Problem 116 - Elementary Algebra

Question

For exercises 115-118, evaluate. On a scientific calculator, use the fraction key. For graphing calculators, use the Frac command to rewrite the decimal answer as a fraction. $$ \frac{5}{34}-\frac{8}{21} $$

Step-by-Step Solution

Answer

\(\frac{5}{34} - \frac{8}{21} = \frac{-167}{714}\)

Step 1: Find a Common Denominator

To subtract two fractions, convert them to have a common denominator. The denominators are 34 and 21. The least common multiple (LCM) of 34 and 21 is 714.

Step 2: Convert Fractions

Convert each fraction so that both have the denominator of 714. This is done by finding equivalent fractions. \(\frac{5}{34} = \frac{5 \times 21}{34 \times 21} = \frac{105}{714}\) \(\frac{8}{21} = \frac{8 \times 34}{21 \times 34} = \frac{272}{714}\)

Step 3: Subtract the Nombrators

Now that the fractions \(\frac{105}{714}\) and \(\frac{272}{714}\) have a common denominator, subtract the numerators: \(\frac{105}{714} - \frac{272}{714} = \frac{105-272}{714} = \frac{-167}{714}\)

Step 4: Simplify the Fraction

Simplify the fraction \(\frac{-167}{714}\). After checking, we find that \(-167\) and 714 do not have common factors other than 1. Therefore \(\frac{-167}{714}\) is already in its simplest form.

Key Concepts

common denominator

To subtract fractions, the first step is to ensure they have a common denominator. The common denominator allows us to compare and operate on the fractions directly.
To find a common denominator, we find a number that both of the original denominators divide into evenly.

In our given problem, the denominators are 34 and 21. To find a shared, larger number that both 34 and 21 can multiply into, we look for their Least Common Multiple (LCM).
This will allow us to convert each fraction so they share this common base, making the subtraction straightforward.

least common multiple

Least Common Multiple (LCM) is the smallest number that is a multiple of two or more denominators. Finding the LCM is crucial when dealing with fractions with different denominators.

For 34 and 21:

• First, we can list multiples of both numbers until we find a common one: Multiples of 34: 34, 68, 102, 136, 170, ... Multiples of 21: 21, 42, 63, 84, 105, ...
• We see the smallest common multiple is 714.

Thus, 714 is the LCM. Now, we can convert each fraction so that both have the same denominator, 714.

equivalent fractions

Once we've found the LCM, the next step is to convert the original fractions to equivalent fractions with the common denominator. This involves multiplying the numerator and the denominator by the same number.

For example, converting \frac{5}{34} to have a denominator of 714 involves multiplying both the numerator and denominator by 21: \(\frac{5*21}{34*21}=\frac{105}{714}\).
Similarly, converting \frac{8}{21} to have a denominator of 714 involves multiplying both the numerator and the denominator by 34: \(\frac{8*34}{21*34}=\frac{272}{714}\).

Now we have two fractions, \(\frac{105}{714}\) and \(\frac{272}{714}\), that we can subtract easily.

simplifying fractions

After subtracting the fractions, the result may need to be simplified. Simplifying a fraction means reducing it to its lowest terms, where the numerator and denominator can no longer be divided by the same number other than 1.

In our exercise, after subtracting the fraction components, we are left with \(\frac{105}{714}-\frac{272}{714}=\frac{-167}{714}\). To check if \(\frac{-167}{714}\) can be simplified, we find the greatest common factor (GCF) of 167 and 714. Since the only common factor is 1, the fraction is already in its simplest form.
Thus, \(\frac{-167}{714}\) is the final, simplified answer.