Question

What is the quotient of \(7 \frac{2}{3} \div 2 \frac{1}{2}\) ? Leave the answer as a reduced improper fraction. Write the correct numerator in the box.

Short answer

The quotient is \(\frac{46}{15}\). The numerator is 46.

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers to improper fractions. For \(7 \frac{2}{3}\), multiply the whole number 7 by the denominator 3, and then add the numerator 2. The result is the new numerator: \(7 \times 3 + 2 = 21 + 2 = 23\). So, \(7 \frac{2}{3} = \frac{23}{3}\).For \(2 \frac{1}{2}\), multiply the whole number 2 by the denominator 2, and then add the numerator 1. The result is the new numerator: \(2 \times 2 + 1 = 4 + 1 = 5\). So, \(2 \frac{1}{2} = \frac{5}{2}\).

Write the Division as a Multiplication

Rewrite the division of fractions as the multiplication of the first fraction by the reciprocal of the second fraction. This means we change \(\frac{23}{3} \div \frac{5}{2}\) to \(\frac{23}{3} \times \frac{2}{5}\).

Multiply the Fractions

To multiply the fractions, multiply the numerators together and the denominators together:\(\frac{23}{3} \times \frac{2}{5} = \frac{23 \times 2}{3 \times 5} = \frac{46}{15}\).

Simplify the Fraction

Check if the fraction can be simplified. The greatest common divisor (GCD) of 46 and 15 is 1, so the fraction \(\frac{46}{15}\) is already in its simplest form. The final answer remains \(\frac{46}{15}\).

Key concepts

mixed numbers

Mixed numbers are numbers that contain both a whole number and a fraction part. For example, in the number \(7 \frac{2}{3}\), 7 is the whole number and \(\frac{2}{3}\) is the fraction part. Mixed numbers are often used in day-to-day life.

To perform arithmetic operations with mixed numbers, it's easier if we convert them into improper fractions. This simplifies calculations. An improper fraction has a numerator larger than its denominator. To convert a mixed number to an improper fraction, follow these steps:

• Multiply the whole number by the denominator of the fraction
• Add the numerator of the fraction to the result
• This gives you the new numerator, which you place over the original denominator

For example, converting \(7 \frac{2}{3}\) to an improper fraction:

• Multiply 7 (whole number) by 3 (denominator): 7 × 3 = 21
• Add the numerator: 21 + 2 = 23
• So, \(7 \frac{2}{3}\) is \(\frac{23}{3}\)

improper fractions

Improper fractions have a numerator that is greater than or equal to the denominator. These fractions represent values greater than or equal to 1. In mathematical operations, improper fractions are highly versatile.

Once a mixed number is converted to an improper fraction, arithmetic calculations become straightforward. For example: \(7 \frac{2}{3}\) is converted to \(\frac{23}{3}\), making it easier to handle in addition, subtraction, multiplication, and division.

Here’s a quick recap:

• Improper fractions simplify complex mixed numbers
• They aid in straightforward arithmetic operations
• They are useful in converting between different types of fractions

reciprocal

A reciprocal of a fraction is simply flipping its numerator and denominator. For instance, the reciprocal of \(\frac{5}{2}\) is \(\frac{2}{5}\).

Reciprocals are especially useful when dividing fractions. When we divide by a fraction, we multiply by its reciprocal. For example:

• The reciprocal of \(\frac{5}{2}\) is \(\frac{2}{5}\)
• So, \(\frac{23}{3} \div \frac{5}{2}\) becomes \(\frac{23}{3} \times \frac{2}{5}\)

This transformation simplifies the division process, turning it into a multiplication problem, which is easier to solve.

simplifying fractions

Simplifying fractions means reducing them to their simplest form. A fraction is in its simplest form if the Greatest Common Divisor (GCD) of its numerator and denominator is 1. For example, if we have \(\frac{46}{15}\), we need to check:

• Find the GCD of 46 and 15, which is 1
• Since the GCD is 1, \(\frac{46}{15}\) is already in its simplest form

Simplification makes fractions easier to understand and work with. Always ensure your final answer is in simplified form to make sure it’s fully correct.