diff --git a/Mathematics/Prealgebra/Mixed and improper fractions.md b/Mathematics/Prealgebra/Mixed and improper fractions.md new file mode 100644 index 0000000..8b05480 --- /dev/null +++ b/Mathematics/Prealgebra/Mixed and improper fractions.md @@ -0,0 +1,30 @@ +--- +tags: + - Mathematics + - Prealgebra + - fractions +--- + +# Mixed and improper fractions + +## Improper fractions + +* Proper fraction: + * The numerator is smaller than the denominator + * E.g. $\frac{2}{3}$, $-\frac{5}{10}$ +* Improper fraction + * The numerator is greater than the denominator + * E.g. $\frac{3}{2}$, $-\frac{5}{4}$ + +## Mixed fractions +A mixed fraction is part whole number, part fraction. For example: $5 \frac{3}{4}$. + +It means, in effect: $5 + \frac{3}{4}$ + +## Converting mixed fractions into improper fractions +Mixed fractions are hard to calculate with. We need some way to convert them to fractions. We can do this by converting them to improper fractions. + +With the example $4 \frac{7}{8}$, we know this means $4 + \frac{7}{8}$. We need to express the amount 4 in terms of eighths. It would be 4 lots of $\frac{8}{8}$ given that 4 is a whole number not a fractional amount. Thus the process would be: +$$ + \frac{8}{8} + \frac{8}{8} + \frac{8}{8} + \frac{8}{8} + \frac{7}{8} +$$ \ No newline at end of file diff --git a/Mathematics/Prealgebra/Multiplying_fractions.md b/Mathematics/Prealgebra/Multiplying_fractions.md index a6ea101..7c717a2 100644 --- a/Mathematics/Prealgebra/Multiplying_fractions.md +++ b/Mathematics/Prealgebra/Multiplying_fractions.md @@ -6,14 +6,16 @@ tags: - multiplication --- +# Multiplying fractions - > - > To find the product of two fractions $\frac{a}{b}$ and $\frac{c}{d}$ multiply their numerators and denominators and then reduce: $$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$ + To find the product of two fractions $\frac{a}{b}$ and $\frac{c}{d}$ multiply their numerators and denominators and then reduce: + + $$\frac{a}{b} \cdot \frac{c}{d} = \frac{a \cdot c}{b \cdot d}$$ -## Example +### Example $$ -\\frac{1}{3} \cdot \frac{2}{5} = \frac{1 \cdot 2}{3 \cdot 5} = \frac{2}{15} +\frac{1}{3} \cdot \frac{2}{5} = \frac{1 \cdot 2}{3 \cdot 5} = \frac{2}{15} $$ ## Prime factorisation in place @@ -21,43 +23,43 @@ $$ The example above did not require a reduction, so here is a more complex example: $$ -\\frac{14}{15} \cdot \frac{30}{140} = \frac{420}{2100} +\frac{14}{15} \cdot \frac{30}{140} = \frac{420}{2100} $$ -It would be laborious to reduce such a large product using factor trees or the repeated application of divisors. We can use a more efficient method. - -This method can be applied at the point at which we conduct the multiplication rather than afterwards once we have the product. We express the the initial multiplicands as factors: +It would be laborious to reduce such a large product using factor trees or the repeated application of divisors, as defined in [reducing fractions](./Reducing_fractions.md). We can use a more efficient method. +This method can be applied at the point at which we conduct the multiplication rather than afterwards once we have the product. We express the the initial multiplicands as prime factors: $$ -\\frac{14}{15} \cdot \frac{30}{140} = \frac{(2 \cdot 7) \cdot (2 \cdot 3 \cdot 5) }{(3 \cdot 5) \cdot (2 \cdot 2 \cdot 7 \cdot 5)} +\frac{14}{15} \cdot \frac{30}{140} = \frac{(2 \cdot 7) \cdot (2 \cdot 3 \cdot 5) }{(3 \cdot 5) \cdot (2 \cdot 2 \cdot 7 \cdot 5)} $$ We now have the product in factorised form before we have applied the multiplication so we can go ahead and cancel: $$ -\\frac{\cancel{2}, \cancel{7}, \cancel{2}, \cancel{3}, \cancel{5}}{\cancel{3}, \cancel{5}, \cancel{2}, \cancel{2}, \cancel{7}, 5} = \frac{1}{5} +\frac{\cancel{2}, \cancel{7}, \cancel{2}, \cancel{3}, \cancel{5}}{\cancel{3}, \cancel{5}, \cancel{2}, \cancel{2}, \cancel{7}, 5} = \frac{1}{5} $$ **Note that in the above case, there was only a single 5 left as a denominator and no value left as a numerator. This is equivalent to there just being "one five" so we write $\frac{1}{5}$** -## Example with negative fractions containing variables +## Example of multiplying fractions with negative fractions containing variables -*Calculate: $$ - \frac{6x}{55y} \cdot - \frac{110y^2}{105x^2} $$* +Calculate: +$$- \frac{6x}{55y} \cdot - \frac{110y^2}{105x^2}$$ First multiply in place: $$ -\\frac{(3 \cdot 2 \cdot x) \cdot (5 \cdot 2 \cdot 11 \cdot y \cdot y)}{(5 \cdot 11 \cdot y) \cdot (7 \cdot 5 \cdot 3 \cdot x \cdot x)} +\frac{(3 \cdot 2 \cdot x) \cdot (5 \cdot 2 \cdot 11 \cdot y \cdot y)}{(5 \cdot 11 \cdot y) \cdot (7 \cdot 5 \cdot 3 \cdot x \cdot x)} $$ Then cancel: $$ -\\frac{(\cancel{3} \cdot 2 \cdot \cancel{x}) \cdot (\cancel{5} \cdot 2 \cdot \cancel{11} \cdot \cancel{y} \cdot y)}{(\cancel{5} \cdot \cancel{11} \cdot \cancel{y}) \cdot (7 \cdot 5 \cdot \cancel{3} \cdot \cancel{x} \cdot x)} = -\\frac{2 \cdot 2 \cdot y}{7 \cdot 5 \cdot x} +\frac{(\cancel{3} \cdot 2 \cdot \cancel{x}) \cdot (\cancel{5} \cdot 2 \cdot \cancel{11} \cdot \cancel{y} \cdot y)}{(\cancel{5} \cdot \cancel{11} \cdot \cancel{y}) \cdot (7 \cdot 5 \cdot \cancel{3} \cdot \cancel{x} \cdot x)} = +\frac{2 \cdot 2 \cdot y}{7 \cdot 5 \cdot x} $$ Then reduce: $$ -\\frac{2 \cdot 2 \cdot y}{7 \cdot 5 \cdot x} = \frac{4y}{35x} +\frac{2 \cdot 2 \cdot y}{7 \cdot 5 \cdot x} = \frac{4y}{35x} $$ diff --git a/pdf/Algorithmic_complexity.pdf b/pdf/Algorithmic_complexity.pdf index e16e237..cf1aa60 100644 Binary files a/pdf/Algorithmic_complexity.pdf and b/pdf/Algorithmic_complexity.pdf differ diff --git a/pdf/Arrays.pdf b/pdf/Arrays.pdf index 0f8adaf..4b480b8 100644 Binary files a/pdf/Arrays.pdf and b/pdf/Arrays.pdf differ diff --git a/pdf/Assembly_programs.pdf b/pdf/Assembly_programs.pdf index 330ef7b..eea2b06 100644 Binary files a/pdf/Assembly_programs.pdf and b/pdf/Assembly_programs.pdf differ diff --git a/pdf/Atomic_and_molecular_sentences.pdf b/pdf/Atomic_and_molecular_sentences.pdf index 856196e..ad4dadb 100644 Binary files a/pdf/Atomic_and_molecular_sentences.pdf and b/pdf/Atomic_and_molecular_sentences.pdf differ diff --git a/pdf/Axioms_of_set_theory.pdf b/pdf/Axioms_of_set_theory.pdf index 42376da..2233a25 100644 Binary files a/pdf/Axioms_of_set_theory.pdf and b/pdf/Axioms_of_set_theory.pdf differ diff --git a/pdf/Basic_database_concepts.pdf b/pdf/Basic_database_concepts.pdf index 2d76cd4..de14ddc 100644 Binary files a/pdf/Basic_database_concepts.pdf and b/pdf/Basic_database_concepts.pdf differ diff --git a/pdf/Basic_properties_of_sets.pdf b/pdf/Basic_properties_of_sets.pdf index 7ffc647..328ef99 100644 Binary files a/pdf/Basic_properties_of_sets.pdf and b/pdf/Basic_properties_of_sets.pdf differ diff --git a/pdf/Biconditional_Elimination.pdf b/pdf/Biconditional_Elimination.pdf index 3ee8bee..493c8db 100644 Binary files a/pdf/Biconditional_Elimination.pdf and b/pdf/Biconditional_Elimination.pdf differ diff --git a/pdf/Biconditional_Introduction.pdf b/pdf/Biconditional_Introduction.pdf index be86a9d..d1aa7f3 100644 Binary files a/pdf/Biconditional_Introduction.pdf and b/pdf/Biconditional_Introduction.pdf differ diff --git a/pdf/Binary_arithmetic.pdf b/pdf/Binary_arithmetic.pdf index 23e04f2..4f9179c 100644 Binary files a/pdf/Binary_arithmetic.pdf and b/pdf/Binary_arithmetic.pdf differ diff --git a/pdf/Binary_arithmetic_with_circuits.pdf b/pdf/Binary_arithmetic_with_circuits.pdf index 4255007..690a0d9 100644 Binary files a/pdf/Binary_arithmetic_with_circuits.pdf and b/pdf/Binary_arithmetic_with_circuits.pdf differ diff --git a/pdf/Bits,_bytes,_nibbles.pdf b/pdf/Bits,_bytes,_nibbles.pdf index 09dea22..aa7c871 100644 Binary files a/pdf/Bits,_bytes,_nibbles.pdf and b/pdf/Bits,_bytes,_nibbles.pdf differ diff --git a/pdf/Compile_from_source.pdf b/pdf/Compile_from_source.pdf index d2eeab2..3669702 100644 Binary files a/pdf/Compile_from_source.pdf and b/pdf/Compile_from_source.pdf differ diff --git a/pdf/Conditional_Elimination.pdf b/pdf/Conditional_Elimination.pdf index 0d7a40f..888ab87 100644 Binary files a/pdf/Conditional_Elimination.pdf and b/pdf/Conditional_Elimination.pdf differ diff --git a/pdf/Conditional_Introduction.pdf b/pdf/Conditional_Introduction.pdf index 583a534..ffc5463 100644 Binary files a/pdf/Conditional_Introduction.pdf and b/pdf/Conditional_Introduction.pdf differ diff --git a/pdf/Conjunction_Elimination.pdf b/pdf/Conjunction_Elimination.pdf index 84ccaa0..d098cb2 100644 Binary files a/pdf/Conjunction_Elimination.pdf and b/pdf/Conjunction_Elimination.pdf differ diff --git a/pdf/Conjunction_Introduction.pdf b/pdf/Conjunction_Introduction.pdf index 0023927..85427f2 100644 Binary files a/pdf/Conjunction_Introduction.pdf and b/pdf/Conjunction_Introduction.pdf differ diff --git a/pdf/Defining_a_computer.pdf b/pdf/Defining_a_computer.pdf index 3b898cf..37d5c7e 100644 Binary files a/pdf/Defining_a_computer.pdf and b/pdf/Defining_a_computer.pdf differ diff --git a/pdf/Design_features_of_language.pdf b/pdf/Design_features_of_language.pdf index eeac4a4..bfb9486 100644 Binary files a/pdf/Design_features_of_language.pdf and b/pdf/Design_features_of_language.pdf differ diff --git a/pdf/Disjunction_Elimination.pdf b/pdf/Disjunction_Elimination.pdf index afefb99..af70ebf 100644 Binary files a/pdf/Disjunction_Elimination.pdf and b/pdf/Disjunction_Elimination.pdf differ diff --git a/pdf/Disjunction_Introduction.pdf b/pdf/Disjunction_Introduction.pdf index 223c833..b8fb893 100644 Binary files a/pdf/Disjunction_Introduction.pdf and b/pdf/Disjunction_Introduction.pdf differ diff --git a/pdf/HTTP_request_types.pdf b/pdf/HTTP_request_types.pdf index 500ea48..51d7cce 100644 Binary files a/pdf/HTTP_request_types.pdf and b/pdf/HTTP_request_types.pdf differ diff --git a/pdf/Indeterminacy.pdf b/pdf/Indeterminacy.pdf index 6820251..183729a 100644 Binary files a/pdf/Indeterminacy.pdf and b/pdf/Indeterminacy.pdf differ diff --git a/pdf/Law_of_the_Excluded_Middle.pdf b/pdf/Law_of_the_Excluded_Middle.pdf index 0931b83..208a91f 100644 Binary files a/pdf/Law_of_the_Excluded_Middle.pdf and b/pdf/Law_of_the_Excluded_Middle.pdf differ diff --git a/pdf/Logic_circuits.pdf b/pdf/Logic_circuits.pdf index a16e16a..2e859ec 100644 Binary files a/pdf/Logic_circuits.pdf and b/pdf/Logic_circuits.pdf differ diff --git a/pdf/Logic_gates.pdf b/pdf/Logic_gates.pdf index f5d998d..11ad7de 100644 Binary files a/pdf/Logic_gates.pdf and b/pdf/Logic_gates.pdf differ diff --git a/pdf/Logical_possibility_and_necessity.pdf b/pdf/Logical_possibility_and_necessity.pdf index 745472a..d63778d 100644 Binary files a/pdf/Logical_possibility_and_necessity.pdf and b/pdf/Logical_possibility_and_necessity.pdf differ diff --git a/pdf/Lovelace_quote.pdf b/pdf/Lovelace_quote.pdf index 77b026b..f1b6538 100644 Binary files a/pdf/Lovelace_quote.pdf and b/pdf/Lovelace_quote.pdf differ diff --git a/pdf/Morphology.pdf b/pdf/Morphology.pdf index fa0e038..a4d850e 100644 Binary files a/pdf/Morphology.pdf and b/pdf/Morphology.pdf differ diff --git a/pdf/Negation_Elimination.pdf b/pdf/Negation_Elimination.pdf index 46e2f01..a9eebd6 100644 Binary files a/pdf/Negation_Elimination.pdf and b/pdf/Negation_Elimination.pdf differ diff --git a/pdf/Negation_Introduction.pdf b/pdf/Negation_Introduction.pdf index 46d728d..0a78c14 100644 Binary files a/pdf/Negation_Introduction.pdf and b/pdf/Negation_Introduction.pdf differ diff --git a/pdf/Object_language_and_meta-language.pdf b/pdf/Object_language_and_meta-language.pdf index 4a2c645..9600780 100644 Binary files a/pdf/Object_language_and_meta-language.pdf and b/pdf/Object_language_and_meta-language.pdf differ diff --git a/pdf/Package_management_in_Arch.pdf b/pdf/Package_management_in_Arch.pdf index 69b921a..ed23f30 100644 Binary files a/pdf/Package_management_in_Arch.pdf and b/pdf/Package_management_in_Arch.pdf differ diff --git a/pdf/Primary_key.pdf b/pdf/Primary_key.pdf index 1c4c92a..d00e2f8 100644 Binary files a/pdf/Primary_key.pdf and b/pdf/Primary_key.pdf differ diff --git a/pdf/Queue.pdf b/pdf/Queue.pdf index f8b6e70..202663f 100644 Binary files a/pdf/Queue.pdf and b/pdf/Queue.pdf differ diff --git a/pdf/RESTful_APIs.pdf b/pdf/RESTful_APIs.pdf index 92c0adc..3776a7e 100644 Binary files a/pdf/RESTful_APIs.pdf and b/pdf/RESTful_APIs.pdf differ diff --git a/pdf/Recursion.pdf b/pdf/Recursion.pdf index 576d0ce..5b7a0b7 100644 Binary files a/pdf/Recursion.pdf and b/pdf/Recursion.pdf differ diff --git a/pdf/Reiteration.pdf b/pdf/Reiteration.pdf index 6e71e9c..e69cb4d 100644 Binary files a/pdf/Reiteration.pdf and b/pdf/Reiteration.pdf differ diff --git a/pdf/SQL_syntax.pdf b/pdf/SQL_syntax.pdf index cca60e7..9ee975e 100644 Binary files a/pdf/SQL_syntax.pdf and b/pdf/SQL_syntax.pdf differ diff --git a/pdf/Semantic_versioning.pdf b/pdf/Semantic_versioning.pdf index dc9551d..90d5e13 100644 Binary files a/pdf/Semantic_versioning.pdf and b/pdf/Semantic_versioning.pdf differ diff --git a/pdf/Soundness.pdf b/pdf/Soundness.pdf index b2125b0..b1f1a4b 100644 Binary files a/pdf/Soundness.pdf and b/pdf/Soundness.pdf differ diff --git a/pdf/Stacks.pdf b/pdf/Stacks.pdf index b96a9f4..ecc43bc 100644 Binary files a/pdf/Stacks.pdf and b/pdf/Stacks.pdf differ diff --git a/pdf/Syllogism.pdf b/pdf/Syllogism.pdf index 6017689..6f6b11d 100644 Binary files a/pdf/Syllogism.pdf and b/pdf/Syllogism.pdf differ diff --git a/pdf/The_Pragmatic_Programmer_1999.pdf b/pdf/The_Pragmatic_Programmer_1999.pdf index 21ef6af..25c6e65 100644 Binary files a/pdf/The_Pragmatic_Programmer_1999.pdf and b/pdf/The_Pragmatic_Programmer_1999.pdf differ diff --git a/pdf/The_binary_number_system.pdf b/pdf/The_binary_number_system.pdf index 5eabdfd..d1cc218 100644 Binary files a/pdf/The_binary_number_system.pdf and b/pdf/The_binary_number_system.pdf differ diff --git a/pdf/Theorems_and_empty_sets.pdf b/pdf/Theorems_and_empty_sets.pdf index 67233ec..c6fe960 100644 Binary files a/pdf/Theorems_and_empty_sets.pdf and b/pdf/Theorems_and_empty_sets.pdf differ diff --git a/pdf/Turing_machines.pdf b/pdf/Turing_machines.pdf index 3bb68b8..83b64ff 100644 Binary files a/pdf/Turing_machines.pdf and b/pdf/Turing_machines.pdf differ diff --git a/pdf/User_management.pdf b/pdf/User_management.pdf index acabab4..9499c34 100644 Binary files a/pdf/User_management.pdf and b/pdf/User_management.pdf differ diff --git a/pdf/Von_Neumann_architecture.pdf b/pdf/Von_Neumann_architecture.pdf index 3ba07ec..a7ddc98 100644 Binary files a/pdf/Von_Neumann_architecture.pdf and b/pdf/Von_Neumann_architecture.pdf differ diff --git a/pdf/Why_computers_use_binary.pdf b/pdf/Why_computers_use_binary.pdf index 0e87d03..079a25f 100644 Binary files a/pdf/Why_computers_use_binary.pdf and b/pdf/Why_computers_use_binary.pdf differ