Question

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Answer 1

$\displaystyle{\sf{\frac{\frac{2}{5}(n-4) }{\frac{3}{4} (n+7)}=\frac{4}{5} }}\\\\\\\displaystyle{\sf{\implies \frac{\frac{2n}{5}-\frac{8}{5} }{\frac{3n}{4}+\frac{21}{4} } }=\frac{4}{5} }}\\\\\\\displaystyle{\sf{\implies \frac{\frac{2n-8}{5} }{\frac{3n+21}{4} }=\frac{4}{5} }}\\\\\\\displaystyle{\sf{\implies \frac{2n-8}{5}\times\frac{4}{3n+21} =\frac{4}{5} }}\\\\\\\displaystyle{\sf{\implies \frac{8n-32}{15n+105}=\frac{4}{5} }}\\\\\\\displaystyle{\sf{\implies 60n+420=40n-160}}$      ...(By cross-multiplication)

$\displaystyle{\sf{\implies 20n+580=0}}\\\\\\\displaystyle{\sf{\implies 20n=-580}}\\\\\\\displaystyle{\sf{\implies n=\frac{-580}{20} }}\\\\\\\underline{\boxed{\boxed{\displaystyle{\sf{\implies n=-29}}}}}$

VERIFICATION :-

$\displaystyle{\sf{\frac{\frac{2}{5}(n-4) }{\frac{3}{4} (n+7)}=\frac{4}{5} }}\\\\\\\displaystyle{\sf{ \implies \frac{\frac{2(-29-4)}{5} }{\frac{3(-29+7)}{4} } =\frac{4}{5} }}\\\\\\\displaystyle{\sf{\implies \frac{\frac{-66}{5} }{\frac{-66}{4} }= \frac{4}{5} }}\\\\\\\textsf{-66 gets cancelled.}\\\\\displaystyle{\sf{\implies \frac{4}{5} =\frac{4}{5} }}\\\\\\\boxed{\sf{LHS=RHS}}$

Answer 2

Answer:

$\Large\boxed{\mathsf{n=-29}}$

Step-by-step explanation:

Isolate n from one side of the equation.

$\displaystyle \mathsf{\frac{2}{5}(n-4)\div\frac{3}{4}(n+7)=\frac{8(n-4)}{15(n+7)} }}$

$\displaystyle \mathsf{\frac{8(n-4)}{15(n+7)}=\frac{4}{5} }}$

Used the cross multiply:

a/b=c/d

8(n-4)*5=15(n+7)*4

Solve.

8*5=40

40(n-4)=60(n+7)

Multiply. (Expand.)

40(n-4)

40*n=40n

60(n+7)

60*n=60n

40*4=160

60*7=420

40n-160

60n+420

Rewrite the problem down.

$\displaystyle \mathsf{40n-160=60n+420}}}$

Add by 160 from both sides.

$\displaystyle \mathsf{40n-160+160=60n+420+160}}$

Solve.

$\displaystyle \mathsf{40n=60n+580}$

Subtract by 60n from both sides.

$\displaystyle \mathsf{40n-60n=60n+580-60n}}$

Solve.

$\displaystyle \mathsf{-20n=580}}$

Then, divide by -20 from both sides of an equation.

$\displaystyle \mathsf{\frac{-20n}{-20}=\frac{580}{-20}}}$

Solve.

$\displaystyle \mathsf{580\div-20=\boxed{\mathsf{-29}}}}$

Verify:

3/4(n+7)=0

Multiply by 4 from both sides.

$\displaystyle \mathsf{4\cdot \frac{3}{4}\left(n+7\right)=0\cdot \:4}}}$

Solve.

$\displaystyle \mathsf{3\left(n+7\right)=0}$

Divide by 3 from both sides of an equation.

$\displaystyle \mathsf{\frac{3\left(n+7\right)}{3}=\frac{0}{3}}}}$

Solve.

$\displaystyle \mathsf{n+7=0}}$

Subtract by 7 from both sides of an equation.

$\displaystyle \mathsf{n+7-7=0-7}}$

Solve.

$\displaystyle \mathsf{n=-7}$

So, the final answer is n=-29.