Solve t+h+1/t+h+4-t+1/t+4/h

$$\frac { \frac { t + h + 1 } { t + h + 4 } - \frac { t + 1 } { t + 4 } } { h }$$

Evaluate

$$\frac{3}{\left(t+4\right)\left(t+h+4\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of t+h+4 and t+4 is \left(t+4\right)\left(t+h+4\right). Multiply \frac{t+h+1}{t+h+4} times \frac{t+4}{t+4}. Multiply \frac{t+1}{t+4} times \frac{t+h+4}{t+h+4}.

$$\frac{\frac{\left(t+h+1\right)\left(t+4\right)}{\left(t+4\right)\left(t+h+4\right)}-\frac{\left(t+1\right)\left(t+h+4\right)}{\left(t+4\right)\left(t+h+4\right)}}{h}$$

Since \frac{\left(t+h+1\right)\left(t+4\right)}{\left(t+4\right)\left(t+h+4\right)} and \frac{\left(t+1\right)\left(t+h+4\right)}{\left(t+4\right)\left(t+h+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(t+h+1\right)\left(t+4\right)-\left(t+1\right)\left(t+h+4\right)}{\left(t+4\right)\left(t+h+4\right)}}{h}$$

Do the multiplications in \left(t+h+1\right)\left(t+4\right)-\left(t+1\right)\left(t+h+4\right).

$$\frac{\frac{t^{2}+4t+ht+4h+t+4-t^{2}-th-4t-t-h-4}{\left(t+4\right)\left(t+h+4\right)}}{h}$$

Combine like terms in t^{2}+4t+ht+4h+t+4-t^{2}-th-4t-t-h-4.

$$\frac{\frac{3h}{\left(t+4\right)\left(t+h+4\right)}}{h}$$

Express \frac{\frac{3h}{\left(t+4\right)\left(t+h+4\right)}}{h} as a single fraction.

$$\frac{3h}{\left(t+4\right)\left(t+h+4\right)h}$$

Cancel out h in both numerator and denominator.

$$\frac{3}{\left(t+4\right)\left(t+h+4\right)}$$

Apply the distributive property by multiplying each term of t+4 by each term of t+h+4.

$$\frac{3}{t^{2}+th+4t+4t+4h+16}$$

Combine 4t and 4t to get 8t.

$$\frac{3}{t^{2}+th+8t+4h+16}$$

Expand

$$\frac{3}{\left(t+4\right)\left(t+h+4\right)}$$

Short Solution Steps

To add or subtract expressions, expand them to make their denominators the same. Least common multiple of t+h+4 and t+4 is \left(t+4\right)\left(t+h+4\right). Multiply \frac{t+h+1}{t+h+4} times \frac{t+4}{t+4}. Multiply \frac{t+1}{t+4} times \frac{t+h+4}{t+h+4}.

$$\frac{\frac{\left(t+h+1\right)\left(t+4\right)}{\left(t+4\right)\left(t+h+4\right)}-\frac{\left(t+1\right)\left(t+h+4\right)}{\left(t+4\right)\left(t+h+4\right)}}{h}$$

Since \frac{\left(t+h+1\right)\left(t+4\right)}{\left(t+4\right)\left(t+h+4\right)} and \frac{\left(t+1\right)\left(t+h+4\right)}{\left(t+4\right)\left(t+h+4\right)} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\frac{\left(t+h+1\right)\left(t+4\right)-\left(t+1\right)\left(t+h+4\right)}{\left(t+4\right)\left(t+h+4\right)}}{h}$$

Do the multiplications in \left(t+h+1\right)\left(t+4\right)-\left(t+1\right)\left(t+h+4\right).

$$\frac{\frac{t^{2}+4t+ht+4h+t+4-t^{2}-th-4t-t-h-4}{\left(t+4\right)\left(t+h+4\right)}}{h}$$

Combine like terms in t^{2}+4t+ht+4h+t+4-t^{2}-th-4t-t-h-4.

$$\frac{\frac{3h}{\left(t+4\right)\left(t+h+4\right)}}{h}$$

Express \frac{\frac{3h}{\left(t+4\right)\left(t+h+4\right)}}{h} as a single fraction.

$$\frac{3h}{\left(t+4\right)\left(t+h+4\right)h}$$

Cancel out h in both numerator and denominator.

$$\frac{3}{\left(t+4\right)\left(t+h+4\right)}$$

Apply the distributive property by multiplying each term of t+4 by each term of t+h+4.

$$\frac{3}{t^{2}+th+4t+4t+4h+16}$$

Combine 4t and 4t to get 8t.

$$\frac{3}{t^{2}+th+8t+4h+16}$$