Problem 53 - Prealgebra

Question

Solve each equation. Check your solution. $$x+\frac{5}{6}=2 \frac{3}{8}$$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{37}{24}\).

Step 1: Convert Mixed Number to Improper Fraction

First, we need to convert the mixed number \(2 \frac{3}{8}\) into an improper fraction. To do this, multiply the whole number by the denominator and add the numerator. \[2 \frac{3}{8} = \frac{2 \times 8 + 3}{8} = \frac{16 + 3}{8} = \frac{19}{8}\]

Step 2: Write the Equation with Fractions

Substitute the improper fraction back into the equation: \[ x + \frac{5}{6} = \frac{19}{8} \]

Step 3: Isolate the Variable

Subtract \(\frac{5}{6}\) from both sides to solve for \(x\): \[x = \frac{19}{8} - \frac{5}{6}\]

Step 4: Find a Common Denominator

In order to subtract the fractions, find a common denominator. Here, the least common denominator for 8 and 6 is 24. Rewrite the fractions: \[\frac{19}{8} = \frac{19 \times 3}{24} = \frac{57}{24} \]\[\frac{5}{6} = \frac{5 \times 4}{24} = \frac{20}{24} \]

Step 5: Subtract the Fractions

Subtract the numerators of the fractions: \[x = \frac{57}{24} - \frac{20}{24} = \frac{37}{24}\]

Step 6: Check the Solution

Substitute \(x = \frac{37}{24}\) back into the original equation to verify. Calculate the left side:\[x + \frac{5}{6} = \frac{37}{24} + \frac{20}{24} = \frac{57}{24} = \frac{19}{8} = 2 \frac{3}{8} \]Since both sides of the equation are equal, our solution is verified.

Key Concepts

Fractions

Fractions are a way to represent numbers that are not whole. They consist of two parts: a numerator and a denominator. The numerator is the top part and tells us how many parts we have, while the denominator is the bottom part and tells us how many equal parts the whole is divided into. For example, in the fraction \(\frac{5}{6}\), 5 is the numerator and 6 is the denominator. This means you have 5 parts out of a total of 6.

Understanding fractions is crucial when solving equations, especially when the equation involves adding or subtracting fractions. This requires understanding how to find a common denominator, which is a shared multiple of the fractions' denominators. Having a strong grasp of fractions allows you to work comfortably with mixed numbers and improper fractions as well.

Mixed Numbers

Mixed numbers are combinations of whole numbers and fractions. They look like \(2 \frac{3}{8}\), which means you have 2 whole parts and \(\frac{3}{8}\) of another part. Mixed numbers are handy because they present numbers in a more intuitive way. However, when solving equations, it’s often easier to convert mixed numbers into improper fractions.

To convert a mixed number to an improper fraction, you multiply the whole number by the denominator and add the numerator. For example:

• Whole number: 2
• Fraction: \(\frac{3}{8}\)
• \(2 \times 8 + 3 = 19\)

This calculation transforms \(2 \frac{3}{8}\) into \(\frac{19}{8}\). Once in improper form, it is easier to manipulate and solve equation problems.

Improper Fractions

Improper fractions are fractions where the numerator is larger than the denominator, such as \(\frac{19}{8}\). These fractions might look less familiar, but they are very useful in mathematical calculations. Turning mixed numbers into improper fractions is a key step in making fractions easier to calculate and solve.

When dealing with improper fractions, always ensure your arithmetic operations (like addition or subtraction) are correct. For example, if you need to add or subtract improper fractions, have a common denominator ready. This will ease the calculation process and ensure you can accurately find solutions to problems and equations involving these types of fractions.

Common Denominator

Finding a common denominator is essential when you add or subtract fractions. This common denominator should be the least common multiple of the denominators in question.

For example, if you have the fractions \(\frac{19}{8}\) and \(\frac{5}{6}\), you need to find a number that both 8 and 6 divide into without leaving a remainder. In this case, the least common denominator would be 24. Once you have a common denominator, you can then rewrite each fraction:

• \(\frac{19}{8} = \frac{57}{24}\)
• \(\frac{5}{6} = \frac{20}{24}\)

With both fractions expressed with a common denominator, it's straightforward to perform calculations such as subtraction or addition. By understanding how to find and use a common denominator, solving fraction problems becomes far simpler and more intuitive. This helps not only in calculations but also in understanding the relationship between different fractional parts.