Problem 74 - Algebra 1: Concepts and Skills

Question

Simplify. $$ \frac{12}{15} \cdot \frac{3}{4} \div \frac{1}{7} $$

Step-by-Step Solution

Answer

\(\frac{21}{5}\) or \(4 \frac{1}{5}\)

Step 1: Simplify the Fractions

Simplify the fraction \(\frac{12}{15}\) to \(\frac{4}{5}\) by dividing the numerator and the denominator by their greatest common divisor, which is 3. Also simplify \(\frac{3}{4}\) as it is, since it cannot be simplified further.

Step 2: Multiply the Simplified Fractions

Multiply the simplified fractions \(\frac{4}{5}\) and \(\frac{3}{4}\) to get \(\frac{12}{20}\), which further simplifies to \(\frac{3}{5}\) again by dividing the numerator and the denominator by their greatest common divisor, which is 4.

Step 3: Divide by a Fraction

Division by a fraction is the same as multiplying by its reciprocal. So, \(\frac{3}{5} \div \frac{1}{7}\), becomes \(\frac{3}{5} \cdot \frac{7}{1}\). Multiply \(\frac{3}{5}\) and \(\frac{7}{1}\) to get \(\frac{21}{5}\), which can be converted to a mixed number if desired. So, the final answer is \(\frac{21}{5}\) or \(4 \frac{1}{5}\).

Key Concepts

Simplifying Fractions

To simplify fractions, you must divide both the numerator and the denominator by their greatest common factor (GCF). This process makes the fraction simpler and easier to work with.

• Take the fraction \( \frac{12}{15} \). Both 12 and 15 are divisible by 3, which is their GCF. When you divide both by 3, you get \( \frac{4}{5} \).
• Important tip: Always check the highest number that divides both the top and bottom accurately without leaving a remainder. This is your GCF.
• A fraction is in its simplest form when no other number keeps it reducible.

Simplifying fractions is a crucial first step whenever you work with fractions in multiplication or division, as it makes calculations much easier and prevents mistakes.

Greatest Common Divisor

The greatest common divisor, or GCD, is the largest number that divides two or more numbers without a remainder. Finding the GCD is essential for simplifying fractions.

• For instance, the GCD of 12 and 15 is 3, because 3 is the highest number that divides both without leaving a remainder.
• To find the GCD:
  • List the factors of each number
  • Identify the common factors
  • Choose the largest common factor

Using the GCD to simplify fractions ensures you don't have to repeat the process and that the fraction is fully reduced.

Reciprocal of a Fraction

The reciprocal of a fraction is obtained by swapping the numerator and the denominator. The reciprocal is particularly useful when dividing fractions.

• For example, the reciprocal of \( \frac{1}{7} \) is \( \frac{7}{1} \).
• When dividing by a fraction, you multiply by its reciprocal. This changes your division problem into a multiplication problem, which is often simpler to solve.
• So, \( \frac{3}{5} \) divided by \( \frac{1}{7} \) turns into multiplying \( \frac{3}{5} \) by the reciprocal, \( \frac{7}{1} \).

Understanding reciprocals allows you to easily handle division of fractions, transforming potentially tricky problems into solvable ones.