Basic Math Examples

$\frac{\frac{1}{3 y + 3 h} - \frac{1}{3 y}}{h}$

Step 1

Factor $3$ out of $3 y + 3 h$ .

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Step 1.1

Factor $3$ out of $3 y$ .

$\frac{\frac{1}{3 (y) + 3 h} - \frac{1}{3 y}}{h}$

Step 1.2

Factor $3$ out of $3 h$ .

$\frac{\frac{1}{3 (y) + 3 (h)} - \frac{1}{3 y}}{h}$

Step 1.3

Factor $3$ out of $3 (y) + 3 (h)$ .

$\frac{\frac{1}{3 (y + h)} - \frac{1}{3 y}}{h}$

Step 2

Simplify the numerator.

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Step 2.1

To write $\frac{1}{3 (y + h)}$ as a fraction with a common denominator, multiply by $\frac{y}{y}$ .

$\frac{\frac{1}{3 (y + h)} \cdot \frac{y}{y} - \frac{1}{3 y}}{h}$

Step 2.2

To write $- \frac{1}{3 y}$ as a fraction with a common denominator, multiply by $\frac{y + h}{y + h}$ .

$\frac{\frac{1}{3 (y + h)} \cdot \frac{y}{y} - \frac{1}{3 y} \cdot \frac{y + h}{y + h}}{h}$

Step 2.3

Write each expression with a common denominator of $3 (y + h) y$ , by multiplying each by an appropriate factor of $1$ .

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Step 2.3.1

Multiply $\frac{1}{3 (y + h)}$ by $\frac{y}{y}$ .

$\frac{\frac{y}{3 (y + h) y} - \frac{1}{3 y} \cdot \frac{y + h}{y + h}}{h}$

Step 2.3.2

Multiply $\frac{1}{3 y}$ by $\frac{y + h}{y + h}$ .

$\frac{\frac{y}{3 (y + h) y} - \frac{y + h}{3 y (y + h)}}{h}$

Step 2.3.3

Reorder the factors of $3 (y + h) y$ .

$\frac{\frac{y}{3 y (y + h)} - \frac{y + h}{3 y (y + h)}}{h}$

Step 2.4

Combine the numerators over the common denominator.

$\frac{\frac{y - (y + h)}{3 y (y + h)}}{h}$

Step 2.5

Rewrite $\frac{y - (y + h)}{3 y (y + h)}$ in a factored form.

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Step 2.5.1

Apply the distributive property.

$\frac{\frac{y - y - h}{3 y (y + h)}}{h}$

Step 2.5.2

Subtract $y$ from $y$ .

$\frac{\frac{0 - h}{3 y (y + h)}}{h}$

Step 2.5.3

Subtract $h$ from $0$ .

$\frac{\frac{- h}{3 y (y + h)}}{h}$

Step 2.6

Move the negative in front of the fraction.

$\frac{- \frac{h}{3 y (y + h)}}{h}$

Step 3

Multiply the numerator by the reciprocal of the denominator.

$- \frac{h}{3 y (y + h)} \cdot \frac{1}{h}$

Step 4

Cancel the common factor of $h$ .

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Step 4.1

Move the leading negative in $- \frac{h}{3 y (y + h)}$ into the numerator.

$\frac{- h}{3 y (y + h)} \cdot \frac{1}{h}$

Step 4.2

Factor $h$ out of $- h$ .

$\frac{h \cdot - 1}{3 y (y + h)} \cdot \frac{1}{h}$

Step 4.3

Cancel the common factor.

$\frac{h \cdot - 1}{3 y (y + h)} \cdot \frac{1}{h}$

Step 4.4

Rewrite the expression.

$\frac{- 1}{3 y (y + h)}$

Step 5

Move the negative in front of the fraction.

$- \frac{1}{3 y (y + h)}$