Question

using distributive property of multiplication of rational numbers over addition/subtraction,simplify the following

( a ) -15/4×(-3/7 + 12/5)
( b ) 2/5×(-3/8 - 4/5)​

Answer 1

Distributive Property over Addition ⇒ $a(b+c)=(a\times b)+(a\times c)$

Distributive Property over Subtraction ⇒ $a(b-c)=(a\times b)-(a\times c)$

a)$\frac{-15}{4}\times (\frac{-3}{7}+\frac{12}{5})$

$\frac{-15}{4}\times (\frac{-3}{7}+\frac{12}{5})= (\frac{-15}{4}\times \frac{-3}{7})+(\frac{-15}{4}\times \frac{12}{5})$

$\frac{-15}{4}\times (\frac{-3\times 5 }{7\times 5 }+\frac{12\times 7 }{5\times 7 })= (\frac{45}{28} )+(-9)$

$\frac{-15}{4}\times (\frac{-15 }{35 }+\frac{84 }{35 })= (\frac{45}{28} )+(\frac{-9\times 28 }{1\times 28 } )$

$\frac{-15}{4}\times (\frac{69 }{35 })= (\frac{45}{28} )+(\frac{-252 }{ 28 } )$

$\frac{-207}{28} =\frac{-207}{28}$

∴ $\frac{-15}{4}\times (\frac{-3}{7}+\frac{12}{5})=\frac{-207}{28}$ using the distributive property.

b) $\frac{2}{5}\times (\frac{-3}{8}-\frac{4}{5})$

$\frac{2}{5}\times (\frac{-3}{8}-\frac{4}{5})=(\frac{2}{5}\times \frac{-3}{8}) -(\frac{2}{5}\times \frac{4}{5})$

$\frac{2}{5}\times (\frac{-3\times5 }{8\times 5 }-\frac{4\times 8 }{5\times 8 })=(\frac{-3}{20}) -(\frac{8}{25})$

$\frac{2}{5}\times (\frac{-15 }{40 }-\frac{32 }{40 })=(\frac{-3\times 5 }{20\times 5 }) -(\frac{8\times 4 }{25\times 4 })$

$\frac{2}{5}\times (\frac{-47 }{40 })=(\frac{-15 }{100}) -(\frac{32}{100 })$

$(\frac{-47 }{100 })=(\frac{-47 }{100})$

∴ $\frac{2}{5}\times (\frac{-3}{8}-\frac{4}{5})=\frac{-47}{100}$ using the distributive property.

Additional Information :)

What is the Distributive Property?

Something that says that multiplying a number by a group of numbers added or subtracted together is the same as doing each multiplication separately is known as the distributive property. It makes it easy for us to solve questions.

For Example -

$2\times (2+5)=(2\times 2)+2\times 5)\\ \\ 2\times7 = 4+10\\ \\ 14=14$

Answer 2

Step-by-step explanation:

Distributive Property over Addition ⇒ a(b+c)=(a\times b)+(a\times c)a(b+c)=(a×b)+(a×c)

Distributive Property over Subtraction ⇒ a(b-c)=(a\times b)-(a\times c)a(b−c)=(a×b)−(a×c)

a)\frac{-15}{4}\times (\frac{-3}{7}+\frac{12}{5})

4

−15

×(

7

−3

+

5

12

)

\frac{-15}{4}\times (\frac{-3}{7}+\frac{12}{5})= (\frac{-15}{4}\times \frac{-3}{7})+(\frac{-15}{4}\times \frac{12}{5})

4

−15

×(

7

−3

+

5

12

)=(

4

−15

×

7

−3

)+(

4

−15

×

5

12

)

\frac{-15}{4}\times (\frac{-3\times 5 }{7\times 5 }+\frac{12\times 7 }{5\times 7 })= (\frac{45}{28} )+(-9)

4

−15

×(

7×5

−3×5

+

5×7

12×7

)=(

28

45

)+(−9)

\frac{-15}{4}\times (\frac{-15 }{35 }+\frac{84 }{35 })= (\frac{45}{28} )+(\frac{-9\times 28 }{1\times 28 } )

4

−15

×(

35

−15

+

35

84

)=(

28

45

)+(

1×28

−9×28

)

\frac{-15}{4}\times (\frac{69 }{35 })= (\frac{45}{28} )+(\frac{-252 }{ 28 } )

4

−15

×(

35

69

)=(

28

45

)+(

28

−252

)

\frac{-207}{28} =\frac{-207}{28}

28

−207

=

28

−207

∴ \frac{-15}{4}\times (\frac{-3}{7}+\frac{12}{5})=\frac{-207}{28}

4

−15

×(

7

−3

+

5

12

)=

28

−207

using the distributive property.

b) \frac{2}{5}\times (\frac{-3}{8}-\frac{4}{5})

5

2

×(

8

−3

−

5

4

)

\frac{2}{5}\times (\frac{-3}{8}-\frac{4}{5})=(\frac{2}{5}\times \frac{-3}{8}) -(\frac{2}{5}\times \frac{4}{5})

5

2

×(

8

−3

−

5

4

)=(

5

2

×

8

−3

)−(

5

2

×

5

4

)

\frac{2}{5}\times (\frac{-3\times5 }{8\times 5 }-\frac{4\times 8 }{5\times 8 })=(\frac{-3}{20}) -(\frac{8}{25})

5

2

×(

8×5

−3×5

−

5×8

4×8

)=(

20

−3

)−(

25

8

)

\frac{2}{5}\times (\frac{-15 }{40 }-\frac{32 }{40 })=(\frac{-3\times 5 }{20\times 5 }) -(\frac{8\times 4 }{25\times 4 })

5

2

×(

40

−15

−

40

32

)=(

20×5

−3×5

)−(

25×4

8×4

)

\frac{2}{5}\times (\frac{-47 }{40 })=(\frac{-15 }{100}) -(\frac{32}{100 })

5

2

×(

40

−47

)=(

100

−15

)−(

100

32

)

(\frac{-47 }{100 })=(\frac{-47 }{100})(

100

−47

)=(

100

−47

)

∴ \frac{2}{5}\times (\frac{-3}{8}-\frac{4}{5})=\frac{-47}{100}

5

2

×(

8

−3

−

5

4

)=

100

−47

using the distributive property.