Finite Math Examples

$\frac{2}{3} \cdot (\frac{5}{8} + 8 y) - \frac{5}{8} = \frac{3}{8}$

Step 1

Simplify $\frac{2}{3} \cdot (\frac{5}{8} + 8 y) - \frac{5}{8}$ .

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Step 1.1

Simplify each term.

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Step 1.1.1

Apply the distributive property.

$\frac{2}{3} \cdot \frac{5}{8} + \frac{2}{3} (8 y) - \frac{5}{8} = \frac{3}{8}$

Step 1.1.2

Cancel the common factor of $2$ .

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Step 1.1.2.1

Factor $2$ out of $8$ .

$\frac{2}{3} \cdot \frac{5}{2 (4)} + \frac{2}{3} (8 y) - \frac{5}{8} = \frac{3}{8}$

Step 1.1.2.2

Cancel the common factor.

$\frac{2}{3} \cdot \frac{5}{2 \cdot 4} + \frac{2}{3} (8 y) - \frac{5}{8} = \frac{3}{8}$

Step 1.1.2.3

Rewrite the expression.

$\frac{1}{3} \cdot \frac{5}{4} + \frac{2}{3} (8 y) - \frac{5}{8} = \frac{3}{8}$

Step 1.1.3

Multiply $\frac{1}{3}$ by $\frac{5}{4}$ .

$\frac{5}{3 \cdot 4} + \frac{2}{3} (8 y) - \frac{5}{8} = \frac{3}{8}$

Step 1.1.4

Multiply $3$ by $4$ .

$\frac{5}{12} + \frac{2}{3} (8 y) - \frac{5}{8} = \frac{3}{8}$

Step 1.1.5

Multiply $\frac{2}{3} (8 y)$ .

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Step 1.1.5.1

Combine $8$ and $\frac{2}{3}$ .

$\frac{5}{12} + \frac{8 \cdot 2}{3} y - \frac{5}{8} = \frac{3}{8}$

Step 1.1.5.2

Multiply $8$ by $2$ .

$\frac{5}{12} + \frac{16}{3} y - \frac{5}{8} = \frac{3}{8}$

Step 1.1.5.3

Combine $\frac{16}{3}$ and $y$ .

$\frac{5}{12} + \frac{16 y}{3} - \frac{5}{8} = \frac{3}{8}$

Step 1.2

To write $\frac{5}{12}$ as a fraction with a common denominator, multiply by $\frac{2}{2}$ .

$\frac{16 y}{3} + \frac{5}{12} \cdot \frac{2}{2} - \frac{5}{8} = \frac{3}{8}$

Step 1.3

To write $- \frac{5}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{16 y}{3} + \frac{5}{12} \cdot \frac{2}{2} - \frac{5}{8} \cdot \frac{3}{3} = \frac{3}{8}$

Step 1.4

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 1.4.1

Multiply $\frac{5}{12}$ by $\frac{2}{2}$ .

$\frac{16 y}{3} + \frac{5 \cdot 2}{12 \cdot 2} - \frac{5}{8} \cdot \frac{3}{3} = \frac{3}{8}$

Step 1.4.2

Multiply $12$ by $2$ .

$\frac{16 y}{3} + \frac{5 \cdot 2}{24} - \frac{5}{8} \cdot \frac{3}{3} = \frac{3}{8}$

Step 1.4.3

Multiply $\frac{5}{8}$ by $\frac{3}{3}$ .

$\frac{16 y}{3} + \frac{5 \cdot 2}{24} - \frac{5 \cdot 3}{8 \cdot 3} = \frac{3}{8}$

Step 1.4.4

Multiply $8$ by $3$ .

$\frac{16 y}{3} + \frac{5 \cdot 2}{24} - \frac{5 \cdot 3}{24} = \frac{3}{8}$

Step 1.5

Combine the numerators over the common denominator.

$\frac{16 y}{3} + \frac{5 \cdot 2 - 5 \cdot 3}{24} = \frac{3}{8}$

Step 1.6

Simplify the numerator.

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Step 1.6.1

Multiply $5$ by $2$ .

$\frac{16 y}{3} + \frac{10 - 5 \cdot 3}{24} = \frac{3}{8}$

Step 1.6.2

Multiply $- 5$ by $3$ .

$\frac{16 y}{3} + \frac{10 - 15}{24} = \frac{3}{8}$

Step 1.6.3

Subtract $15$ from $10$ .

$\frac{16 y}{3} + \frac{- 5}{24} = \frac{3}{8}$

Step 1.7

Move the negative in front of the fraction.

$\frac{16 y}{3} - \frac{5}{24} = \frac{3}{8}$

Step 2

Move all terms not containing $y$ to the right side of the equation.

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Step 2.1

Add $\frac{5}{24}$ to both sides of the equation.

$\frac{16 y}{3} = \frac{3}{8} + \frac{5}{24}$

Step 2.2

To write $\frac{3}{8}$ as a fraction with a common denominator, multiply by $\frac{3}{3}$ .

$\frac{16 y}{3} = \frac{3}{8} \cdot \frac{3}{3} + \frac{5}{24}$

Step 2.3

Write each expression with a common denominator of $24$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{3}{8}$ by $\frac{3}{3}$ .

$\frac{16 y}{3} = \frac{3 \cdot 3}{8 \cdot 3} + \frac{5}{24}$

Step 2.3.2

Multiply $8$ by $3$ .

$\frac{16 y}{3} = \frac{3 \cdot 3}{24} + \frac{5}{24}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{16 y}{3} = \frac{3 \cdot 3 + 5}{24}$

Step 2.5

Simplify the numerator.

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Step 2.5.1

Multiply $3$ by $3$ .

$\frac{16 y}{3} = \frac{9 + 5}{24}$

Step 2.5.2

Add $9$ and $5$ .

$\frac{16 y}{3} = \frac{14}{24}$

Step 2.6

Cancel the common factor of $14$ and $24$ .

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Step 2.6.1

Factor $2$ out of $14$ .

$\frac{16 y}{3} = \frac{2 (7)}{24}$

Step 2.6.2

Cancel the common factors.

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Step 2.6.2.1

Factor $2$ out of $24$ .

$\frac{16 y}{3} = \frac{2 \cdot 7}{2 \cdot 12}$

Step 2.6.2.2

Cancel the common factor.

$\frac{16 y}{3} = \frac{2 \cdot 7}{2 \cdot 12}$

Step 2.6.2.3

Rewrite the expression.

$\frac{16 y}{3} = \frac{7}{12}$

Step 3

Multiply both sides of the equation by $\frac{3}{16}$ .

$\frac{3}{16} \cdot \frac{16 y}{3} = \frac{3}{16} \cdot \frac{7}{12}$

Step 4

Simplify both sides of the equation.

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Step 4.1

Simplify the left side.

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Step 4.1.1

Simplify $\frac{3}{16} \cdot \frac{16 y}{3}$ .

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Step 4.1.1.1

Cancel the common factor of $3$ .

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Step 4.1.1.1.1

Cancel the common factor.

$\frac{3}{16} \cdot \frac{16 y}{3} = \frac{3}{16} \cdot \frac{7}{12}$

Step 4.1.1.1.2

Rewrite the expression.

$\frac{1}{16} (16 y) = \frac{3}{16} \cdot \frac{7}{12}$

Step 4.1.1.2

Cancel the common factor of $16$ .

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Step 4.1.1.2.1

Factor $16$ out of $16 y$ .

$\frac{1}{16} (16 (y)) = \frac{3}{16} \cdot \frac{7}{12}$

Step 4.1.1.2.2

Cancel the common factor.

$\frac{1}{16} (16 y) = \frac{3}{16} \cdot \frac{7}{12}$

Step 4.1.1.2.3

Rewrite the expression.

$y = \frac{3}{16} \cdot \frac{7}{12}$

Step 4.2

Simplify the right side.

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Step 4.2.1

Simplify $\frac{3}{16} \cdot \frac{7}{12}$ .

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Step 4.2.1.1

Cancel the common factor of $3$ .

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Step 4.2.1.1.1

Factor $3$ out of $12$ .

$y = \frac{3}{16} \cdot \frac{7}{3 (4)}$

Step 4.2.1.1.2

Cancel the common factor.

$y = \frac{3}{16} \cdot \frac{7}{3 \cdot 4}$

Step 4.2.1.1.3

Rewrite the expression.

$y = \frac{1}{16} \cdot \frac{7}{4}$

Step 4.2.1.2

Multiply $\frac{1}{16}$ by $\frac{7}{4}$ .

$y = \frac{7}{16 \cdot 4}$

Step 4.2.1.3

Multiply $16$ by $4$ .

$y = \frac{7}{64}$

Step 5

The result can be shown in multiple forms.

Exact Form:

$y = \frac{7}{64}$

Decimal Form:

$y = 0.109375$