Problem 39 - Fundamentals of Mathematics

Question

What part of \(\frac{9}{10}\) is \(3 \frac{3}{5}\) ?

Step-by-Step Solution

Answer

The part of \(\frac{9}{10}\) that is \(3 \frac{3}{5}\) is 4.

Step 1: Convert Mixed Number to Improper Fraction

First, convert the mixed number \(3 \frac{3}{5}\) to an improper fraction. \(3 \frac{3}{5}\) can be written as \(\frac{18}{5}\) because \(3\) is equivalent to \(\frac{15}{5}\), and adding \(\frac{3}{5}\) gives \(\frac{18}{5}\).

Step 2: Divide Fraction by Fraction

The problem asks what part of \(\frac{9}{10}\) is \(\frac{18}{5}\). To find this, divide \(\frac{18}{5}\) by \(\frac{9}{10}\).

Step 3: Change Division to Multiplication

Division by a fraction is the same as multiplying by its reciprocal. Therefore, change \(\frac{18}{5} \div \frac{9}{10}\) to \(\frac{18}{5} \times \frac{10}{9}\).

Step 4: Multiply the Fractions

Multiply the numerators and denominators: \(\frac{18 \times 10}{5 \times 9} = \frac{180}{45}\).

Step 5: Simplify the Resulting Fraction

Simplify \(\frac{180}{45}\) by dividing the numerator and the denominator by their greatest common divisor, which is \(45\). This gives \(\frac{180 \div 45}{45 \div 45} = \frac{4}{1}\).

Step 6: Express the Answer as a Whole Number

Since \(\frac{4}{1}\) is equal to \(4\), the solution is a whole number.

Key Concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. This means it is a number that lies between two whole numbers. For instance, in the exercise, the mixed number is \(3 \frac{3}{5}\). Here, \(3\) is the whole number, and \(\frac{3}{5}\) is the fraction. Mixed numbers are often used because they are simpler to read and interpret in certain contexts. For example, \(3 \frac{3}{5}\) is easier to visualize in terms of real-world applications like measurements.To work with mixed numbers in calculations, it is important to convert them to improper fractions. This process involves multiplying the whole number by the denominator of the fraction and adding it to the numerator. This gives a single fraction that is greater than one. In our example, we convert \(3 \frac{3}{5}\) as follows:

• Multiply the whole number \(3\) by the denominator \(5\) to get \(15\).
• Add the numerator \(3\) to \(15\) to get \(18\).
• Place \(18\) over the original denominator \(5\), resulting in \(\frac{18}{5}\).

Improper Fractions

Improper fractions have a numerator that is larger than or equal to the denominator. This is in contrast to a proper fraction, where the numerator is smaller than the denominator. When dealing with improper fractions, it means that the value is greater than or equal to one.In the context of our exercise, the mixed number \(3 \frac{3}{5}\) was converted into an improper fraction \(\frac{18}{5}\). This conversion is essential because it allows us to perform division operations more efficiently. Once a fraction is improper, it can easily be used in arithmetic calculations like multiplication and division without the difficulty of handling whole number components separately. Remember, an improper fraction can be converted back to a mixed number for easier interpretation or end results.

Reciprocal

The reciprocal of a fraction is simply a flipped version where the numerator becomes the denominator and the denominator becomes the numerator. Understanding and using reciprocals is crucial, particularly when dealing with division of fractions. This is because dividing by a fraction is equivalent to multiplying by its reciprocal.For example, if you have the fraction \(\frac{9}{10}\), its reciprocal is \(\frac{10}{9}\). In the exercise, when dividing \(\frac{18}{5}\) by \(\frac{9}{10}\), you actually multiply \(\frac{18}{5}\) by \(\frac{10}{9}\). This switch from division to multiplication using the reciprocal simplifies the calculation process significantly. Remember:

• To find a reciprocal, just interchange the numerator and the denominator.
• Multiplying by the reciprocal is the same as dividing by the original fraction.

Simplifying Fractions

Simplifying fractions is the process of reducing fractions to their simplest form so they are easier to read and use. This involves dividing the numerator and the denominator by their greatest common divisor (GCD).In the given exercise, after multiplying the fractions \(\frac{18}{5}\) and \(\frac{10}{9}\), the result was \(\frac{180}{45}\). This fraction can be simplified by identifying the GCD of \(180\) and \(45\), which is \(45\). By dividing both the numerator and the denominator by \(45\), we simplify:

• \(180 \div 45 = 4\)
• \(45 \div 45 = 1\)
• Resulting in the simplified fraction \(\frac{4}{1}\), or simply \(4\).

Simplifying not only provides a cleaner answer but also ensures that calculations remain manageable, particularly in larger computations.