Chapter Two — Rational Numbers

The Humongous Book of Algebra Problems

26

In this case, the repeated string is consists of one digit (5). To convert into a

fraction, divide the repeated string by 9.

The rationale behind this shortcut is omitted here, as it is based on skills

not discussed until Chapter 4. This technique, in its more rigorous form, is

explained in greater detail in Problems 4.26–4.28.

2.21 Express as a fraction.

The repeated string of 0.727272 … consists of two digits, so divide the repeated

string (72) by two nines (99) and reduce the fraction to lowest terms.

Remember that this technique applies only when the repeated string of digits

begins immediately to the left of the decimal point. If the repeated string

begins farther left in the decimal, you should apply the technique described in

Problems 4.26–4.28.

Combining Fractions

Add, subtract, multiply, and divide fractions

2.22 Explain what is meant by a least common denominator.

Equivalent fractions might have different denominators. For instance, Problem

2.13 demonstrated that and have the same value, as is expressed

in lowest terms. It is often useful to rewrite one or more fractions so that their

denominators are equal. Usually, there are numerous options from which you

can choose a common denominator, and the least common denominator is the

smallest of those options.

Note: Problems 2.23–2.25 refer to the fractions and .

2.23 Identify the least common denominator of the fractions.

Begin by identifying the largest of the given denominators; here, the largest

denominator is 10. Because the other denominator (2) is a factor of 10,

then 10 is the least common denominator (LCD). The LCD is never

smaller than the largest denominator.

If the

repeated string

is two digits long,

divide by 99. If it’s

three digits long,

divide by 999.

In other

words,

has

no remainder.

Chapter Two — Rational Numbers

The Humongous Book of Algebra Problems

27

Note: Problems 2.23–2.25 refer to the fractions and .

2.24 Generate equivalent fractions using the least common denominator.

To rewrite using the least common denominator, divide the LCD by the

current denominator: . Multiply the numerator and denominator of

by that result.

Because already contains the least common denominator, it does not need

to be rewritten.

Note: Problems 2.23–2.25 refer to the fractions and .

2.25 Calculate the sum of the fractions.

To calculate the sum or difference of fractions, those fractions must have a

common denominator. According to Problem 2.24, .

Add the numerators of the fractions, but not the denominators.

Unless otherwise directed, you should always reduce answers to lowest terms.

Note: Problems 2.26–2.27 refer to the fractions , , and .

2.26 Identify the least common denominator.

The largest denominator of the three fractions is 9. However, both of the

remaining denominators are not factors of 9, so 9 is not the LCD. To identify

another potential LCD candidate, multiply the largest denominator by 2:

. All the denominators (3, 6, and 9) are factors of 18, so it is the LCD.

Multiplying

the top and

bottom of the

fraction by 5 is

like multiplying the

entire fraction by

. You’re allowed

to do that because

, and

multiplying any number

by 1 doesn’t change

it, according to the

multiplicative

identity property

in Problem

1.36.

In other

words, if you

want to ADD

or SUBTRACT

fractions ….

If 18 didn’t

work, you’d test

to see if

were the LCD. If

27 didn’t work, you’d

multiply 9 by 4, then

5, then 6, and so on,

until nally all the

denominators

divided in

evenly.

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