9. Improper Fractions and Mixed Numerals
An improper fraction may be regarded as multiple of a unit-numerator
fraction: one half, one third, one quarter, one fifth and so one. But in
those fractions, the
numerator = a whole number × the denominator + a remainder
where the remainder is less than the denominator.
Identify each group of 2 halves, three
thirds, four quarters and in general N times an N-ths
with the whole number one. Or, we may regard the number one
as a group 2 halves, three thirds, four quarters and in general
N times an N-ths. There in lies the basis for regrouping
and so rewriting an improper fraction as a mixed number - a whole
number plus a proper fraction.
Example (A)
The improper fraction
46
11
|
=
|
4×11 + 3
11
|
|
|
|
|
=
|
4 ×
|
11
11
|
+
|
3
11
|
|
=
|
4 ×
|
1
|
+
|
3
11
|
|
=
|
4
|
+
|
3
11
|
|
|
=
|
4
|
3
11
|
|
|
Thus 46 elevenths may be expressed as mixed counts of ones and elevenths,
where the number of 11-ths is less than 11. So we are rewriting or
re-forming 46 elevenths hundredths as a count of ones and plus a
count of elevenths. That gives a mixed number, a whole number plus a
fraction.
Remark 1 - A hazard: The notation for mixed numbers is an
exception to the rule in algebra that two numbers written side by side
indicates a multiplication. Another exception is given by multi-digit
decimal notation.
Example (B)
350
100
|
=
|
7×50
2 ×50
|
|
=
|
7
2
|
|
=
|
3×2 + 1
2
|
|
=
|
3 + ½
|
|
=
|
3½
|
Now 350 one hundredths
350
100
|
=
|
3 ones + one half
|
has the same value as 3 ones (one be a hundred hundreths) and one half -
the latter being 50 one hundredths. So we are rewriting or re-forming 350
hundredths as a count of ones and halves.
Example (B) Revisited
We may like wise say the improper fraction
350
100
|
=
|
3×100 + 50
100
|
=
|
3 + ½
|
=
|
3½
|
|
|
3 ×
|
100
100
|
+
|
50
100
|
|
|
|
3 ×
|
1
|
+
|
50
100
|
|
|
=
|
3
|
+
|
50
100
|
|
|
|
=
|
3
|
50
100
|
|
|
|
|
=
|
3
|
1
2
|
|
|
|
Here we following the route of converting an improper fraction
(numerator greater than denominator) to a mixed number before reducing
terms in the resulting proper fraction part. . The proper
fraction with a smaller numerator may be easier to factor than
the numerator in the improper fraction.
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Teachers & Tutors: Site pages offer better or best practices for providing skills -
simpler than expected & comprehensive but for exercises. For your charges, your duty is to study them alone or in
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Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
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Arithmetic
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Algebra
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Calculus Starter Lessons
Calculus Lessons Elsewhere:
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They cover basic topics in ways likely to complement your
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Unsolicited Advice
Learning to do and high marks if it comes to easy is often
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Appetite.
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