Problem 77 - Algebra 1: Concepts and Skills

Question

Find the reciprocal of the mixed number. Write your answer in lowest terms. $$ 1 \frac{3}{10} $$

Step-by-Step Solution

Answer

The reciprocal of the mixed number \(1 \frac{3}{10}\) in lowest terms is \( \frac{10}{13} \).

Step 1: Converting Mixed Number into Improper Fraction

First, the mixed number is converted into an improper fraction. The formula for this conversion is \( \frac{(whole \ number \times denominator) + numerator}{denominator} \). Applying this to \(1 \frac{3}{10}\), we have \( \frac{(1 \times 10) + 3}{10} \) which equals \( \frac{13}{10} \).

Step 2: Finding the Reciprocal

The reciprocal of the fraction is found by switching the numerator and the denominator. So the reciprocal of \( \frac{13}{10} \) is \( \frac{10}{13} \) as both numbers are prime numbers, this cannot be simplified further.

Key Concepts

Mixed Numbers

When learning about fractions, a mixed number is one of the first types of numbers you'll encounter. A mixed number is simply a whole number combined with a fraction, such as 1 3/10. This format of expressing numbers is quite handy when dealing with quantities that aren't perfect wholes but are more than one.

Mixed numbers are particularly prevalent in everyday life, like when measuring ingredients in cooking or marking measurements in construction. To do more complex operations with them, such as finding reciprocals, we often need to convert mixed numbers into improper fractions. It's this step that allows us to manage calculations more comfortably within the realm of fractions.

Improper Fractions

Improper fractions are another key concept in the study of fractions. These are simply fractions where the numerator (the top number) is larger than the denominator (the bottom number), like 13/10. These types of fractions may look incorrect at a glance, but they are perfectly valid and often more useful in mathematical calculations than their mixed number counterparts.

To convert a mixed number to an improper fraction, you multiply the whole number by the denominator of the fraction and then add the numerator. For example, for the mixed number 1 3/10, you'd multiply 1 (the whole number) by 10 (the denominator) and then add 3 (the numerator), getting 13/10, the improper fraction equivalent.

Simplifying Fractions

Simplifying fractions is an essential skill in math. This process involves reducing a fraction to its smallest possible form—where both the numerator and the denominator are as small as they can be, yet still represent the same value. For instance, 2/4 is the same as 1/2 when simplified because both represent the same quantity.

To simplify a fraction, you find the greatest common divisor (GCD) for both the numerator and the denominator, and then divide both by that number. However, when a fraction's numerator and denominator are primes with respect to each other, such as with 10/13, the fraction is already in its simplest form and cannot be simplified further.

Reciprocals

Finally, the concept of reciprocals is crucial when working with fractions. A reciprocal is simply the inverse of a number. In terms of fractions, to find a reciprocal, you flip the numerator and denominator. For example, the reciprocal of 13/10 is 10/13.

This is an important operation when dividing fractions or solving equations because multiplying by the reciprocal is the same as dividing by the original number. It's key to note that the reciprocals of mixed numbers first necessitate the conversion to improper fractions, as was done in our original example where 1 3/10 became 13/10 before we found its reciprocal.