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The Humongous Book of Algebra Problems by W. Michael Kelley

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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 4.264.28.
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 4.264.28.
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
. Youre allowed
to do that because
, and
multiplying any number
by 1 doesnt 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 didnt
work, youd test
to see if
were the LCD. If
27 didnt work, youd
multiply 9 by 4, then
5, then 6, and so on,
until nally all the
denominators
divided in
evenly.

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