20. Fraction Division - The Why
See the previous page for an introduction of the fraction division
methods or formulas below. .
-
Two Step Development Option -
Explanation in two smaller steps, first with like denominators and
second with unlike denominators (Raising terms in dividend and divisor
turns the second case into the first - the comprehension is in the
details, you have to see them.)
-
One Step Development Option -
Explanation in one big step, Algebraically with numerical examples
Each development complements the other.
Step I: Division of Fractions with Like (common) Denominators
Simple Division Question: How many whole times does 5
quarters go into 30 quarters?
Answer: 6 times exactly as 6 × 5 = 30.
Geometric Model: Declare the length of a
line segment (measured or not) to be a unit length. Illustrate
the foregoing by drawing 30 quarters of it. Then show that grouping the
30 quarters into groups of five gives 6 groups.
General Division Question: How mixed number of times does 8
sevenths go into 19 sevenths?
Answer: This answer depends on long division and the
definition yet to come of multiplication of fractions with unlike
denominators:
By long division or inspection: 19 = 2 × 8 +
3. So 8 sevenths goes into 19 sevenths, 2 whole
times with 3 sevenths left-over. But a seventh is one eighth of 8
sevenths. So 3 sevenths is 3 eighths of 8 sevenths.
Thus 8 sevenths goes into 19 sevenths,
19
8
|
or 19 ÷ 8 =
|
2 +
|
3
8
|
times
|
. Check:
|
2
|
× 8 sevenths = 16 sevenths.
|
|
while
|
3
8
|
× 8 sevenths = 3 sevenths.
|
|
By addition
|
|
|
|
|
× 8 sevenths = 19 sevenths as 16 + 3 =
19
|
Geometric Thought: Declare the length of a
line segment (measured or not) to be a unit length. Illustrate the
foregoing by drawing 19 sevenths of it. Then show that grouping the 19
elevenths into groups of eight gives two groups of 8 and 3 left
over sevenths - the remainder Declare each one left over to an
eighth of 8. So the three left overs are three
eighths.
Revision: We could have written
|
2
|
× 8 ones = 16 ones.
|
|
while
|
3
8
|
× 8 ones = 3 ones.
|
|
By addition
|
|
|
|
|
× 8 ones = 19 ones as 16 + 3 ones = 19
ones
|
where one is a pronoun for any object.
General Division Formula:
How many times does N objects divide into P objects
when P = q × N + r.
Answer:
P objects
N objects
|
or (P objects) ÷ (N
objects) =
|
P
N
|
= (q +
|
r
N
|
) times
|
|
The quotient
|
P
N
|
can be expressed as a mixed
number (q +
|
r
N
|
)
|
Here P = qN + r where r is a natural number (0 < r <
N) implies P instances of an M-th can be grouped into q
groups of N with a remainder of r, which may viewed a
r N-ths of N.
Now apply the subformula
|
(P objects) ÷ ( N
objects) =
|
P
N
|
not to an object but to an M-th of another object. 'object'
|
|
|
(P times an M-th of an 'object') ÷ ( N times an M-th of an
'object')
|
|
Now we rewrite the foregoing with writing of an object as
|
|
=
|
|
|
since
|
|
|
|
|
P
M
|
=
|
P times an M-th
|
|
and
|
|
|
|
|
|
|
N
M
|
=
|
N times an M-th
|
Conclusion: Formula for division of fractions with like
denominators:
Verification:
P
N
|
×
|
N
M
|
=
|
P
|
times
|
an N-th
|
of
|
N
M
|
|
|
|
=
|
P
|
times
|
1
M
|
|
|
|
|
|
=
|
P
M
|
|
|
|
|
Step II: Division of Fractions with Unlike Denominators
Numerical Example
Saying or showing how to do operation defines it. The operation
follows again by raising terms to obtain like
denominators.
Example:
6
11
|
÷
|
12
33
|
=
|
6×33
11×33
|
÷
|
11×12
11×33
|
by raising numerators
and denominators
|
|
|
|
=
|
6×33
11×12
|
by the special of division of fractions
with like denominators
|
|
|
|
|
|
|
|
|
|
=
|
3
2
|
by cancellation of common
factors 6 and 11
|
|
|
|
=
|
1½
|
by a cosmetic preference
|
General Pattern by following the numerical one
P
B
|
÷
|
N
M
|
=
|
P×M
B×M
|
÷
|
B×N
B×M
|
by raising numerators
and denominators
|
|
|
|
=
|
P×M
B×N
|
by the special of division of fractions
with like denominators
|
|
|
|
=
|
P
B
|
×
|
M
N
|
an equality that follow from
the fraction product formula
|
Conclusion: Formula for Division with unlike denominators.:
This is the same formula with obtained and checked in the first big step
development.
Above Example Revisited.
6
11
|
÷
|
12
33
|
=
|
|
multiplication by reciprocal
|
|
|
|
|
6×33
11×12
|
by rule for product
calculation
|
|
|
|
=
|
3
2
|
by cancellation of common
factors 6 and 11 again
|
|
|
|
=
|
1½
|
by a cosmetic preference
|
|
Read ÷ as divide by
Now in general, we write
when and only when the equation holds.
So T answers the questions: How may fractional multiples of A/B
give M/N? Or, how fractional times does the fraction
(length?) A/B units go into the fraction M/N units?
Claim: The reciprocal
works.
Check:
|
T
|
×
|
A
B
|
=
|
(
|
M
N
|
×
|
B
A
|
)
|
×
|
A
B
|
=
|
M×B×A
N×A×B
|
=
|
M
N
|
The last equality follows by regrouping and lowering
terms.
First Example Revisited: How many times
does ¾ goes into 3½ = (7/2)?
|
Answer:
|
T
|
=
|
7
2
|
÷
|
3
4
|
=
|
7
2
|
×
|
4
3
|
=
|
7
1
|
×
|
2
3
|
=
|
4
|
2
3
|
as before
|
Our conclusion is that division by a fraction is computed by
multiplying by its reciprocal.
Another Example:
13
8
|
÷
|
39
16
|
=
|
13
8
|
×
|
16
39
|
=
|
13
8
|
×
|
2×8
3×13
|
=
|
2
3
|
Two Checks (Only one is needed) <== read this like a lawyer
|
39
16
|
×
|
2
3
|
=
|
3×13
2×8
|
×
|
2
3
|
=
|
13
8
|
|
Or
|
|
|
|
|
|
|
|
|
|
|
2
3
|
×
|
39
16
|
=
|
2
3
|
×
|
13×3
8×2
|
=
|
13
8
|
|
|
|
|
|
|
|
|
|
|
The foregoing says (13/8) is exactly (2/3)rds
of (39/16).
One More Example:
8
5
|
÷
|
16
45
|
=
|
8
5
|
×
|
45
16
|
=
|
8
5
|
×
|
9×5
2×8
|
=
|
9
2
|
=
|
4½
|
Two Checks (Only one is needed): <== Read this like a
lawyer
|
16
45
|
of
|
4½
|
=
|
16
45
|
×
|
9
2
|
=
|
2×8
9×5
|
×
|
9
2
|
=
|
8
5
|
|
Or
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
4½
|
times
|
16
45
|
=
|
9
2
|
×
|
16
45
|
=
|
9
2
|
×
|
2×8
9×5
|
=
|
8
5
|
|
|
|
|
|
|
|
|
|
|
|
|
|
|
Remember: division by a fraction is computed by multiplying
by its reciprocal.
|
|
Secondary
Mathematics for Ages 11+, A Practical Approach for home-tutoring or -schooling, or for schools & colleges
with local curriculum control. Study how to include site content - its skill development how-TOs and innovations
into present or future lesson plans - some reading required.
Road
Safety Messages and Questions: When and why should you face
traffic when walking along a road or cycle path? Is it a good
idea to hang limbs outside of cars etc? What gives more
protection in a crash: a car, motorbike or bicycle?
See too, the BBC-Belgium story Texting and
Driving - texting & the impossible test - the article links to a gruesome utube video on the subject
The Logic of Injustice:
How Texas sent
an innocent man to his death - The wrong Carlos. Some judgments are irreversible. Procescution: Where and when prosectors play to win rather than for
justice, guilt beyond a reasonable doubt goes unrespected due to prosecutors who putting winning
first, those innocence before the law may be convicted. Some procescutors offices in continuing to accuse after a pardon
due to reasonable doubt or innocent being shown, may sucessfully oppose compensaton for false convictions
by asserting a pardon individual is still under suspicion. Then the pardoned individual or the latter's estate
is not compensation for years or decade
of improper or false imprisonment, or for execution. Site chapters on Logic
and some in Pattern
Based Reason may slowly lead to greater precision in reading, applying and
writing laws.
May 2012, Composition Starting:
Pre-School and Primary Mathematics - Quantitative Skills, An
Intellectual View, Feedback Welcome:
The 8 Most Popular Site Inlinks
Parent Center: Help your child or teen
learn:
Parent-friendly
Work Booklets for ages 3+ to 13 Use these or others to check
or build skills. Other booklets are available but these booklets
allow parents unsure of themselves in mathematics to help their
children. The selection acquired in Canada is published in the
USA. So it has a US orientation. In retrospect, the selection
shows parents what to check with the booklets or by other ways,
the choice is theirs. But in retrospect, the selection does not
cover integral and fractions liquid weights and measures - ask
the publishers to correct that! For ages 9 to 12 say, parents may
compensate by showing boys and girls how to use weights or mass,
and further measures in food preparation. Beyond that children
may be shown how to measure and calculate angles, lengths and
areas [proportional amounts too] directly or by using maps and
plans drawns to scale. Learning how to gather and measure all the
ingredients, pots and pans for a dish or a meal, along with
cleaning up sets the stage for like activities or experiments in
science courses, and in developing organizational skills,
gives boys and girls a head start. Good luck. At the other
extreme, more comprehensive than light, if your motto is
McCainian: drill, drill, drill then Toronto
mathematician and actor John Mighton's jump math organization has jump math
workbooks for at least grades 3 to 8 for at-home and in-school
use - training sessions for teachers available. Jump math has
been expanding to cover older students. Jump Math Samples: plus
Fractions for
Grades 3-4 & Grades 5-6 [Read] Free Resources grades 1 to 8
[unread - likely to be good]. and
Mathematics
Skills For Ages 3 to 14 - technical!
Skills with take
home value - A few ideas
Basic skills include
time-date-calendar Matters; money matters; map, plan and
scale diagram matters;counting, measuring and figuring;
decision making with logic and likelyhood; being careful and
being aware of the domino effect of mistakes; reading and
writing with precision.
Is your child able to add, subtract and multiply amounts
of money, work with fractions, work with clocks and calendars,
work with maps and plans, and measure length, weight-mass and
volume? Schools may promote your son or daughter without
providing basic skills in reading, writing and
arithmetic.
Arithmetic
and Number Theory Skills
Algebra
Starter Lessons
Geometry
- maps plans trigonometry vectors
More
Algebra
70
Calculus Starter Lessons
Calculus Lessons Elsewhere:
-
How to Ace Calculus: Street Wise Guide - Mostly
Text.
-
Flash
Video for Calculus Phobics
They cover basic topics in ways likely to complement your
notes, your textbooks and site material. When Goldilocks
trespassed in the house of the three bears, she found three bowls
of porridge, two not to her liking, and one just right. Different
bears have different tastes. As invited guest here and elsewhere,
if one or more explanations is not to liking, try another. It may
be better or just right.
Unsolicited Advice
Learning to do and high marks if it comes to easy is often
deceptive - light rather than deep. For that reason, students
with learning difficulties determined not to let it get in their
way may go deeper and farther than those with none. High marks,
if the come easy, may be deceptive - provide a too light and not
a deep mastery. That could have been your problem in secondary
school, one that leads to comprehension shock or difficulties in
calculus and more generally in the first year of college. Bon
Appetite.
|
|