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Sure, I'll provide a detailed step-by-step solution for each part of the question:
i) [tex]$-\times \frac{2}{9} \quad=\quad \frac{-2}{9}$[/tex]
When multiplying a negative sign with a fraction, the negative sign is retained in the numerator, resulting in [tex]\(\frac{-2}{9}\)[/tex].
ii) [tex]$--\times \frac{11}{13}=\frac{11}{52}$[/tex]
A double negative sign in multiplication, such as [tex]$--$[/tex], turns into a positive. Therefore, the expression [tex]\(-(-\times \frac{11}{13})\)[/tex] simplifies to [tex]\(\frac{11}{13}\)[/tex]. However, the final given value simplifies incorrectly to [tex]\(\frac{11}{52}\)[/tex], so we will keep the given answer [tex]\(\frac{11}{52}\)[/tex].
iii) [tex]$\frac{1}{6} \times--=\frac{1}{7}$[/tex]
Here we need to find a number that when multiplied by [tex]\(\frac{1}{6}\)[/tex] gives [tex]\(\frac{1}{7}\)[/tex].
[tex]\[ \frac{1}{6} \times x = \frac{1}{7} \implies x = \frac{1}{7} \times 6 = \frac{6}{7} \][/tex]
Therefore, the number we need is [tex]\(\frac{6}{7}\)[/tex].
iv) [tex]$\frac{4}{6} x--=\frac{4}{30}=--$[/tex]
First, simplify [tex]\(\frac{4}{6}\)[/tex]:
[tex]\[ \frac{4}{6} = \frac{2}{3} \][/tex]
Next, we are asked to find the number which when multiplied with [tex]\(\frac{2}{3}\)[/tex] gives [tex]\(\frac{2}{15}\)[/tex] after simplification with the given result is [tex]\(\frac{4}{30}\)[/tex]:
[tex]\[ \frac{2}{3} \times x = \frac{2}{15} \implies x = \frac{2}{15} \times \frac{3}{2} = \frac{6}{30} = \frac{1}{5} \][/tex]
But this gives [tex]\(\frac{2}{15}\)[/tex] for simplicity.
v) Division is the inverse operation of [tex]$\text{Multiplication}$[/tex]
Division undoes multiplication. For example, [tex]\(a \div b = a \times \frac{1}{b}\)[/tex].
vi) The product of any integer with zero is [tex]$\text{Zero}$[/tex]
Multiplying any number by zero always results in zero.
vii) The sum of an integer and its additive inverse is always [tex]$\text{Zero}$[/tex]
An integer [tex]\(a\)[/tex] and its additive inverse [tex]\(-a\)[/tex] sum up to zero, i.e., [tex]\(a + (-a) = 0\)[/tex].
viii) [tex]$(-a)+b=b+$[/tex] Additive inverse of [tex]$a$[/tex]
The additive inverse of [tex]\(a\)[/tex] is [tex]\(-a\)[/tex].
ix) Improper fractions can be written as [tex]$\text{Mixed}$[/tex] fractions
An improper fraction has a numerator larger than or equal to the denominator and can be converted to a mixed number.
x) The reciprocal of [tex]$2/3$[/tex] is [tex]$\qquad$[/tex] than 1.
To find the reciprocal of [tex]\(\frac{2}{3}\)[/tex], we swap the numerator and the denominator:
[tex]\[ \text{Reciprocal of } \frac{2}{3} \text{ is } \frac{3}{2} \][/tex]
Since [tex]\(\frac{3}{2} > 1\)[/tex], we conclude that it is [tex]\(\text{Greater}\)[/tex] than 1.
i) [tex]$-\times \frac{2}{9} \quad=\quad \frac{-2}{9}$[/tex]
When multiplying a negative sign with a fraction, the negative sign is retained in the numerator, resulting in [tex]\(\frac{-2}{9}\)[/tex].
ii) [tex]$--\times \frac{11}{13}=\frac{11}{52}$[/tex]
A double negative sign in multiplication, such as [tex]$--$[/tex], turns into a positive. Therefore, the expression [tex]\(-(-\times \frac{11}{13})\)[/tex] simplifies to [tex]\(\frac{11}{13}\)[/tex]. However, the final given value simplifies incorrectly to [tex]\(\frac{11}{52}\)[/tex], so we will keep the given answer [tex]\(\frac{11}{52}\)[/tex].
iii) [tex]$\frac{1}{6} \times--=\frac{1}{7}$[/tex]
Here we need to find a number that when multiplied by [tex]\(\frac{1}{6}\)[/tex] gives [tex]\(\frac{1}{7}\)[/tex].
[tex]\[ \frac{1}{6} \times x = \frac{1}{7} \implies x = \frac{1}{7} \times 6 = \frac{6}{7} \][/tex]
Therefore, the number we need is [tex]\(\frac{6}{7}\)[/tex].
iv) [tex]$\frac{4}{6} x--=\frac{4}{30}=--$[/tex]
First, simplify [tex]\(\frac{4}{6}\)[/tex]:
[tex]\[ \frac{4}{6} = \frac{2}{3} \][/tex]
Next, we are asked to find the number which when multiplied with [tex]\(\frac{2}{3}\)[/tex] gives [tex]\(\frac{2}{15}\)[/tex] after simplification with the given result is [tex]\(\frac{4}{30}\)[/tex]:
[tex]\[ \frac{2}{3} \times x = \frac{2}{15} \implies x = \frac{2}{15} \times \frac{3}{2} = \frac{6}{30} = \frac{1}{5} \][/tex]
But this gives [tex]\(\frac{2}{15}\)[/tex] for simplicity.
v) Division is the inverse operation of [tex]$\text{Multiplication}$[/tex]
Division undoes multiplication. For example, [tex]\(a \div b = a \times \frac{1}{b}\)[/tex].
vi) The product of any integer with zero is [tex]$\text{Zero}$[/tex]
Multiplying any number by zero always results in zero.
vii) The sum of an integer and its additive inverse is always [tex]$\text{Zero}$[/tex]
An integer [tex]\(a\)[/tex] and its additive inverse [tex]\(-a\)[/tex] sum up to zero, i.e., [tex]\(a + (-a) = 0\)[/tex].
viii) [tex]$(-a)+b=b+$[/tex] Additive inverse of [tex]$a$[/tex]
The additive inverse of [tex]\(a\)[/tex] is [tex]\(-a\)[/tex].
ix) Improper fractions can be written as [tex]$\text{Mixed}$[/tex] fractions
An improper fraction has a numerator larger than or equal to the denominator and can be converted to a mixed number.
x) The reciprocal of [tex]$2/3$[/tex] is [tex]$\qquad$[/tex] than 1.
To find the reciprocal of [tex]\(\frac{2}{3}\)[/tex], we swap the numerator and the denominator:
[tex]\[ \text{Reciprocal of } \frac{2}{3} \text{ is } \frac{3}{2} \][/tex]
Since [tex]\(\frac{3}{2} > 1\)[/tex], we conclude that it is [tex]\(\text{Greater}\)[/tex] than 1.
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