Problem 29 - Algebra 1

Question

Solve the equation. $$r+3 \frac{1}{4}=2 \frac{1}{2}$$

Step-by-Step Solution

Answer

The value of \(r\) is \(\frac{-3}{4}\)

Step 1: Convert Mixed Fractions to Improper Fractions

A mixed fraction can be converted to an improper fraction using the formula: \(a \frac{b}{c} = \frac{ac + b}{c}\), where \(a\) is the whole part, \(b\) is the numerator and \(c\) is the denominator. Thus, \(r + 3 \frac{1}{4} = r + \frac{13}{4}\) and \(2 \frac{1}{2} = \frac{5}{2}\). The equation becomes \(r + \frac{13}{4} = \frac{5}{2}\).

Step 2: Simplify the Equation

Subtract \(\frac{13}{4}\) from both sides to isolate \(r\). The equation then becomes \(r = \frac{5}{2} - \frac{13}{4} \). Calculate the right side to find the value of \(r\).

Step 3: Solve for \(r\)

We need to take note that before we subtract fractions, they must have the same denominator. We already have the same denominator, so we just subtract the numerators: \(r = \frac{5*2 - 13*1}{4} = \frac{-3}{4}\).

Key Concepts

Fractions

Fractions are a crucial part of mathematics that help us represent numbers that are not whole. They consist of two parts: the numerator and the denominator. The numerator sits at the top and tells us how many parts we have. The denominator is at the bottom and indicates into how many parts the whole is divided.

• Example: In the fraction \(\frac{3}{4}\), the numerator is 3, and the denominator is 4.
• Fractions represent parts of a whole, like a slice of pizza from an entire pizza.

To work with fractions, you need to learn operations like addition, subtraction, multiplication, and division. When dealing with these operations, it's often necessary to find a common denominator, especially in addition and subtraction.

Fractions can also be transformed into decimal or percentage forms. For example, \(\frac{1}{2}\) converts to 0.5 as a decimal and 50% as a percentage. Understanding these conversions helps solve equations involving fractions.

Mixed Numbers

Mixed numbers are numbers composed of a whole part and a fraction, like \(2 \frac{1}{2}\). They show a quantity more than a whole number but less than another whole number. Mixed numbers often appear in daily life, like when measuring ingredients in cooking.

• Example: \(3 \frac{1}{4}\) means you have 3 whole units plus a quarter of another.
• The mixed numeral \(a \frac{b}{c}\) can be rewritten as an improper fraction through the formula: \(\frac{ac + b}{c}\).

To handle mixed numbers in calculations, especially with equations, they are usually converted to improper fractions. This conversion helps simplify the operations since improper fractions eliminate the whole number component, allowing for straightforward arithmetic manipulation.

Always remember to convert them back to mixed numbers or simplest terms if needed, so the result fits the context of the problem better.

Improper Fractions

Improper fractions have numerators greater than or equal to their denominators, making them a mix between a whole number and a fraction, like \(\frac{9}{4}\). This means the fraction represents a number greater than 1.

• Example: \(\frac{7}{3}\) is an improper fraction, as 7 is greater than 3.
• They can be very handy when solving mathematical problems involving fractions.

Improper fractions are especially useful because they simplify mathematical operations compared to mixed numbers. Adding, subtracting, or multiplying is easier with improper fractions because they follow simple arithmetic rules.

Whenever you encounter a mixed number in a problem, converting it to an improper fraction is a crucial step. Post calculation, you might convert it back to a mixed number for the final answer to make it more understandable. Understanding this conversion helps improve your skills in solving equations involving fractions.