Question

simplify.
\frac{\frac{5x + 20}{16x}}{\frac{3x + 12}{4}}

Explanation:

Step1: Rewrite as division of fractions

Dividing by a fraction is multiplying by its reciprocal, so \(\frac{\frac{5x + 20}{16x}}{\frac{3x + 12}{4}}=\frac{5x + 20}{16x}\times\frac{4}{3x + 12}\)

Step2: Factor numerators and denominators

Factor \(5\) from \(5x + 20\): \(5x+20 = 5(x + 4)\)
Factor \(3\) from \(3x + 12\): \(3x + 12=3(x + 4)\)
Now the expression is \(\frac{5(x + 4)}{16x}\times\frac{4}{3(x + 4)}\)

Step3: Cancel common factors

Cancel \((x + 4)\) (for \(x
eq - 4\)) and cancel \(4\) and \(16\) ( \(16\div4 = 4\), so \(16x\) becomes \(4x\)):
\(\frac{5\cancel{(x + 4)}}{4x\times4}\times\frac{\cancel{4}}{3\cancel{(x + 4)}}=\frac{5}{4x}\times\frac{1}{3}\) (Wait, correction: Wait, \(16x\) and \(4\): \(4\) in numerator and \(16x\) in denominator: \(4/16x=1/(4x)\)? No, wait, original after factoring: \(\frac{5(x + 4)}{16x}\times\frac{4}{3(x + 4)}\). So \(4\) and \(16\) have a common factor of \(4\): \(4\div4 = 1\), \(16\div4=4\). So it becomes \(\frac{5(x + 4)}{4x\times4}\times\frac{1}{3(x + 4)}\)? No, better: \(\frac{5(x + 4)\times4}{16x\times3(x + 4)}\). Now cancel \((x + 4)\) and \(4\) with \(16\): \(4/16 = 1/4\). So \(\frac{5\times1}{4x\times3}=\frac{5}{12x}\)? Wait, no: Wait, \(16x\) is \(4x\times4\), and the numerator has \(4\). So \(4\) in numerator and \(16x\) (which is \(4\times4x\)): so \(4/(16x)=1/(4x)\). Then we have \(5(x + 4)\times1/(4x)\times1/(3(x + 4))\). Then cancel \((x + 4)\): \(5\times1/(4x)\times1/3=\frac{5}{12x}\)? Wait, no, let's do it again:

\(\frac{5(x + 4)}{16x}\times\frac{4}{3(x + 4)}=\frac{5\cancel{(x + 4)}\times4}{16x\times3\cancel{(x + 4)}}\)

Now, \(4\) and \(16\) can be simplified: \(4\div4 = 1\), \(16\div4 = 4\). So now we have \(\frac{5\times1}{4x\times3}=\frac{5}{12x}\)? Wait, no, \(4\) in numerator and \(16\) in denominator: \(4/16 = 1/4\), so:

\(\frac{5\times1}{4x\times3}=\frac{5}{12x}\)? Wait, no, \(5\times4 = 20\), \(16x\times3 = 48x\), then \(20/48x = 5/12x\) (dividing numerator and denominator by 4: \(20\div4 = 5\), \(48x\div4 = 12x\)). Yes, that's correct.

Answer:

\(\frac{5}{12x}\) (for \(x
eq - 4,0\))