Solving Equations A Step-by-Step Guide With Example

Solve the equation for s: $1 \frac{1}{4} = \frac{1}{12} + s$. Simplify the answer completely.

Understanding the Equation and the Goal

In this mathematical problem, we aim to solve the equation $1 \frac{1}{4} = \frac{1}{12} + s$ for the unknown variable s. Solving an equation means isolating the variable on one side of the equation to determine its value. To achieve this, we need to perform algebraic manipulations while maintaining the equality of both sides. The equation involves fractions, so we will need to work with them effectively. Fractions represent parts of a whole, and operations with fractions require a common denominator for addition and subtraction. Additionally, we have a mixed number ($1 \frac{1}{4}$), which we will convert to an improper fraction for easier calculation. Understanding these basic concepts is crucial for solving this equation successfully. The ultimate goal is to find the value of s that makes the equation true. This involves several steps, including converting the mixed number to an improper fraction, finding a common denominator, and isolating s by performing the inverse operation. By carefully following these steps, we can arrive at a simplified and accurate solution for s. Furthermore, once we find a solution, it's always a good practice to substitute it back into the original equation to verify its correctness. This ensures that our algebraic manipulations were accurate and that we have indeed found the correct value for s. This process is fundamental in algebra and forms the basis for solving more complex equations in the future. Mastering the techniques for solving such equations is a key skill in mathematics and has applications in various fields.

Step 1: Convert the Mixed Number to an Improper Fraction

The first step in solving the equation is to convert the mixed number, $1 \frac{1}{4}$, into an improper fraction. An improper fraction is a fraction where the numerator is greater than or equal to the denominator. This conversion is necessary because it simplifies the subsequent arithmetic operations. To convert a mixed number to an improper fraction, we multiply the whole number part by the denominator and add the numerator. This result becomes the new numerator, and the denominator remains the same. In this case, we have $1 \frac{1}{4}$. We multiply the whole number (1) by the denominator (4), which gives us 4. Then, we add the numerator (1) to this result, yielding 5. Thus, the new numerator is 5, and the denominator remains 4. Therefore, the improper fraction equivalent of $1 \frac{1}{4}$ is $\frac{5}{4}$. This conversion is crucial because it allows us to perform addition and subtraction with fractions more easily. When dealing with mixed numbers, it is often simpler to convert them into improper fractions before proceeding with any calculations. Now that we have converted the mixed number to an improper fraction, our equation becomes $\frac{5}{4} = \frac{1}{12} + s$. This form of the equation is easier to work with, and we can proceed to the next step, which involves isolating the variable s. By correctly converting the mixed number to an improper fraction, we have laid the groundwork for solving the equation and finding the value of s. Understanding this conversion process is essential for working with fractions and mixed numbers in various mathematical contexts.

Step 2: Find a Common Denominator

Now that we have the equation $\frac{5}{4} = \frac{1}{12} + s$, the next step is to isolate s. To do this, we need to subtract $\frac{1}{12}$ from both sides of the equation. However, before we can subtract fractions, they must have a common denominator. A common denominator is a multiple that the denominators of both fractions share. In our case, the denominators are 4 and 12. The least common multiple (LCM) of 4 and 12 is 12. This means that 12 is the smallest number that both 4 and 12 can divide into evenly. To make $\frac{5}{4}$ have a denominator of 12, we need to multiply both the numerator and the denominator by the same number. Since 4 multiplied by 3 equals 12, we multiply both the numerator and the denominator of $\frac{5}{4}$ by 3. This gives us $\frac{5 \times 3}{4 \times 3} = \frac{15}{12}$. Now, our equation looks like $\frac{15}{12} = \frac{1}{12} + s$. Both fractions have the same denominator, which allows us to perform the subtraction. Finding a common denominator is a fundamental skill in fraction arithmetic. It ensures that we are comparing and combining equal-sized parts of a whole. Without a common denominator, the fractions cannot be directly added or subtracted. By correctly identifying and applying the common denominator, we can proceed to the next step in solving for s, which involves subtracting $\frac{1}{12}$ from both sides of the equation. This step brings us closer to isolating the variable and finding its value.

Step 3: Isolate the Variable s

With the fractions having a common denominator, the equation is now $\frac{15}{12} = \frac{1}{12} + s$. To isolate the variable s, we need to subtract $\frac{1}{12}$ from both sides of the equation. This is based on the principle that performing the same operation on both sides of an equation maintains the equality. Subtracting $\frac{1}{12}$ from the right side of the equation will cancel out the term, leaving s by itself. On the left side, we subtract $\frac{1}{12}$ from $\frac{15}{12}$. Since the fractions have the same denominator, we simply subtract the numerators: $\frac{15}{12} - \frac{1}{12} = \frac{15 - 1}{12} = \frac{14}{12}$. Therefore, the equation becomes $\frac{14}{12} = s$. This means that the value of s is $\frac{14}{12}$. However, this fraction can be simplified further. Isolating the variable is a crucial step in solving equations. It involves using inverse operations to undo any operations that are being performed on the variable. In this case, since $\frac{1}{12}$ is being added to s, we use subtraction as the inverse operation. By subtracting $\frac{1}{12}$ from both sides, we effectively remove it from the right side, leaving s alone. This process is fundamental to solving various types of equations, from simple linear equations to more complex algebraic expressions. Now that we have found the value of s, we can proceed to simplify the fraction to its lowest terms. Simplifying fractions is important for presenting the solution in its most concise and understandable form.

Step 4: Simplify the Fraction

We have found that $s = \frac{14}{12}$. Now, we need to simplify this fraction completely. Simplifying a fraction means reducing it to its lowest terms, where the numerator and denominator have no common factors other than 1. To simplify $\frac{14}{12}$, we need to find the greatest common divisor (GCD) of 14 and 12. The GCD is the largest number that divides both 14 and 12 without leaving a remainder. The factors of 14 are 1, 2, 7, and 14. The factors of 12 are 1, 2, 3, 4, 6, and 12. The greatest common factor of 14 and 12 is 2. To simplify the fraction, we divide both the numerator and the denominator by their GCD, which is 2. Dividing 14 by 2 gives us 7, and dividing 12 by 2 gives us 6. Therefore, the simplified fraction is $\frac{7}{6}$. So, $s = \frac{7}{6}$. This is an improper fraction, where the numerator is greater than the denominator. We can also express this as a mixed number. To convert $\frac{7}{6}$ to a mixed number, we divide 7 by 6. The quotient is 1, and the remainder is 1. So, the mixed number is $1 \frac{1}{6}$. Thus, the simplified answer for s is $\frac{7}{6}$ or $1 \frac{1}{6}$. Simplifying fractions is an essential skill in mathematics. It helps to present the answer in its simplest and most understandable form. By finding the greatest common divisor and dividing both the numerator and denominator by it, we ensure that the fraction is in its lowest terms. This makes the solution easier to interpret and use in further calculations.

Final Answer

Therefore, the simplified solution for the equation $1 \frac{1}{4} = \frac{1}{12} + s$ is:

$s = \frac{7}{6}$ or $s = 1 \frac{1}{6}$

This completes the process of solving the equation. We converted the mixed number to an improper fraction, found a common denominator, isolated the variable s, and simplified the resulting fraction. The final answer represents the value of s that makes the equation true. Always remember to simplify your answers completely to ensure clarity and accuracy in mathematical solutions. By following these steps, you can confidently solve similar equations involving fractions and mixed numbers. The ability to solve equations is a fundamental skill in mathematics, applicable in various contexts and fields. Understanding the underlying principles and practicing the techniques will enhance your mathematical proficiency and problem-solving abilities. The solution $s = \frac{7}{6}$ or $s = 1 \frac{1}{6}$ is the final result, providing a clear and concise answer to the given equation. This demonstrates the application of algebraic principles and fraction manipulation to arrive at a simplified and accurate solution.