Problem 90 - Fundamentals of Mathematics

Question

Perform each multiplication and division. $$17 \div 4 \frac{1}{4}$$

Step-by-Step Solution

Answer

The result is 4.

Step 1: Convert Mixed Number to Improper Fraction

To solve this problem, start by converting the mixed number \(4 \frac{1}{4}\) into an improper fraction. Multiply the whole number (4) by the denominator (4) of the fractional part, and then add the numerator (1). This gives you: \(4 \times 4 + 1 = 17\). So the improper fraction is \(\frac{17}{4}\).

Step 2: Rewrite the Division as Multiplication with Reciprocal

When dividing by a fraction, you can rewrite the expression by multiplying by the reciprocal of that fraction. Hence, \(17 \div \frac{17}{4}\) becomes \(17 \times \frac{4}{17}\).

Step 3: Set Up the Multiplication

The multiplication now is between a whole number and a fraction: \(17 \times \frac{4}{17}\).

Step 4: Simplify the Multiplication

When multiplying integer by a fraction, multiply the integer by the numerator of the fraction. However, since both the numerator of the fraction (4) shares a common factor with the integer 17 in the denominator, you can cancel them out directly, leading to \(1\).

Step 5: Calculate Final Result

After canceling out the common factors, you are left with \(4\) in the numerator and \(1\) in the denominator, leading to the simplified result of \(4\).

Key Concepts

Converting Mixed Numbers to Improper Fractions

When dealing with mixed numbers, the process of converting them to improper fractions is essential for calculations like division or multiplication. A mixed number consists of a whole number and a fraction. For example, let's consider the mixed number \(4 \frac{1}{4}\). To convert this into an improper fraction:

• Multiply the whole number by the denominator of the fractional part. Here, it's \(4 \times 4\).
• Add the result to the numerator of the fractional part. That’s \((4 \times 4) + 1 = 17\).

This results in the improper fraction \(\frac{17}{4}\). By transposing the whole number and its associated fraction into one single fraction with a numerator greater than the denominator, it becomes easier to manage in multiplication and division scenarios.
Remember, converting mixed numbers is a key step before performing operations with fractions.

Reciprocal in Division

The concept of using a reciprocal in division is foundational in fraction division. To divide by a fraction, multiply by its reciprocal instead. For instance, the problem \(17 \div \frac{17}{4}\) can be changed to a multiplication problem: \(17 \times \frac{4}{17}\).
Here's how it works:

• Identify the fraction you want to divide by—in this case, \(\frac{17}{4}\).
• Flip the fraction to find its reciprocal. The numerator becomes the denominator and vice versa, resulting in \(\frac{4}{17}\).
• Multiply the original number by this reciprocal. The division problem then becomes a much simpler multiplication problem.

Utilizing reciprocals simplifies division, making computations with fractions more straightforward and less prone to error.This approach is particularly useful when simplifying complex expressions.

Simplifying Fractions

Simplifying fractions involves reducing them to their simplest form, making them easier to interpret and work with in further calculations. When you multiply a whole number by a fraction and both share a common factor, you can simplify the calculation by canceling common factors in the numerator and denominator.
In the expression \(17 \times \frac{4}{17}\), observe:

• Both the whole number 17 and the denominator 17 can cancel each other out because they are the same.
• After canceling, you are left with \(4 \times 1\), which is \(4\).

The reduced multiplication shows the result of \(4\), simplifying the expression to its lowest terms.
By identifying and eliminating common factors early on, we can quickly arrive at the simplest form, ensuring clear and concise results.