the same value, we can use the following principle which is sometimes called cross-multiplication:


Example 18
Check that 154/182 = 11/13.154 × 13 = 2,002 and 182 × 11 = 2,002. Therefore the fractions are equal. (Note that there is no particular significance to the number 2,002, only that you get the same value two different ways.)
If two or more fractions have the same denominator we say that they have a common denominator. To add or subtract fractions with a common denominator, simply add or subtract their numerators and keep the common denominator.

Example 19
Combine as a single fraction: 2/13 + 5/13 – 3/13. Since the fractions have a common denominator, 13, we add and subtract the numerators: 2 + 5 – 3 = 4. The answer is 4/13.

Example 20
Combine as a single fraction: 1/20 + 3/20 – 7/20 + 17/20. Since 1 + 3 – 7 + 17 = 14, the answer is 14/20, which can be reduced to 7/10.
To combine fractions with different denominators, we must convert each fraction into an equivalent fraction with a common denominator. If the denominators have no common factors, i.e., they cannot both be divided by the same whole number, the common denominator is the product of the two denominators.

Example 21
To add 2/3 + 3/5 we first have to find a common denominator. In this problem the common denominator is 5 × 3 = 15. By the fundamental principle, 2/3 = 10/15 and 3/5 = 9/15 so 2/3 + 3/5 = 10/15 + 9/15 = 19/15.
We can obtain a common denominator by multiplying the individual denominators, but this often leads to large numbers which are difficult to work with. It is sometimes possible to obtain a smaller common denominator which will work more efficiently. The smallest possible common denominator is called the least common denominator (LCD).

Example 22
Add 5/24 + 7/36.
To determine the LCD, obtain the prime factorizations of the denominators: 24 and 36. Then compute the highest power (exponent) of each prime and multiply.
For this example, 24 = 23 · 31 and 36 = 22 · 32. The highest power of 2 is 23 and the highest power of 3 is 32. Thus our LCD is 23 · 32 = 72. Since 5/24 = 15/72 and 7/36 = 14/72 by the fundamental principle, 5/24 + 7/36 = 15/72 + 14/72 = 29/72.

Example 23
Combine as a single fraction: 5/12 + 11/18 – 7/15 – 1/10.

To multiply two or more fractions, multiply their numerators and multiply their denominators. Then reduce, if possible.

Example 24
Multiply 3/5 by 6/7:


Example 25
Multiply 2/9, 3/4, and 1/2:

To divide two fractions, invert the second fraction and multiply.

Example 26
Divide 3/5 by 6/7.
First, 3/5 ÷ 6/7 is rewritten as 3/5 × 7/6. Multiplication yields 21/30 = 7/10.
A mixed number is a combination of whole number and fraction. The number is an example of a mixed number. To convert a mixed number to a fraction, multiply the whole number by the denominator of the fraction, add the numerator, and place that result over the original denominator.

Example 27
Convert to a fraction: 3 × 5 + 2 = 17, so the fraction is 17/5.
An improper fraction is a fraction whose numerator is larger than its denominator. To convert an improper fraction to a mixed number, divide numerator by denominator. This determines the whole number part. The remainder, if not 0, is placed over the denominator to determine the fractional part of the mixed number.

Example 28
Convert 67/12 to a mixed number. If you divide 67 by 12 the quotient is 5 with remainder 7. Thus the mixed number is .

Solved Problems

1.7   Express the fraction 5/7 three equivalent ways.
Solution
If we multiply the numerator and denominator by the same number, the value of the fraction does not change. Multiply, for example, by 2, 3, and 4:


1.8   Reduce to lowest terms.

Solution


1.9   Check that

Solution
1,575 × 7 = 11,025    2,205 × 5 = 11,025
Since we get the same number, 11,025, both ways, the fractions are equivalent.

1.10  Combine as a single fraction: 1/11 – 3/11 + 5/11 – 2/11.
Solution
Since the denominators are the same, we combine the numerators and place the result over the denominator: (1 – 3 + 5 – 2)/ll = 1/11.

1.11  Add 5/12 + 7/18.
Solution
The LCD is 36: 5/12 = 15/36 and 7/18 = 14/36, so 5/12 + 7/18 = 15/36 + 14/36 = 29/36.

1.12  Combine as a single fraction: 1/5 + 1/3 – 1/7.
Solution
Since the denominators are all primes, the LCD is simply 5 × 3 × 7 = 105. Now, 1/5 = 21/105, 1/3 = 35/105, and 1/7 = 15/105, so 1/5 + 1/3 – 1/7 = 21/105 + 35/105 – 15/105 = (21 + 35 – 15)/105 = 41/105.

1.13  Combine as a single fraction: 5/12 + 7/18 – 3/20.
Solution
To determine the LCD, factor each denominator into primes. 12 = 22 · 31,18 = 21 · 32, and 20 = 22 · 51. Take the highest exponent of each prime factor and multiply to get LCD = 22 · 32 · 51 = 180.
Next, convert each fraction to an equivalent fraction having a common denominator of 180. 5/12 = 75/180, 7/18 = 70/180, and 3/20 = 27/180. Now add: 75/180 + 70/180 – 27/180 = 118/180. This can be reduced to 59/90 by dividing numerator and denominator by 2.

1.14  Multiply 4/7 by 5/8.
Solution
4/7 × 5/8 = 20/56
which can be reduced to 5/14.

1.15  Divide 3/11 by 6/7.
Solution


1.16  Convert the mixed number 5 2/3 to a fraction.
Solution


1.17  Convert the fraction 23/5 to a mixed number.
Solution
Divide 23 by 5 to get a quotient of 4 with a remainder of 3. 23/5 =