Problem 7 - Fundamentals of Mathematics

Question

Find the reciprocal of each number. $$10 \frac{3}{16}$$

Step-by-Step Solution

Answer

The reciprocal of \(10 \frac{3}{16}\) is \(\frac{16}{163}\).

Step 1: Convert Mixed Number to Improper Fraction

The number given is a mixed number, \(10 \frac{3}{16}\). First, convert this to an improper fraction: Multiply the whole number (10) by the denominator (16) and add the numerator (3). \(10 \times 16 + 3 = 160 + 3 = 163\). Thus, \(10 \frac{3}{16} = \frac{163}{16}\).

Step 2: Find the Reciprocal

The reciprocal of a fraction is obtained by inverting the fraction. Therefore, the reciprocal of \(\frac{163}{16}\) is \(\frac{16}{163}\).

Step 3: Verify the Reciprocal

To verify, multiply \(\frac{163}{16}\) by its reciprocal, \(\frac{16}{163}\), to check if the result is 1: \(\frac{163}{16} \times \frac{16}{163} = \frac{163 \times 16}{16 \times 163} = \frac{2608}{2608} = 1\). The multiplication confirms that \(\frac{16}{163}\) is indeed the reciprocal.

Key Concepts

Mixed Number

A mixed number is a number that combines a whole number and a fraction. For example, in the mixed number \(10 \frac{3}{16}\), the 10 is the whole number and \(\frac{3}{16}\) is the fractional part. Mixed numbers are often used to express values that are more than one but not whole, making them very convenient in everyday settings. To work with mixed numbers mathematically, it's frequently useful to convert them into improper fractions. This is because improper fractions offer a straightforward way to perform operations like multiplication and division. The conversion is simple: multiply the whole number by the denominator and then add the numerator. This total becomes the numerator of the improper fraction, with the original denominator remaining the same.By understanding mixed numbers and their conversions, you can easily handle various arithmetic operations, much like the one in our example.

Improper Fraction

An improper fraction is a fraction where the numerator is larger than or equal to the denominator, such as \(\frac{163}{16}\). This type of fraction represents values that are at least as large as one whole unit. When dealing with improper fractions, the number can often be easier to manipulate in multiplication and division, as all the values are expressed in terms of a single fraction.In the example given, \(10 \frac{3}{16}\) was converted to \(\frac{163}{16}\). The conversion process took the whole number (10) and bundled it with the fractional part (\(\frac{3}{16}\)), leading to an improper fraction that's straightforward to use in further calculations. Using improper fractions, you gain a solid foundation for understanding more complex mathematical concepts, including finding reciprocals.

Verify Reciprocal

The concept of reciprocal refers to swapping the numerator and the denominator of a fraction. For any fraction \(\frac{a}{b}\), its reciprocal is \(\frac{b}{a}\). This concept is key to many mathematical operations, particularly in division and solving equations involving fractions. For the number to be its true reciprocal, multiplying the fraction by its reciprocal should result in 1.In this example, the reciprocal of \(\frac{163}{16}\) was found to be \(\frac{16}{163}\). The verification involved multiplying the original fraction with this reciprocal: \(\frac{163}{16} \times \frac{16}{163} = \frac{2608}{2608} = 1\). As the product equals 1, this confirms that \(\frac{16}{163}\) is indeed the correct reciprocal.Verifying reciprocals is a useful skill for ensuring the accuracy of your work, especially when working with equations and complex calculations.