Problem 64 - Algebra 2

Question

Find each sum if \(a=\frac{1}{2}, b=\frac{1}{6}, c=\frac{2}{3},\) and \(d=\frac{3}{4}\) $$ a+d $$

Step-by-Step Solution

Answer

\( \frac{5}{4} \) or \( 1 \frac{1}{4} \)

Step 1: Identify the Variables

We start with the values given in the problem: \( a = \frac{1}{2} \), \( b = \frac{1}{6} \), \( c = \frac{2}{3} \), and \( d = \frac{3}{4} \), and we need to find the sum \( a + d \).

Step 2: Align the Fractions for Addition

To add the fractions, \( a = \frac{1}{2} \) and \( d = \frac{3}{4} \), first ensure they have a common denominator. The denominators here are 2 and 4. The least common denominator is 4.

Step 3: Convert to a Common Denominator

Convert \( \frac{1}{2} \) into a fraction with a denominator of 4. Multiply both the numerator and denominator by 2: \( \frac{1}{2} \times \frac{2}{2} = \frac{2}{4} \). Now, \( d = \frac{3}{4} \) is already with denominator 4.

Step 4: Add the Fractions

With common denominators, add \( \frac{2}{4} \) and \( \frac{3}{4} \) together: \[ \frac{2}{4} + \frac{3}{4} = \frac{2+3}{4} = \frac{5}{4}. \]

Step 5: Simplify the Fraction Result

The sum \( \frac{5}{4} \) is an improper fraction. It can be expressed as a mixed number: \( 1 \frac{1}{4} \).

Key Concepts

Improper Fractions

An improper fraction is a fraction where the numerator (the top number) is larger than the denominator (the bottom number). This can look a bit strange because we're used to seeing fractions with smaller numerators, but improper fractions can be really useful for calculations. They tell us something is more than one whole. Imagine you have a pizza cut into 4 slices.

• If you have 5 slices, it means you have more than one whole pizza.
• This is represented as \( \frac{5}{4} \).

It's important to recognize improper fractions because they frequently appear when adding fractions with a common denominator. Improper fractions can also be converted into mixed numbers, which can help make them easier to understand.

Common Denominator

The common denominator is crucial when adding or subtracting fractions. Fractions must have the same denominator to be correctly added or subtracted. This makes it possible to combine the numerators directly. In our example, we needed to add \( \frac{1}{2} \) and \( \frac{3}{4} \). The denominators were originally 2 and 4.

• To find the least common denominator, identify the smallest number that both denominators can divide into evenly.
• Here, the least common denominator is 4 because both 2 and 4 can fit into it.

Once you have a common denominator, you can convert all fractions to that denominator, making them ready to add together smoothly.

Mixed Numbers

A mixed number combines a whole number with a proper fraction. Imagine you've borrowed 1 full book and another half a book. This can be represented as \( 1 \frac{1}{2} \), which means 1 whole plus a fraction.When we add fractions and end up with an improper fraction, it's often helpful to convert it to a mixed number:

• This gives us a clearer picture of how much more than one whole we have.
• For instance, \( \frac{5}{4} \) can be rewritten as \( 1 \frac{1}{4} \).

Using mixed numbers can simplify understanding quantities, especially when they are close to or beyond a whole number.

Converting Fractions

Converting fractions is a fundamental skill that helps simplify expressions and problems in math. Whether you're converting proper fractions to improper ones, improper fractions to mixed numbers, or vice versa, it's critical to understand how conversion works.

• To convert \( \frac{1}{2} \) to have a denominator of 4, you multiply both the numerator and denominator by 2, yielding \( \frac{2}{4} \).
• To change an improper fraction like \( \frac{5}{4} \) into a mixed number, divide the numerator by the denominator. The quotient becomes the whole number, and the remainder becomes the new numerator over the original denominator, creating \( 1 \frac{1}{4} \).

Understanding these conversions will help in performing calculations quickly and accurately, making math problems less daunting.