Solve 7/24(9/14a+4b-8c)+5/18(9/8a-3b+frac{6}{5}c)

$$\frac { 7 } { 24 } ( \frac { 9 } { 14 } a + 4 b - 8 c ) + \frac { 5 } { 18 } ( \frac { 9 } { 8 } a - 3 b + \frac { 6 } { 5 } c )$$

Evaluate

$$\frac{a}{2}+\frac{b}{3}-2c$$

Solution Steps

Use the distributive property to multiply \frac{7}{24} by \frac{9}{14}a+4b-8c.

$$\frac{7}{24}\times \frac{9}{14}a+\frac{7}{24}\times 4b+\frac{7}{24}\left(-8\right)c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Multiply \frac{7}{24} times \frac{9}{14} by multiplying numerator times numerator and denominator times denominator.

$$\frac{7\times 9}{24\times 14}a+\frac{7}{24}\times 4b+\frac{7}{24}\left(-8\right)c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Do the multiplications in the fraction \frac{7\times 9}{24\times 14}.

$$\frac{63}{336}a+\frac{7}{24}\times 4b+\frac{7}{24}\left(-8\right)c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Reduce the fraction \frac{63}{336} to lowest terms by extracting and canceling out 21.

$$\frac{3}{16}a+\frac{7}{24}\times 4b+\frac{7}{24}\left(-8\right)c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Express \frac{7}{24}\times 4 as a single fraction.

$$\frac{3}{16}a+\frac{7\times 4}{24}b+\frac{7}{24}\left(-8\right)c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Multiply 7 and 4 to get 28.

$$\frac{3}{16}a+\frac{28}{24}b+\frac{7}{24}\left(-8\right)c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Reduce the fraction \frac{28}{24} to lowest terms by extracting and canceling out 4.

$$\frac{3}{16}a+\frac{7}{6}b+\frac{7}{24}\left(-8\right)c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Express \frac{7}{24}\left(-8\right) as a single fraction.

$$\frac{3}{16}a+\frac{7}{6}b+\frac{7\left(-8\right)}{24}c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Multiply 7 and -8 to get -56.

$$\frac{3}{16}a+\frac{7}{6}b+\frac{-56}{24}c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Reduce the fraction \frac{-56}{24} to lowest terms by extracting and canceling out 8.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Use the distributive property to multiply \frac{5}{18} by \frac{9}{8}a-3b+\frac{6}{5}c.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{18}\times \frac{9}{8}a+\frac{5}{18}\left(-3\right)b+\frac{5}{18}\times \frac{6}{5}c$$

Multiply \frac{5}{18} times \frac{9}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5\times 9}{18\times 8}a+\frac{5}{18}\left(-3\right)b+\frac{5}{18}\times \frac{6}{5}c$$

Do the multiplications in the fraction \frac{5\times 9}{18\times 8}.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{45}{144}a+\frac{5}{18}\left(-3\right)b+\frac{5}{18}\times \frac{6}{5}c$$

Reduce the fraction \frac{45}{144} to lowest terms by extracting and canceling out 9.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{16}a+\frac{5}{18}\left(-3\right)b+\frac{5}{18}\times \frac{6}{5}c$$

Express \frac{5}{18}\left(-3\right) as a single fraction.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{16}a+\frac{5\left(-3\right)}{18}b+\frac{5}{18}\times \frac{6}{5}c$$

Multiply 5 and -3 to get -15.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{16}a+\frac{-15}{18}b+\frac{5}{18}\times \frac{6}{5}c$$

Reduce the fraction \frac{-15}{18} to lowest terms by extracting and canceling out 3.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{16}a-\frac{5}{6}b+\frac{5}{18}\times \frac{6}{5}c$$

Multiply \frac{5}{18} times \frac{6}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{16}a-\frac{5}{6}b+\frac{5\times 6}{18\times 5}c$$

Cancel out 5 in both numerator and denominator.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{16}a-\frac{5}{6}b+\frac{6}{18}c$$

Reduce the fraction \frac{6}{18} to lowest terms by extracting and canceling out 6.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{16}a-\frac{5}{6}b+\frac{1}{3}c$$

Combine \frac{3}{16}a and \frac{5}{16}a to get \frac{1}{2}a.

$$\frac{1}{2}a+\frac{7}{6}b-\frac{7}{3}c-\frac{5}{6}b+\frac{1}{3}c$$

Combine \frac{7}{6}b and -\frac{5}{6}b to get \frac{1}{3}b.

$$\frac{1}{2}a+\frac{1}{3}b-\frac{7}{3}c+\frac{1}{3}c$$

Combine -\frac{7}{3}c and \frac{1}{3}c to get -2c.

$$\frac{1}{2}a+\frac{1}{3}b-2c$$

Expand

$$\frac{a}{2}+\frac{b}{3}-2c$$

Solution Steps

Use the distributive property to multiply \frac{7}{24} by \frac{9}{14}a+4b-8c.

$$\frac{7}{24}\times \frac{9}{14}a+\frac{7}{24}\times 4b+\frac{7}{24}\left(-8\right)c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Multiply \frac{7}{24} times \frac{9}{14} by multiplying numerator times numerator and denominator times denominator.

$$\frac{7\times 9}{24\times 14}a+\frac{7}{24}\times 4b+\frac{7}{24}\left(-8\right)c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Do the multiplications in the fraction \frac{7\times 9}{24\times 14}.

$$\frac{63}{336}a+\frac{7}{24}\times 4b+\frac{7}{24}\left(-8\right)c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Reduce the fraction \frac{63}{336} to lowest terms by extracting and canceling out 21.

$$\frac{3}{16}a+\frac{7}{24}\times 4b+\frac{7}{24}\left(-8\right)c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Express \frac{7}{24}\times 4 as a single fraction.

$$\frac{3}{16}a+\frac{7\times 4}{24}b+\frac{7}{24}\left(-8\right)c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Multiply 7 and 4 to get 28.

$$\frac{3}{16}a+\frac{28}{24}b+\frac{7}{24}\left(-8\right)c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Reduce the fraction \frac{28}{24} to lowest terms by extracting and canceling out 4.

$$\frac{3}{16}a+\frac{7}{6}b+\frac{7}{24}\left(-8\right)c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Express \frac{7}{24}\left(-8\right) as a single fraction.

$$\frac{3}{16}a+\frac{7}{6}b+\frac{7\left(-8\right)}{24}c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Multiply 7 and -8 to get -56.

$$\frac{3}{16}a+\frac{7}{6}b+\frac{-56}{24}c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Reduce the fraction \frac{-56}{24} to lowest terms by extracting and canceling out 8.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{18}\left(\frac{9}{8}a-3b+\frac{6}{5}c\right)$$

Use the distributive property to multiply \frac{5}{18} by \frac{9}{8}a-3b+\frac{6}{5}c.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{18}\times \frac{9}{8}a+\frac{5}{18}\left(-3\right)b+\frac{5}{18}\times \frac{6}{5}c$$

Multiply \frac{5}{18} times \frac{9}{8} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5\times 9}{18\times 8}a+\frac{5}{18}\left(-3\right)b+\frac{5}{18}\times \frac{6}{5}c$$

Do the multiplications in the fraction \frac{5\times 9}{18\times 8}.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{45}{144}a+\frac{5}{18}\left(-3\right)b+\frac{5}{18}\times \frac{6}{5}c$$

Reduce the fraction \frac{45}{144} to lowest terms by extracting and canceling out 9.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{16}a+\frac{5}{18}\left(-3\right)b+\frac{5}{18}\times \frac{6}{5}c$$

Express \frac{5}{18}\left(-3\right) as a single fraction.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{16}a+\frac{5\left(-3\right)}{18}b+\frac{5}{18}\times \frac{6}{5}c$$

Multiply 5 and -3 to get -15.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{16}a+\frac{-15}{18}b+\frac{5}{18}\times \frac{6}{5}c$$

Reduce the fraction \frac{-15}{18} to lowest terms by extracting and canceling out 3.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{16}a-\frac{5}{6}b+\frac{5}{18}\times \frac{6}{5}c$$

Multiply \frac{5}{18} times \frac{6}{5} by multiplying numerator times numerator and denominator times denominator.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{16}a-\frac{5}{6}b+\frac{5\times 6}{18\times 5}c$$

Cancel out 5 in both numerator and denominator.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{16}a-\frac{5}{6}b+\frac{6}{18}c$$

Reduce the fraction \frac{6}{18} to lowest terms by extracting and canceling out 6.

$$\frac{3}{16}a+\frac{7}{6}b-\frac{7}{3}c+\frac{5}{16}a-\frac{5}{6}b+\frac{1}{3}c$$

Combine \frac{3}{16}a and \frac{5}{16}a to get \frac{1}{2}a.

$$\frac{1}{2}a+\frac{7}{6}b-\frac{7}{3}c-\frac{5}{6}b+\frac{1}{3}c$$

Combine \frac{7}{6}b and -\frac{5}{6}b to get \frac{1}{3}b.

$$\frac{1}{2}a+\frac{1}{3}b-\frac{7}{3}c+\frac{1}{3}c$$

Combine -\frac{7}{3}c and \frac{1}{3}c to get -2c.

$$\frac{1}{2}a+\frac{1}{3}b-2c$$