Answer:
[tex] \boxed{\bf \: \dfrac{ - 1}{6} \times \dfrac{4}{7} + \dfrac{1}{2} - \dfrac{5}{7} \times \dfrac{1}{6} = \dfrac{2}{7} \: } \\ [/tex]
Step-by-step explanation:
[tex]\large\underline{\sf{Solution-}}[/tex]
Given expression is
[tex]\sf \: \dfrac{ - 1}{6} \times \dfrac{4}{7} + \dfrac{1}{2} - \dfrac{5}{7} \times \dfrac{1}{6} \\ [/tex]
can be re-arranged as
[tex]\sf \: = \: \dfrac{ - 1}{6} \times \dfrac{4}{7} - \dfrac{5}{7} \times \dfrac{1}{6} + \dfrac{1}{2} \\ [/tex]
can be rewritten as
[tex]\sf \: = \: \dfrac{ - 1}{6} \times \dfrac{4}{7} + \dfrac{5}{7} \times \dfrac{ - 1}{6} + \dfrac{1}{2} \\ [/tex]
We know,
Distributive Property :- If x, y, z are rational numbers, then
[tex]\boxed{ \rm{ \:x \times y + x \times z = x \times (y + z) \: }} \\ [/tex]
So, using this property, we get
[tex]\sf \: = \: \dfrac{ - 1}{6} \times \left(\dfrac{4}{7} + \dfrac{5}{7} \right) + \dfrac{1}{2} \\ [/tex]
[tex]\sf \: = \: \dfrac{ - 1}{6} \times \left(\dfrac{4 + 5}{7} \right) + \dfrac{1}{2} \\ [/tex]
[tex]\sf \: = \: \dfrac{ - 1}{6} \times \left(\dfrac{9}{7} \right) + \dfrac{1}{2} \\ [/tex]
[tex]\sf \: = \: \dfrac{ - 1}{2} \times \left(\dfrac{3}{7} \right) + \dfrac{1}{2} \\ [/tex]
[tex]\sf \: = \: \dfrac{ - 3}{14} + \dfrac{1}{2} \\ [/tex]
[tex]\sf \: = \: \dfrac{ - 3}{14} + \dfrac{1 \times 7}{2 \times 7} \\ [/tex]
[tex]\sf \: = \: \dfrac{ - 3}{14} + \dfrac{7}{14} \\ [/tex]
[tex]\sf \: = \: \dfrac{ - 3 + 7}{14} \\ [/tex]
[tex]\sf \: = \: \dfrac{4}{14} \\ [/tex]
[tex]\sf \: = \: \dfrac{2}{7} \\ [/tex]
Hence,
[tex]\implies\sf \: \boxed{\bf \: \dfrac{ - 1}{6} \times \dfrac{4}{7} + \dfrac{1}{2} - \dfrac{5}{7} \times \dfrac{1}{6} = \dfrac{2}{7} \: } \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
1. Commutative Property of Addition.
[tex]\sf \: \boxed{ \sf{ \:a + b = b + a \: }} \\ \\ [/tex]
2. Associative Property of Addition
[tex]\sf \: \boxed{ \sf{ \:(a + b) + c = a + (b + c) \: }} \\ \\[/tex]
3. Additive Identity
[tex]\sf \: \boxed{ \sf{ \:x + 0 = 0 + x = x \: }} \\ \\ [/tex]
4. Commutative Property of Multiplication
[tex]\sf \: \boxed{ \sf{ \:a \times b = b \times a \: }} \\ \\ [/tex]
5. Associative Property of Multiplication
[tex]\sf \: \boxed{ \sf{ \:(a \times b) \times c = a \times (b \times c) \: }} \\ \\[/tex]
6. Multiplicative Identity
[tex]\sf \: \boxed{ \sf{ \:x \times 1 = 1 \times x = x \: }} \\ \\ [/tex]
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Answers & Comments
Answer:
[tex] \boxed{\bf \: \dfrac{ - 1}{6} \times \dfrac{4}{7} + \dfrac{1}{2} - \dfrac{5}{7} \times \dfrac{1}{6} = \dfrac{2}{7} \: } \\ [/tex]
Step-by-step explanation:
[tex]\large\underline{\sf{Solution-}}[/tex]
Given expression is
[tex]\sf \: \dfrac{ - 1}{6} \times \dfrac{4}{7} + \dfrac{1}{2} - \dfrac{5}{7} \times \dfrac{1}{6} \\ [/tex]
can be re-arranged as
[tex]\sf \: = \: \dfrac{ - 1}{6} \times \dfrac{4}{7} - \dfrac{5}{7} \times \dfrac{1}{6} + \dfrac{1}{2} \\ [/tex]
can be rewritten as
[tex]\sf \: = \: \dfrac{ - 1}{6} \times \dfrac{4}{7} + \dfrac{5}{7} \times \dfrac{ - 1}{6} + \dfrac{1}{2} \\ [/tex]
We know,
Distributive Property :- If x, y, z are rational numbers, then
[tex]\boxed{ \rm{ \:x \times y + x \times z = x \times (y + z) \: }} \\ [/tex]
So, using this property, we get
[tex]\sf \: = \: \dfrac{ - 1}{6} \times \left(\dfrac{4}{7} + \dfrac{5}{7} \right) + \dfrac{1}{2} \\ [/tex]
[tex]\sf \: = \: \dfrac{ - 1}{6} \times \left(\dfrac{4 + 5}{7} \right) + \dfrac{1}{2} \\ [/tex]
[tex]\sf \: = \: \dfrac{ - 1}{6} \times \left(\dfrac{9}{7} \right) + \dfrac{1}{2} \\ [/tex]
[tex]\sf \: = \: \dfrac{ - 1}{2} \times \left(\dfrac{3}{7} \right) + \dfrac{1}{2} \\ [/tex]
[tex]\sf \: = \: \dfrac{ - 3}{14} + \dfrac{1}{2} \\ [/tex]
[tex]\sf \: = \: \dfrac{ - 3}{14} + \dfrac{1 \times 7}{2 \times 7} \\ [/tex]
[tex]\sf \: = \: \dfrac{ - 3}{14} + \dfrac{7}{14} \\ [/tex]
[tex]\sf \: = \: \dfrac{ - 3 + 7}{14} \\ [/tex]
[tex]\sf \: = \: \dfrac{4}{14} \\ [/tex]
[tex]\sf \: = \: \dfrac{2}{7} \\ [/tex]
Hence,
[tex]\implies\sf \: \boxed{\bf \: \dfrac{ - 1}{6} \times \dfrac{4}{7} + \dfrac{1}{2} - \dfrac{5}{7} \times \dfrac{1}{6} = \dfrac{2}{7} \: } \\ [/tex]
[tex]\rule{190pt}{2pt}[/tex]
Additional Information
1. Commutative Property of Addition.
[tex]\sf \: \boxed{ \sf{ \:a + b = b + a \: }} \\ \\ [/tex]
2. Associative Property of Addition
[tex]\sf \: \boxed{ \sf{ \:(a + b) + c = a + (b + c) \: }} \\ \\[/tex]
3. Additive Identity
[tex]\sf \: \boxed{ \sf{ \:x + 0 = 0 + x = x \: }} \\ \\ [/tex]
4. Commutative Property of Multiplication
[tex]\sf \: \boxed{ \sf{ \:a \times b = b \times a \: }} \\ \\ [/tex]
5. Associative Property of Multiplication
[tex]\sf \: \boxed{ \sf{ \:(a \times b) \times c = a \times (b \times c) \: }} \\ \\[/tex]
6. Multiplicative Identity
[tex]\sf \: \boxed{ \sf{ \:x \times 1 = 1 \times x = x \: }} \\ \\ [/tex]