Question

Use a calculator to solve the equation. (Round your solution to three decimal places.) \(\frac{x}{2.625}+\frac{x}{4.875}=1\)

Short answer

The value of \(x\) is approximately 1.706.

Step-by-step solution

Rewrite the Equation

Rewrite the equation by blowing up the denominator from each fraction. The equation would then look like this: \[2.625x+4.875x=2.625*4.875\].

Add Like terms

Add the like terms on the left-hand side of the equation: \[7.5x = 2.625*4.875\]. It simplifies to \[7.5x = 12.796875\].

Solve for x

Divide each side of the equation by 7.5 to isolate x, which gives \(x = \frac{12.796875}{7.5}\).

Approximate the value of x

To get the value of \(x\), we divide 12.796875 by 7.5. Using the calculator, the value resulted is approximately 1.706, when rounded to three decimal places.

Key concepts

Understanding Fractions

Fractions are a way to express a part of a whole. They consist of a numerator (top part) and a denominator (bottom part). In the equation \( \frac{x}{2.625} + \frac{x}{4.875} = 1 \), these fractions represent parts of the variable \( x \) divided by constants.

Handling fractions in equations often involves finding a common denominator or eliminating the denominators, as was done when the equation was rewritten. By multiplying through by the least common multiple, which was the product of the denominators (2.625 and 4.875 in this case), we clear the fractions, making the equation easier to solve. This step simplifies calculations significantly.

Remember, managing fractions in equations can seem tough at first but recognizing how to manipulate them by multiplying through can turn complex fractions into simple terms.

Combining Like Terms

Like terms in algebra are terms that have identical variable parts. Combining like terms is a crucial step in simplifying equations. In this exercise, after eliminating fractions, the equation becomes \(2.625x + 4.875x = 2.625 \cdot 4.875\). Both terms on the left side of the equation, \(2.625x\) and \(4.875x\), are like terms because they both involve the variable \(x\) simply multiplied by different coefficients.

• Add the coefficients: 2.625 and 4.875.
• Perform the arithmetic to combine them into a single term.

Combining these like terms results in \(7.5x = 2.625 \cdot 4.875\).

This step not only simplifies the equation but also sets the stage for isolating the variable to eventually solve the equation.

The Process of Approximation

Approximation is the process of finding a value that is close enough to the correct answer for practical purposes, often by rounding. In solving equations like this one, when the answer is a long decimal, we round to a specified number of decimal places. This is important when dealing with real-world scenarios where exact precision is less critical.

In our exercise, after dividing the product 12.796875 by 7.5, we get a very detailed decimal number. Rounding to three decimal places provides a balance between accuracy and simplicity. Therefore, the number 1.706 is an approximation of the exact solution.

Knowing when and how to round is key, especially when following instructions that specify the precision of the solution, like in this exercise.

Using Calculators Effectively

Calculators are powerful tools for performing arithmetic operations quickly and accurately. They become invaluable when dealing with complex calculations or those involving fractions and decimals, like multiplying or dividing long decimal numbers.

• Multiply or divide the numbers as needed using the calculator.
• Make sure to input the correct numbers and check the solution is rounded correctly to the specified places.

In this exercise, the calculator was used to solve \( \frac{12.796875}{7.5} \), providing an exact answer that was then rounded. Calculators help ensure that our arithmetic is accurate, even if we might not be fully comfortable with manual calculations. Learning to leverage calculators correctly can greatly enhance your problem-solving skills and efficiency.