Problem 83 - Algebra 1: Concepts and Skills

Question

Write the answer as a fraction or as a mixed number in lowest terms. (Skills Review p. 764) $$\frac{3}{4}+\frac{8}{9}$$

Step-by-Step Solution

Answer

1 \(\frac{23}{36}\)

Step 1: Finding a Common Denominator

The aim is to find a common denominator for the two fractions \( \frac{3}{4} \) and \( \frac{8}{9} \). A common denominator can be found by finding the least common multiple (LCM) of the two denominators, which are 4 and 9. The LCM of 4 and 9 is 36. This will be the common denominator

Step 2: Making the Fractions Comparable

The original fractions \( \frac{3}{4} \) and \( \frac{8}{9} \) need to be expressed in terms of the common denominator, 36. For the first fraction \( \frac{3}{4} \), multiply the numerator and denominator by 9 to get \( \frac{27}{36} \). For the second fraction \( \frac{8}{9} \), multiply the numerator and denominator by 4 to get \( \frac{32}{36} \).

Step 3: Adding the Fractions

Now that the fractions have the same denominator, they can be added. Add the two fractions to get \( \frac{27}{36} + \frac{32}{36} = \frac{59}{36} \).

Step 4: Simplifying the Result

The result, \( \frac{59}{36} \), is an improper fraction. It can be converted to a mixed number by dividing the numerator by the denominator.

Step 5: Converting the Result to a Mixed Number

Dividing 59 by 36 gives 1 with a remainder of 23. So, \( \frac{59}{36} \) equals to \( 1 \frac{23}{36} \).

Key Concepts

Least Common Multiple

One of the key steps in adding fractions is finding the Least Common Multiple (LCM) of their denominators. The LCM is the smallest number that both denominators divide into without leaving a remainder.
For our fractions, \( \frac{3}{4} \) and \( \frac{8}{9} \), the denominators are 4 and 9.

• List multiples of 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, ...
• List multiples of 9: 9, 18, 27, 36, ...

The smallest multiple they both share is 36, so 36 is the LCM. Using the LCM as a common denominator allows us to easily add the fractions by converting each fraction so that they share the same denominator.

Simplifying Fractions

After adding fractions, it's often necessary to simplify the result. Simplifying means making the fraction as simple as possible, typically by reducing it.
Once you reach an answer, check if both the numerator and the denominator have any common factors. If they do, divide them by their greatest common factor.
In our addition of \( \frac{3}{4} + \frac{8}{9} \), the result was \( \frac{59}{36} \).
In this case, 59 and 36 do not have common factors other than 1, so \( \frac{59}{36} \) is already in its simplest form.

Improper Fractions

An improper fraction is a fraction where the numerator (the top number) is greater than or equal to the denominator (the bottom number).
For example, \( \frac{59}{36} \) is improper because 59 is larger than 36.

• Improper fractions can also be expressed as mixed numbers, which can be easier to understand and visualize.
• To convert an improper fraction to a mixed number, divide the numerator by the denominator.

In our solution, \( \frac{59}{36} \) gives us 1 with a remainder of 23, which becomes the mixed number \( 1 \frac{23}{36} \).

Mixed Numbers

A mixed number combines a whole number and a fraction. It's a convenient way to express quantities that exceed a whole unit.
Mixed numbers aid in visualizing amounts more clearly, especially when fractions are involved.
When converting an improper fraction like \( \frac{59}{36} \) into a mixed number, you perform a division: 59 divided by 36 gives us:

• The whole number part: 1
• The remainder 23 becomes the numerator of the fractional part.
• The denominator remains the same: 36.

Thus, \( \frac{59}{36} \) converts to the mixed number \( 1 \frac{23}{36} \), which balances the equation in a more manageable form.