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1995-03-22
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êê SUBTRACTING FRACTIONS, ELEMENTARY LEVEL
è In this section we will be looking at subtracting positive frac-
tions.ïIn every case at this level, we will subtract a smaller frac-
tion from a larger fraction.ïThis means that the answer will al-
ways be a positive fraction in return.ïWe will begin our study of sub-
tracting by considering two cases.ïThe first case will involve sub-
tracting positive fractions that have the same denominators.ïCase two,
a more general problem type, includes subtracting positive fractions
that have different denominators.ïThe positive fractions and zero are
described in the following list.
êêêêPositive Fractions
#êêêï╚è╔è╩è╦è╠è═è╬ ...
êêêï1è1è1è1è1è1è1
#êêêï╚è╔è╩è╦è╠è═è╬ ...
êêêï2è2è2è2è2è2è2
#êêêï╚è╔è╩è╦è╠è═è╬ ...
êêêï3è3è3è3è3è3è3
êêêè .
êêêè .
êêêè .
Case 1)èSubtracting Positive Fractions With the Same Denominators
è To subtract positive fractions that have the same denominators, you
should write down the common denominator once and then subtract the nu-
merators.
Example 1)
è To subtract 3/8 from 7/8, you should write down the common denomi-
nator once then subtract the numerators.ïWith subtraction it is very
important which fraction is written first and which is second.ïThe
fraction doing the subtracting always comes second.ïIn this problem,
since 3/8 is being subtracted from 7/8, 3/8 must be written second.
êê 7è3êï7 - 3êï4ê 1
#êê ─ - ─è =è ─────è =è ─è =è─
êê 8è8êè 8êè 8ê 2
The answer is reduced to lowest form.
Case 2)ïSubtracting Positive Fractions with Different Denominators
è To subtract positive fractions that have different denominators, it
is first necessary to express the denominators in prime factored form.
To write a denominator in prime factored form, you should break it down
into products of prime numbers.ïThe prime numbers are described in the
following list.
êêè 2, 3, 5, 7, 11, 13, 17, 19,...
Each of ç numbers has the property that the only factors of each
number are "1" and the number itself.
Example 2)ïExpress the number, 6, in prime factored form.ïSince the
number, 6, can be factored into the product of the two prime numbers
2 and 3, the prime factorization of 6 is 2∙3.
Example 3)ïExpress the number, 18, in prime factored form.ïYou should
start with the smallest prime number, 2, and see if it divides evenly
into 18.ïSince 2 goes into 18 nine times, you can express 18 as 2∙9.
Also, since the next smallest prime number, 3, divides evenly into 9,
18 can be expressed as 2∙3∙3.ïSince ç factors are all prime num-
bers, the prime factorization of 18 is 2∙3∙3.
Example 4)
è To subtract the fractions, 3/4 - 2/3, it is first necessary to
express the denominators in prime factored form.
êêêè3ë2êè3ë 2
#êêêè─ï-ï─è =è ───ï-ï─
êêêè4ë3êï2∙2ë3
At this point you can see that the second fraction is missing a factor
of "2∙2" in its denominator, and the first fraction is missing a factor
of 3.ïIt is necessary to multiply both the top and bottom of the first
fraction by 3, and the second fraction by 2∙2.
êêê3ë 2êè3ï3è 2è2∙2
#êêë ───ï-ï─è =è ─── ─ +ï─ ∙ ───
êêë 2∙2ë3êï2∙2∙3è 3è2∙2
Now, both denominators have the same factors, and you can multiply to
simplify the form of the problem.
êêë3ï3ë2è2∙2êè9ë 8
#êêè ─── ─ï-ï─ ∙ ─ ─è =è ──ï-ï──
êêè 2∙2∙3ë3è2∙2êï12ë12
Since the two fractions have the same denominators, you can write down
the denominator once and subtract the numerators like we did in Exam-
ple 1.
êêê 9ë 8êï9 - 8êè1
#êêê──ï-ï──è =è ─────è =è ──
êêê12ë12êè 12êè12
Thus, the difference between 3/4 and 2/3 is 1/12.
è Another way to subtract fractions is to add them in a column.ïLeon
the Fraction Wizard prefers to use the method in the above examples, and
his method should be considered to be correct and general in the sense
that it always works no matter how big the numbers.èMany people, how-
ever, prefer to subtract fractions in a column.ïIt is still necessary
to find the least common denominator when you subtract fractions in a
column, and it is perfectly alright to just write down the least common
denominator if you can identify it by inspection.ïYou can always go
back to the prime factorization method if the numbers are too large to
identify the least common denominator by inspection.
Example 5)
êSubtract the fraction 2/3 from 3/4 using the column approach.
First, you should identify the least common denominator and write it
down next to the original problem.ïThen, you can find the missing nu-
merators by dividing and multiplying.ïFinally, the resulting fractions
should be subtracted.
êêêï3êêê9
#êêêï─è =è ──è =è ──
êêêï4êï12êï12
êêêï2êêê8
#êêë-è─è =è ──è =è ──
êêêï3êï12êï12
#êêê────ê────ê────
êêêêêêè1
#êêêêêêï──
êêêêêêï12