Problem 101 - Fundamentals of Mathematics

Question

What part of \(4 \frac{1}{16}\) is \(3 \frac{3}{4} ?\)

Step-by-Step Solution

Answer

\(3 \frac{3}{4}\) is \(\frac{12}{13}\) of \(4 \frac{1}{16}\).

Step 1: Convert Mixed Numbers to Improper Fractions

First, we'll convert the mixed numbers into improper fractions. For \(4 \frac{1}{16}\), multiply the whole number 4 by the denominator 16 and add the numerator 1. This gives \(4 \times 16 + 1 = 65\), so \(4 \frac{1}{16} = \frac{65}{16}\). For \(3 \frac{3}{4}\), multiply the whole number 3 by the denominator 4 and add the numerator 3. This gives \(3 \times 4 + 3 = 15\), so \(3 \frac{3}{4} = \frac{15}{4}\).

Step 2: Calculate the Part

To find what part \(3 \frac{3}{4}\) is of \(4 \frac{1}{16}\), divide \(\frac{15}{4}\) by \(\frac{65}{16}\). This is equivalent to multiplying \(\frac{15}{4}\) by the reciprocal of \(\frac{65}{16}\), which is \(\frac{16}{65}\). Therefore, compute \(\frac{15}{4} \times \frac{16}{65}\).

Step 3: Multiply Fractions

Multiply the numerators: \(15 \times 16 = 240\), and multiply the denominators: \(4 \times 65 = 260\). This gives us \(\frac{240}{260}\).

Step 4: Simplify the Fraction

Simplify \(\frac{240}{260}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 20. This results in \(\frac{12}{13}\).

Key Concepts

Understanding Improper Fractions

An improper fraction is one where the numerator, which is the top part of the fraction, is larger than the denominator, the bottom part. This means the fraction represents more than one whole unit. Improper fractions might seem odd at first, but they are quite useful.
For example, in the problem above, the mixed number \(4 \frac{1}{16}\) is converted to an improper fraction \(\frac{65}{16}\). To do this, multiply the whole number by the denominator and then add the numerator: \(4 \times 16 + 1 = 65\). The resulting fraction, \(\frac{65}{16}\), shows exactly how many equal parts are within that mixed number.
This conversion is essential as it allows you to perform arithmetic operations like addition, subtraction, multiplication, or division of fractions efficiently.

The Importance of Reciprocals in Fraction Operations

Reciprocals are one of the key concepts when working with fractions, particularly when dividing them. When you take the reciprocal of a fraction, you simply flip the numerator and the denominator.
For instance, the reciprocal of \(\frac{65}{16}\) is \(\frac{16}{65}\). Why is this important? In division problems involving fractions, you multiply by the reciprocal of the divisor.
So, dividing \(\frac{15}{4}\) by \(\frac{65}{16}\) means multiplying \(\frac{15}{4}\) by \(\frac{16}{65}\). By using reciprocals, we transform the division into a multiplication problem, which is often easier to solve.
This step is crucial as it paves the way for simplifying complex fraction calculations.

Simplifying Fractions

Simplifying fractions means reducing them to their simplest form. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD). A fully simplified fraction is the easiest to understand.
In the exercise solution, once the multiplication was complete, we ended up with \(\frac{240}{260}\). To simplify it, determine the greatest common divisor of 240 and 260, which is 20.
After dividing the numerator and denominator by their GCD, the fraction becomes \(\frac{12}{13}\). This final fraction provides a clear and concise result of the original problem.
Always remember, simplifying fractions often makes them easier to interpret and applies to all types of arithmetic operations.

Finding the Greatest Common Divisor (GCD)

To simplify fractions effectively, understanding how to find the greatest common divisor is essential. The GCD of two numbers is the largest number that divides both without leaving a remainder.
In our exercise, to simplify \(\frac{240}{260}\), we needed to find the GCD of 240 and 260.
Here are steps to find GCD:

• List the factors of each number.
• Identify the common factors.
• Select the largest common factor—this is the GCD.

Using these steps, we find that 20 is the GCD of 240 and 260. Once you have the GCD, simplifying becomes a straightforward task by dividing the numerator and the denominator by this number.
Learning to determine the GCD will enable you to simplify any fraction quickly and accurately, which is an invaluable skill in both mathematics and applied sciences.