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YOU are better than YOU think. Show yourself how:
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Logic
Mastery Do not leave here without it - Logic mastery will develops critical thinking, improve reading and writing, and give a firmer base for work and studies at many levels. Good luck.
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-/[]\- After logic, (a) continue reading Three Skills for Algebra, chapters 8 to 14 and do so alongside site area on solving linear2007 Equations ; or (b) see this calculus starter lesson and Volume 3, Why Slopes & More Math, chapters 2 to 6; For online automated help in senior high school maths & calculus, visit quickmath.com For Automatic Calculus and Algebra Help with derivatives, integrals, graphs, linear equations, matrix algebra, visit calc101.com With overlap, each site quickmath & calc101offers a different range of services, some free, some not, all based on webmathematica. Good luck. |
12. Division of Fractions and Compound Fractions
In comparing lengths, we may ask how many times a shorter length goes into a longer one. The result may be a whole number plus a part left over. When the latter is a fraction of the shorter length, the number of times the shorter goes into the longer is a mixed number, a mixed number equivalent to an improper fraction. If the shorter length is a unit length, then the number of times the shorter length goes into the longer length may be measured or estimated with a ruler or tape measure.
In comparing lengths, we may ask how many times a longer length goes into a shorter one. The number of times may be a fraction. The shorter length may be say a half, a quarter or seven tenths or another fraction of or times the longer length.
Division
The following diagram indicates that the fraction ¾ goes into 3½ units, 4 full times with ½ left over. The ½ is two-thirds of ¾.
| 1 | 2 | 3 | 3½ | 4 | 5 |
|
3½ |
|||||||||||||||||||
We see that
| 4 | 2 3 |
* | 3 4 |
= | 3+ | 2 3 |
* | 3 4 |
= | 3+ | 2 4 |
= | 3½ |
So we put
| 3½ ¾ | = | 4 | 2 3 |
We say 3½ divided by ¾ is
| 4 | 2 3 |
We also say 3½ is ¾ is of
| 4 | 2 3 |
Algebraic Shorthand Description of Ideas
Read ÷ as divide by
Now in general, we write| M N |
÷ | A B |
= | T |
when and only when
| T | * | A B |
= | M N |
Here the reciprocal
| T | = | M N |
* | B A |
works.
Check:
| T | * | A B |
= | ( | M N |
* | B A |
) | * | A B |
= | M*B*A N*A*B |
= | M N |
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 Examples:
| 13 8 |
÷ | 39 16 |
= | 13 8 |
* | 16 39 |
= | * | 2* 3* |
= | 2 3 |
Check:
| 39 16 |
* | 2 3 |
= | * | 3 |
= | 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 |
= | * | 9* 2* |
= | 9 2 |
= | 4½ |
Check:
| 16 45 |
of | 4½ | = | 16 45 |
* | 9 2 |
= | * | = | 8 5 |
Remember: division by a fraction is computed by multiplying by its reciprocal.
Compound Fractions
Instead of writing
| 8 5 |
÷ | 16 45 |
we may write
| 8 5 16 45 |
= | 8 5 |
÷ | 16 45 |
= | 8 5 |
* | 45 16 |
Expressions of the form
| 8 5 16 45 |
where the numerator and denominators are given by fractions or mixed numbers provide compound fractions. They are evaluated by multiplying the numerator by the reciprocal of the denominator in accordance with the rule or pattern for division of one fraction by another. In mathematics, an expression is defined by saying how to evaluate it. The foregoing tells us how to evaluate compound fractions.
Notation: In compound fractions, the division bar between the fraction numerator (top) and the fraction giving the denominator (bottom) should be longer and thicker than the fraction bars in the numerator and denominator. Use parentheses to indicate the order of operations if you wish to depart from this convention.
Shorthand Description
| A B C D |
= | A B |
÷ | C D |
= | A B |
* | D C |
Remark:
www.whyslopes.com
Fractions, Ratios, Units, Rates & ProportionalityFraction Starter Lesson
(simplify, multiply, divide & then add or subtract)
An Alternative Starter Lesson
(take your pick, or try both)
Area Map & Intro Fraction Starter Lesson A Fraction Starter Lesson B 1 What is a Fraction 2 Multiplication I 3 Multiplication II 4 Multiplication III 5 Equivalent Fractions 6. Mixed Numbers 7 Comparison 8 Addition I 9 Addition II 10 Addition III 11 Multiplication IV 12 Division 13 Two Term Ratios 14 Implied Ratios 15 Multiple Ratios 16 Units in Arithmetic 16 Longer Explanation 16 Change Units 16 Products of Quantities 16. Fractions with Units 16. Division+Reciprocals 17 Proportionality 17 Examples 18 Rates & Slopes EGs 18 Constant Rate 18 Varying Rate 18 Velocity Calc., EGs 18 Changing Units 18 Slopes and Units 18 Slopes, No Units 19 RealPlayer Videos Links Arithmetic Videos - Real Player Format
Decimal Addition
Methods
Decimal Subtraction Methods
Decimal Multiplication Methods
Decimal Division
Methods
Fractions
Primes
Greatest Common
Divisors
Least Common Multiples
Square Root
Simplification
Area Content Summary
- Fraction Starter Lesson
- Real Player Videos on Operations with Primes and Fractions
- Continuous Ruler & Line Segment
model for fractions and operations on fractions - Number Theory Area points to the general model. - Distinction between Ratios and Fractions, a nuance: While binary ratios a:b may be identified with a fraction, triple ratios a:b:c and further multiple ratios cannot.
- Saying how to add and subtract like monomials in units and their powers, and saying how multiply and divide like and unlike monomials leads to fraction like expressions involving units and a framework for discussion rates - ratios of quantities - a framework for handling proportionality constants, and framework for carrying units through calculation in quantitative disciplines
Hint: See site area on solving linear equations to strengthen fraction sense and algebra skills together. Good luck.