Problem 65 - Prealgebra

Question

Find quotient. Write in simplest form. \(\frac{4}{7} \cdot 2 \frac{1}{3}\)

Step-by-Step Solution

Answer

The quotient in simplest form is \(\frac{4}{3}\).

Step 1: Convert the mixed number to an improper fraction

The mixed number is \(2 \frac{1}{3}\). To convert it to an improper fraction, multiply the whole number by the denominator and add the numerator. Thus, \(2 \cdot 3 + 1 = 7\). So, \(2 \frac{1}{3} = \frac{7}{3}\).

Step 2: Set up the multiplication of fractions

Write the expression as the multiplication of fractions: \(\frac{4}{7} \cdot \frac{7}{3}\).

Step 3: Multiply the numerators and denominators

Multiply the numerators: \(4 \times 7 = 28\). Multiply the denominators: \(7 \times 3 = 21\). Thus, the product is \(\frac{28}{21}\).

Step 4: Simplify the fraction

To simplify \(\frac{28}{21}\), find the greatest common divisor (GCD) of 28 and 21, which is 7. Divide both numerator and denominator by 7: \(\frac{28 \div 7}{21 \div 7} = \frac{4}{3}\).

Step 5: Final Answer

The simplified form of \(\frac{4}{7} \cdot 2 \frac{1}{3}\) is \(\frac{4}{3}\).

Key Concepts

Simplifying Fractions

Simplifying fractions is an essential skill in mathematics that helps express fractions in their simplest form. A fraction consists of a numerator, the number on the top, and a denominator, the number on the bottom. To simplify a fraction, you find the greatest common divisor (GCD) of both the numerator and the denominator. The GCD is the largest number that can divide both numbers without leaving a remainder.

• For example, in the fraction \(\frac{28}{21}\), the GCD is 7.
• By dividing both the numerator and the denominator by 7, the fraction simplifies to \(\frac{4}{3}\).

This process ensures that the fraction is expressed in its most reduced form, making it easier to understand and compare with other numbers. Simplified fractions provide a clearer picture of actual quantities.

Multiplication of Fractions

Multiplying fractions is a straightforward process that involves multiplying the numerators together and the denominators together to form a new fraction. Start by setting up the fractions side by side and ensure all are in fraction form, like converting any mixed numbers to improper fractions beforehand.

• For example, multiply \(\frac{4}{7}\) by \(\frac{7}{3}\).
• Multiply the numerators: \(4 \times 7 = 28\).
• Multiply the denominators: \(7 \times 3 = 21\).
• The product is \(\frac{28}{21}\).

Always remember to simplify the resulting fraction for a neater final answer. Multiplying fractions is simpler when the fractions are reduced along the way, making the final simplification step easier.

Mixed Numbers

Mixed numbers combine a whole number with a fraction, commonly used to describe numbers larger than 1 but not whole. An example of a mixed number is \(2 \frac{1}{3}\), which consists of the whole number 2 and the fraction \(\frac{1}{3}\). Before performing mathematical operations like multiplication, it's helpful to convert mixed numbers to improper fractions. This conversion simplifies calculations and aligns them with operations that require consistent formats.

• Multiply the whole number by the denominator: \(2 \times 3 = 6\).
• Add the numerator: \(6 + 1 = 7\).
• The resulting improper fraction is \(\frac{7}{3}\).

This step facilitates smoother mathematical computations, particularly those involving fractions.

Improper Fractions

An improper fraction is a type of fraction where the numerator is greater than or equal to the denominator, which means its value is always 1 or greater. These fractions are critical when performing operations that involve whole and fractional numbers. An improper fraction can be expressed as a mixed number or left in its current form depending on the context of the problem.

• For instance, \(\frac{7}{3}\) is an improper fraction, resulting from converting \(2 \frac{1}{3}\).
• Improper fractions are especially helpful for performing arithmetic operations, as their format aligns better with equations and models in mathematics.

Being able to convert between improper fractions and mixed numbers ensures flexibility and efficiency in solving mathematical problems.