124 lines
3.2 KiB
Markdown
124 lines
3.2 KiB
Markdown
---
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categories:
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- Mathematics
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tags:
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- prealgebra
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- fractions
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---
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# Adding and subtracting fractions
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## Adding/ subracting fractions with common denominators
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For two fractions $\frac{a}{c}$ and $\frac{b}{c}$ with a common denominator, their sum is defined as:
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$$
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\frac{a}{c} + \frac{b}{c} = \frac{a + b}{c}
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$$
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For example:
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$$
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\frac{2}{8} + \frac{3}{8} = \frac{5}{8}
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$$
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The same applies to subtraction:
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$$
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\frac{a}{c} - \frac{b}{c} = \frac{a - b}{c}
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$$
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## Adding/ subracting fractions without common denominators
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- Find the lowest common denominator for the two fractions
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- Use this to create two equivalent fractions
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- Add/subtract
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- Reduce
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### Lowest common denominator and lowest common multiple
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Given the symmetry between [factors and divisors](/Mathematics/Prealgebra/Factors_and_divisors.md) these properties are related. Note however that the LCM is more generic: it applies to any set of numbers not just fractions. Whereas the LCD is explicitly to do with fractions (hence 'denominator').
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- For two fractions $a, b$ (or a set), the LCD is the smallest number divisble by both the denominator of $a$ and the denominator of $b$ (or each member of the set).
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- For two fractions $a, b$ (or a set), the LCM is the smallest number that is a multiple of the denominator of $a$ and the denominator of $b$ (or each member of the set).
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In order to find the LCM of the set $\{12, 16\}$ we list the multiples of both:
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$$
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12, 24, 36, 48 \\
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16, 32, 48
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$$
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Until we identify the smallest number common to both lists. In this case it is 48. Thus the LCM of 12 and 16 is 48.
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The relationship between LCM and LCD is that _the least common denominator is the least common multiple of the fractions' denomintors_.
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### Demonstration: addition
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We can now use this to calculate the addition of two fractions without common denominators: $\frac{4}{9} + \frac{1}{6}$.
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First identify the common multiples of 9 and 6:
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$$
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9, 18, ... \\
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6, 12, 18, ...
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$$
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The least common multiple is 18.
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We then think: what do we need to multiply each denominator by to get 18?
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In the case of the first fraction ($\frac{4}{9}$) it is 2:
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$$
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\frac{4}{9 \cdot 2} = \frac{4}{18}
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$$
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But what we do to the denominator, we must also do to the numerator, hence:
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$$
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\frac{4 \cdot 2}{9 \cdot 2} = \frac{8}{18}
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$$
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We then do the same to the second fraction ($\frac{1}{6}$). We need to multiply its denominator by 3 to get 18 and we apply this also to the numerator.
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$$
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\frac{1 \cdot 3}{6 \cdot 3} = \frac{3}{18}
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$$
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We now have two fractions that share a common denominator so we can sum:
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$$
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\frac{8}{18} + \frac{3}{18} = \frac{11}{18}
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$$
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### Demonstration: subtraction
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Calculate:
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$$
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\frac{3}{5} - \frac{2}{3}
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$$
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Once again we need to find the least common denominator for the two fractions. We start by listing the common multiples for the two denominators 5 and 3:
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$$
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5, 10, 15, ... \\
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3, 6, 9, 12, 15,...
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$$
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The lowest common multiple is 15. From the first fraction we get 15 by multiplying by 3. With the second fraction we get 15 by multiplying by 5. Thus:
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$$
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\frac{3 \cdot 3}{5 \cdot 3} = \frac{9}{15}
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$$
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$$
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\frac{2 \cdot 5}{3 \cdot 5} = \frac{10}{15}
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$$
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We can now carry out the subtraction:
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$$
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\frac{9}{15} - \frac{10}{15} = -\frac{1}{15}
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$$
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