Problem 40 - Introductory Algebra for College Students

Question

Divide as indicated. $$\frac{x+1}{3} \div \frac{3 x+3}{7}$$

Step-by-Step Solution

Answer

The result of the division \(\frac{x+1}{3} \div \frac{3x+3}{7}\) is 7.

Step 1: Transform the division into multiplication

The division operation \(\frac{x+1}{3} \div \frac{3x+3}{7}\) can be transformed into a multiplication problem by multiplying by the reciprocal of the second fraction. That becomes \(\frac{x+1}{3} * \frac{7}{3x+3}\).

Step 2: Simplify Before Multiplying

The second fraction of the multiplication \(\frac{7}{3x+3}\) can be simplified. The numerator and the denominator of the fraction share a common factor of 3 that can be factored out. This simplifies the fraction to \(\frac{7}{3*(x+1)}\). Now the multiplication is \(\frac{x+1}{3} * \frac{7}{3*(x+1)}\).

Step 3: Cancel Out Common Terms

Now, there are common terms in the numerators and denominators that can be canceled out. (x+1) in the numerator of the first fraction can be canceled out with (x+1) in the denominator of the second fraction. 3 in the denominator of the first fraction also cancels out with 3 in the denominator of the second fraction. This leaves us with \(\frac{1}{1} * \frac{7}{1} = \frac{7}{1}\).

Step 4: Perform the Multiplication

Now that the fractions have been simplified, perform the multiplication to find the final result. \(\frac{7}{1}\) simplifies to 7.