Section Topics
Fraction,
Fraction with Units, Fractions & Ratios; and Proportionality
forwards & backwards.
Section Pages
Fraction How-TOs
1 What is a Fraction
2 Fraction Multiplication I
3 Fraction Multiplication II
4 Fraction Multiplication III
5 Equivalent Fractions
6 - Products Algebraically
7. Mixed Numbers Etc.,
8. Fraction Comparison, Etc
9 Fraction Addition I
10. Fraction Addition II
11. Add, Subtract, Compare Similarities
12. Fraction Addition III
13 Fraction Multiplication IV
14. Fraction Division & Reciprocals
15. Division Formulas Justified
16. Fraction Webvideos
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Fraction Division Methods Explained
See the previous page for an introduction of the fraction division methods or
formulas below. .
- One Step Development Option - Explanation in one big step, Algebraically with
numerical examples
- 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.)
Each development complements the other.
Short Development - Division of Fractions in One Big Step
(Algebraic Development)
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 |
|
|
|
|
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|
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Remember: division by a fraction is computed by multiplying by its
reciprocal.
|
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 |
| |
|
|
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