Problem 6 - Fundamentals of Mathematics

Question

Find the reciprocal of each number. $$5 \frac{1}{4}$$

Step-by-Step Solution

Answer

The reciprocal of \(5 \frac{1}{4}\) is \(\frac{4}{21}\).

Step 1: Convert the Mixed Number to an Improper Fraction

The first step is to convert the mixed number into an improper fraction. A mixed number consists of a whole number and a fraction. For the mixed number \(5 \frac{1}{4}\), multiply the whole number 5 by the denominator 4 and add the numerator 1. This gives us \(5 \times 4 + 1 = 21\), so \(5 \frac{1}{4}\) is equivalent to \(\frac{21}{4}\).

Step 2: Find the Reciprocal of the Fraction

To find the reciprocal of a fraction, simply swap the numerator and the denominator. For the fraction \(\frac{21}{4}\), the reciprocal is \(\frac{4}{21}\).

Step 3: Verify the Reciprocation

Verify that the computed reciprocal is correct by ensuring that multiplying the fraction \(\frac{21}{4}\) with its reciprocal \(\frac{4}{21}\) results in 1. Multiplying these gives \(\frac{21}{4} \times \frac{4}{21} = \frac{21 \times 4}{4 \times 21} = \frac{84}{84} = 1\). Thus, the reciprocal calculation is correct.

Key Concepts

Mixed Number

A mixed number is a way of expressing a number that includes both a whole number and a fraction. It's a friendly way of representing numbers greater than one in a way that's often easier to comprehend. For example, in the mixed number \(5 \frac{1}{4}\), the \(5\) is the whole number, and \(\frac{1}{4}\) represents the fractional part. Mixed numbers are common in everyday situations, like measuring ingredients in cooking.

• Whole number and a fraction together
• Easy representation for numbers larger than one

Converting a mixed number to an improper fraction is often necessary for mathematical operations, especially when finding reciprocals. We do this by multiplying the whole number by the fraction's denominator and then adding the fractional numerator. This conversion is essential to ensure that you can easily perform other operations such as multiplying or dividing fractions.
Understanding how to work with mixed numbers is a crucial skill for various math problems and practical scenarios. Practice converting mixed numbers, and you'll find many math problems become simpler.

Improper Fraction

An improper fraction is where the numerator is larger than or equal to the denominator. It's called 'improper' because traditionally, fractions have a smaller numerator than the denominator, thus representing less than one. For instance, when we convert the mixed number \(5 \frac{1}{4}\) into an improper fraction, it becomes \(\frac{21}{4}\). Here, 21 is greater than 4, indicating a value greater than one.

• Numerator larger than the denominator
• Represents values greater than or equal to one

Improper fractions are useful for calculations like multiplications or divisions and for finding reciprocals. They streamline these processes because they maintain a consistent format. To convert back and forth between mixed numbers and improper fractions, remember to multiply and add to get to the improper fraction form, while dividing gives you the mixed number.
Understanding improper fractions unlocks the ability to perform more complex arithmetic operations, making them an essential part of your math toolkit.

Multiply Fractions

Multiplying fractions is one of the fundamental operations you must understand. It involves multiplying the numerators together and the denominators together. When fractions are multiplied, the result is a product of the numerators over the product of the denominators. For example, multiplying \(\frac{21}{4}\) by its reciprocal \(\frac{4}{21}\) leads to:\[\frac{21}{4} \times \frac{4}{21} = \frac{21 \times 4}{4 \times 21} = \frac{84}{84} = 1\]This operation, when done with reciprocals, is a great way to verify that you have found the correct reciprocal.

• Multiply numerators together
• Multiply denominators together
• The product of reciprocal fractions equals one

Multiplying fractions directly requires you to handle the basics of numerators and denominators confidently. Always simplify the fraction if possible, and remember the special property that any fraction times its reciprocal equals one. Mastering these basics helps in effortlessly navigating through fraction-related problems.