Question

Solve for $x$ in the following proportions. Carry division two decimal places as necessary. $\frac{x}{26}=\frac{10.1}{13}$

Short answer

x ≈ 20.20

Step-by-step solution

Set Up the Equation

We are given the proportion $\frac{x}{26}=\frac{10.1}{13}$. Our goal is to solve for $x$. In a proportion, the cross products of the fractions are equal. We'll use this property to set up an equation.

Apply Cross Multiplication

Cross multiply to eliminate the fractions. This gives us: $$x\times 13=26\times 10.1$$

Perform Multiplication

Calculate the product of the numbers on the right-hand side of the equation: $26\times 10.1=262.6$. Now our equation is: $$13x=262.6$$

Solve for x

Divide both sides of the equation by 13 to get $x$ by itself: $$x=\frac{262.6}{13}$$ Calculate this division to two decimal places: $x\approx 20.20$.

Key concepts

Cross Multiplication

Cross multiplication is a powerful technique used primarily in proportion problems in algebra. It helps to eliminate fractions from an equation and is particularly useful when we want to solve for one of the unknowns. The concept relies on the equality of the cross products of two fractions.Imagine you have a proportion set up as $\frac{a}{b}=\frac{c}{d}$. To cross multiply, you multiply the numerator of the first fraction by the denominator of the second fraction and set it equal to the product of the denominator of the first fraction and the numerator of the second fraction.

• This results in the equation: $a\cdot d=b\cdot c$.
• This method is straightforward and is the backbone of solving many types of proportion problems.

Let's see it in action. Given $\frac{x}{26}=\frac{10.1}{13}$, to cross multiply, you would perform:

• Multiply $x$ by 13.
• Multiply 26 by 10.1.

This gives us the equation $x\times 13=26\times 10.1$, simplifying the problem significantly.

Solving Equations

Solving equations involves finding the unknown value that makes an equation true. After setting up an equation using cross multiplication, the next step is to isolate the variable you're solving for.In the equation $13x=262.6$, we want to isolate $x$. To do this, we need to "undo" the multiplication of $x$ by 13. This is done through division.

• Divide both sides of the equation by 13.
• This operation gives us $x=\frac{262.6}{13}$.

Performing the division will provide you with the value for $x$. Solving equations is all about balancing both sides and systematically reducing the operations to find the unknown.

Decimal Places

Decimal places refer to the number of digits to the right of the decimal point in a number. When dealing with division problems or when a precise answer is required as in most math problems, it's important to be aware of how many decimal places you need to consider.In the context of mathematics education, rounding a number to a specific number of decimal places is a common practice. When you have $x=\frac{262.6}{13}$ and perform the division, you obtain a result that might have more than two decimal places.

• The instruction might specify rounding or cutting off after two decimal places, which is a common standard for precision in many fields.
• In this case, you round $x\approx 20.20$ up to two decimal places.

Rounding ensures that your results are not only precise but also practical for further calculations or interpretations.

Mathematics Education

Mathematics education involves imparting knowledge about concepts like proportions, cross multiplication, and precision in calculations. These topics serve as foundational skills for solving everyday problems and academic exercises. Understanding proportion problems helps students relate real-world situations to mathematical equations. The application of cross multiplication not only simplifies complex algebraic expressions but also enhances logical thinking.

• Learning how to solve equations boosts problem-solving skills and progresses students from simple arithmetic to more advanced topics.
• Developing an understanding of decimal places ensures that students appreciate the importance of accuracy and precision, which is crucial in scientific, financial, and everyday scenarios.

Education in these concepts equips students for future academic challenges and cultivates a strong analytical mindset necessary for various professional fields.