Solving Mathematical Problems with Algebra: A Case Study on Quantities and Fractions
Solving Mathematical Problems with Algebra: A Case Study on Quantities and Fractions
Mathematics is a fundamental subject that requires a clear understanding of various concepts. One such concept is the use of algebra to solve problems involving quantities and fractions. Let's explore a specific problem to demonstrate how algebra can be used to find the solution.
Problem Statement
The problem at hand involves a quantity and its fifth part added together to become 21. The challenge is to find the value of the quantity. We will walk through the solution step-by-step to ensure a thorough understanding of the process.
Formulating the Equation
Let's denote the unknown quantity by the variable a. According to the problem statement, the quantity and its fifth part added together equal 21. Therefore, we can write the equation as:
a a/5 21
Step-by-Step Solution
Step 1: Combine the terms on the left side of the equation.
Combine a and a/5. To do this, we need a common denominator, which is 5. Thus, the equation becomes:
a/1 a/5 21
Converting a to a fraction with a denominator of 5, we get:
5a/5 a/5 21
Combining the fractions, we have:
(5a a)/5 21
Simplify the numerator:
6a/5 21
Step 2: Isolate the variable.
To isolate a, we need to eliminate the denominator. Multiply both sides of the equation by 5 to get rid of the denominator:
6a/5 × 5 21 × 5
This simplifies to:
6a 105
Step 3: Solve for a.
To solve for a, we need to divide both sides of the equation by 6:
6a/6 105/6
This simplifies to:
a 105/6
Divide 105 by 6:
a 17.5
Verification
To verify our solution, let's substitute a 17.5 back into the original equation:
17.5 17.5/5 21
Calculate 17.5/5:
17.5/5 3.5
Now, add 17.5 and 3.5:
17.5 3.5 21
This confirms that our solution is correct.
Conclusion
Through the use of algebra, we were able to solve the problem of finding the quantity and its fifth part that sum to 21. The final answer is 17.5. This example demonstrates the power of algebra in solving mathematical problems involving quantities and fractions.