Problem 79 - Algebra 1: Concepts and Skills

Question

Find the reciprocal of the mixed number. Write your answer in lowest terms. $$ 1 \frac{7}{50} $$

Step-by-Step Solution

Answer

The reciprocal of the mixed number \(1 \frac{7}{50}\) in lowest terms is \(\frac{50}{57}\).

Step 1: Convert Mixed Number to an Improper Fraction

An improper fraction is a fraction where the numerator is greater than the denominator. To convert a mixed number into an improper fraction, the whole number part of the mixed number is multiplied by the denominator of the fractional part, and then this product is added to the numerator of the fractional part. For the given mixed number \(1 \frac{7}{50}\), multiply the whole number 1 by the denominator 50, and add this product to the numerator 7. This gives:\(50*1 + 7 = 57\).So, the improper fraction form of mixed number \(1 \frac{7}{50}\) is \(\frac{57}{50}\).

Step 2: Find the Reciprocal of the Fraction

The reciprocal of a fraction is the fraction obtained by interchanging the numerator and the denominator. Hence, the reciprocal of the fraction \(\frac{57}{50}\) is \(\frac{50}{57}\).

Step 3: Express the Result in Lowest Terms

To express the fraction in lowest terms, find the greatest common divisor (gcd) of the numerator and the denominator. If the gcd is 1, the fraction is already in the lowest term. In this case, the gcd of 50 and 57 is 1. Hence, the fraction \(\frac{50}{57}\) is the reciprocal of \(1 \frac{7}{50}\) in lowest terms.

Key Concepts

Mixed Numbers

Mixed numbers are a way to express numbers that include both a whole number and a fractional part. This format is often used because it is easier to understand in everyday life compared to improper fractions. For example, if you have one and a half apples, you would write that as a mixed number:

• Whole Number: Represents complete units. For example, in the mixed number \(1 \frac{7}{50}\), "1" is the whole number part.
• Fractional Part: This symbolizes the part of a whole. The fraction in \(\frac{7}{50}\) indicates that 7 parts out of 50 are being considered.

To work with fractions mathematically, we often convert mixed numbers into improper fractions. This helps with operations like reciprocal finding.

Improper Fractions

An improper fraction is a type of fraction where the numerator — that's the number on top — is either equal to or greater than the denominator, which is the bottom number.

• It's "improper" because it's not the conventional way to interpret fractions. Normally, the "part" (numerator) should be less than the "whole" (denominator).
• In math, improper fractions are handy, especially for calculations like multiplication and finding reciprocals.

For example, the mixed number \(1 \frac{7}{50}\) turns into the improper fraction \(\frac{57}{50}\) by multiplying the whole part by the denominator and adding the numerator: 1. Multiply: \(1 \times 50 = 50\)2. Add Numerator: \(50 + 7 = 57\)Now, the improper fraction lets us easily find the reciprocal or engage in other calculations.

Lowest Terms

When a fraction is expressed in its simplest form, with no common factors between the numerator and the denominator other than 1, it is said to be in "lowest terms." Simplifying fractions is crucial because it makes them easier to understand and compare.

• Finding the lowest terms involves division using the greatest common divisor (GCD). If the GCD of two numbers is 1, they are already in lowest terms.
• This process sometimes drastically changes the appearance of the fraction, while maintaining its value.

For instance, the fraction \(\frac{50}{57}\) obtained from our exercise is already in lowest terms, because the GCD of 50 and 57 is 1. Here, no further simplification is needed, which confirms that it's expressed as simply as possible.

Greatest Common Divisor

The greatest common divisor (GCD) is a key concept in simplifying fractions. It is the largest number that divides exactly into two or more numbers without leaving a remainder. Finding the GCD helps to bring fractions into their lowest terms.

• By identifying the GCD, you determine how much you can "shrink" or simplify a fraction.
• This not only impacts computational ease but also ensures clarity in communication of numerical information.

To find the GCD, you can use multiple methods like the Euclidean algorithm or by listing factors. In our example, the GCD of 50 and 57 is indeed 1, complemented by the fact that both numbers share no common factors besides 1. Consequently, this means \(\frac{50}{57}\) is already in its simplest form.