Solve (1/40+1/40+1/40+1/2)varphi=frac{1}{20}+frac{1}{20}+frac{1}{20}

$$( \frac { 1 } { 40 } + \frac { 1 } { 40 } + \frac { 1 } { 40 } + \frac { 1 } { 2 } ) \varphi = \frac { 1 } { 20 } + \frac { 1 } { 20 } + \frac { 1 } { 20 }$$

Solve for φ

$$\phi =\frac{6}{23}\approx 0.260869565$$

Steps for Solving Linear Equation

Since \frac{1}{40} and \frac{1}{40} have the same denominator, add them by adding their numerators.

$$\left(\frac{1+1}{40}+\frac{1}{40}+\frac{1}{2}\right)\phi =\frac{1}{20}+\frac{1}{20}+\frac{1}{20}$$

Add 1 and 1 to get 2.

$$\left(\frac{2}{40}+\frac{1}{40}+\frac{1}{2}\right)\phi =\frac{1}{20}+\frac{1}{20}+\frac{1}{20}$$

Reduce the fraction \frac{2}{40} to lowest terms by extracting and canceling out 2.

$$\left(\frac{1}{20}+\frac{1}{40}+\frac{1}{2}\right)\phi =\frac{1}{20}+\frac{1}{20}+\frac{1}{20}$$

Least common multiple of 20 and 40 is 40. Convert \frac{1}{20} and \frac{1}{40} to fractions with denominator 40.

$$\left(\frac{2}{40}+\frac{1}{40}+\frac{1}{2}\right)\phi =\frac{1}{20}+\frac{1}{20}+\frac{1}{20}$$

Since \frac{2}{40} and \frac{1}{40} have the same denominator, add them by adding their numerators.

$$\left(\frac{2+1}{40}+\frac{1}{2}\right)\phi =\frac{1}{20}+\frac{1}{20}+\frac{1}{20}$$

Add 2 and 1 to get 3.

$$\left(\frac{3}{40}+\frac{1}{2}\right)\phi =\frac{1}{20}+\frac{1}{20}+\frac{1}{20}$$

Least common multiple of 40 and 2 is 40. Convert \frac{3}{40} and \frac{1}{2} to fractions with denominator 40.

$$\left(\frac{3}{40}+\frac{20}{40}\right)\phi =\frac{1}{20}+\frac{1}{20}+\frac{1}{20}$$

Since \frac{3}{40} and \frac{20}{40} have the same denominator, add them by adding their numerators.

$$\frac{3+20}{40}\phi =\frac{1}{20}+\frac{1}{20}+\frac{1}{20}$$

Add 3 and 20 to get 23.

$$\frac{23}{40}\phi =\frac{1}{20}+\frac{1}{20}+\frac{1}{20}$$

Since \frac{1}{20} and \frac{1}{20} have the same denominator, add them by adding their numerators.

$$\frac{23}{40}\phi =\frac{1+1}{20}+\frac{1}{20}$$

Add 1 and 1 to get 2.

$$\frac{23}{40}\phi =\frac{2}{20}+\frac{1}{20}$$

Reduce the fraction \frac{2}{20} to lowest terms by extracting and canceling out 2.

$$\frac{23}{40}\phi =\frac{1}{10}+\frac{1}{20}$$

Least common multiple of 10 and 20 is 20. Convert \frac{1}{10} and \frac{1}{20} to fractions with denominator 20.

$$\frac{23}{40}\phi =\frac{2}{20}+\frac{1}{20}$$

Since \frac{2}{20} and \frac{1}{20} have the same denominator, add them by adding their numerators.

$$\frac{23}{40}\phi =\frac{2+1}{20}$$

Add 2 and 1 to get 3.

$$\frac{23}{40}\phi =\frac{3}{20}$$

Multiply both sides by \frac{40}{23}, the reciprocal of \frac{23}{40}.

$$\phi =\frac{3}{20}\times \frac{40}{23}$$

Multiply \frac{3}{20} times \frac{40}{23} by multiplying numerator times numerator and denominator times denominator.

$$\phi =\frac{3\times 40}{20\times 23}$$

Do the multiplications in the fraction \frac{3\times 40}{20\times 23}.

$$\phi =\frac{120}{460}$$

Reduce the fraction \frac{120}{460} to lowest terms by extracting and canceling out 20.

$$\phi =\frac{6}{23}$$