5. Equivalent Fractions
Answer the following questions:
- what is the difference between six eighths and three
quarters?
- what is the difference between two quarters and one half of a
dollar?
- what is the difference between two quarters and one half of a
length, of a weight (or mass), and of an hour?
This lesson introduces and provides motivation for equivalent
fractions.
Example:
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Three Quarters
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3
4
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|
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Each box is a quarter
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1
4
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1
4
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1
4
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1
4
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Each box is a half of a quarter
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|
|
|
|
|
|
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Each quarter is two eighths.. Hence 3 quarters is 6 = 3×2 eighths.
Hence
The foregoing equation may be read forwards or backwards. The
fractions
3 and 6
4
8
are equivalent.
Numerals and Fractionals
Numerals: Whole numbers may be written in different
ways:
XXII = two tens and three = 2 × 10 + 3 = 23
Thus there are different expressions (numerals) for each and every
whole number.
Fractionals: In the English language, a fraction refers to a
part of a whole. There may be different ways (fractionals) to
describe the same fraction.
Some parts, fractions or fractionals, are given exactly by (A) a
half, a third, a quarter, a fifth, a sixth, a seventh or a unit
numerator fractions; and some further fractions are given exactly
by whole number multiples of unit numerator fractions, for
example two thirds, two quarters, three quarters, two fifths,
three fifths, four fifths, two sixths, three sixths, four sixths,
five sixths and so on.
Consider an example.
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1
|
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1
3
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1
3
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1
3
|
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_1_
12
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_1_
12
|
_1_
12
|
_1_
12
|
|
|
|
|
_1_
12
|
_1_
12
|
_1_
12
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_1_
12
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Let us go over that again.
In the diagram, we see a single third can be divided into four
parts of equal size and value. Each of those parts is a
twelfth and equals a fourth of a third. Four quarters
of a divisible object is the object. So one third is four
times a quarter of itself. That is
Whence two thirds would be twice as much:
Thus different fractions
|
1
|
2
3
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1
3
|
|
1
3
|
1
3
|
1
3
|
|
_1_
12
|
_1_
12
|
_1_
12
|
_1_
12
|
|
|
|
|
_1_
12
|
_1_
12
|
_1_
12
|
_1_
12
|
Here the same part of the length, namely 2 thirds may be described
in two different ways
Thus different fractionals may describe the same part or fraction
of a whole. When they do, the fractional are said to be
equivalent. That is, they have the same value.
Note: Same value and same meaning are slightly
different. The fraction 2 thirds and 8 twelfths have the
same value for many purposes. Taking two thirds of a cake
(literally dividing it into thirds and taking two of those
thirds) and taking 8 twelfths of a cake, the same cake, may
physically represent different operations.
Extension: In the above diagram, we see a single length 1
can be divided into three parts of equal size and value. Each
of those parts is a third. Three thirds has the same
size as one. Thus
Thus 1 has the same value as the fraction 3 thirds. Thus a
single fractional may be also be a numeral, that is a different way
to express a whole number, here the number one.
Note Again: Same value and same meaning are slightly
different. The fraction 3 thirds and the number one
have the same value for many purposes. But three thirds of a
apple (literally dividing it into thirds and taking all three of
those thirds) and taking a whole apple are different.
A whole apple with its skin intact will last longer than three
thirds, each with the interior of the apple exposed to the
air.
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Algebraic Shorthand Description of ideas
for reading as part of algebra skill development - optional
reading for now
- Assume N = 4 and M = 5 and B = 3 on first reading below.
We may use the property of divisible objects (fractions included)
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N
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× (
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1
N
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of an object
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)
|
= the object
|
Thus if we have a fraction
B
M
of an object then
|
N
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× (
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1
N
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of
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B
M
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of an object
|
) =
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B
M
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of the object
|
In shorthand we see
or equivalently
The latter in turn gives the common factor cancellation property
The left and right hand side in foregoing equation are said to be
equivalent fractions. Replacing the left hand side by the
right hand side in a calculation is called a simplification, a reduction,
a cancellation or a lowering of terms. On the other hand, replacing
the right hand side by the left hand side is called raising terms.
Raising of terms is useful in the addition and multiplication of
fractions.
View the following RealPlayer videos after this lesson
-
[Play
Video] 3-4 minutes. Equivalent fractions - Lowering and
raising terms (the values of numerators and denominators) to obtain
equivalent fractions. Simplification involves lowering terms -
cancelling common factors or divisors on top and bottom. Addition &
subtraction of fractions may involve raising terms to obtain a common
denominators. See below.
-
[Play
Video] 2-3 minutes A few examples of Simplifying Fractions -
lowering terms by canceling common factors until there are no more
common factors, so that the numerator and denominator are relatively
prime, that is there prime decompositions have no primes in common.
Question (involving fractions of a set)
- Tom see one hundred pennies. He takes a tenth of them. Jane
takes a ninth of what remains. Then Andrew takes an eight of what is left
after Janes pick. Jean takes a tenth of the rest? Finally, Tom
takes two ninths of the what remains? Who has the least number of
pennies?
|
Secondary
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Arithmetic
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Algebra
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Calculus Starter Lessons
Calculus Lessons Elsewhere:
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How to Ace Calculus: Street Wise Guide - Mostly
Text.
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Flash
Video for Calculus Phobics
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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