Mateo Is Making A School Spirit Flag. He Has { \frac{1}{3}$}$ As Many Yards Of Red Fabric As Blue Fabric. He Buys ${ 2 \frac{2}{3}\$} Yards More Red Fabric. Now He Has Equal Amounts Of Red And Blue Fabric. Use { X$}$ To
Solving the School Spirit Flag Problem: A Math Adventure
In this article, we will delve into a real-world problem that involves mathematics. Mateo is making a school spirit flag, and he needs to determine the amount of fabric he needs to buy. The problem involves fractions, algebra, and critical thinking. We will break down the problem step by step and provide a solution to help Mateo and others like him.
Mateo has {\frac{1}{3}$}$ as many yards of red fabric as blue fabric. This means that if he has {x$}$ yards of blue fabric, he has {\frac{1}{3}x$}$ yards of red fabric. Let's represent the amount of blue fabric as {x$}$ and the amount of red fabric as {\frac{1}{3}x$}$.
Mateo buys ${2 \frac{2}{3}\$} yards more red fabric. Now he has equal amounts of red and blue fabric. We need to find the value of {x$}$, which represents the amount of blue fabric.
Step 1: Representing the Additional Red Fabric
Mateo buys ${2 \frac{2}{3}\$} yards more red fabric. To represent this as an improper fraction, we can convert the mixed number to an improper fraction:
${$2 \frac{2}{3} = \frac{8}{3}$]
So, Mateo buys [\frac{8}{3}\$} yards more red fabric.
Step 2: Setting Up the Equation
Since Mateo now has equal amounts of red and blue fabric, we can set up an equation to represent this situation:
{\frac{1}{3}x + \frac{8}{3} = x$]
Step 3: Solving the Equation
To solve the equation, we can start by multiplying both sides by 3 to eliminate the fraction:
[$x + 8 = 3x$]
Next, we can subtract [x\$} from both sides to get:
${$8 = 2x$]
Finally, we can divide both sides by 2 to solve for [x\$}:
{x = 4$]
Mateo has [4\$} yards of blue fabric. This means that he has {\frac{1}{3} \times 4 = \frac{4}{3}$}$ yards of red fabric initially. After buying ${2 \frac{2}{3}\$} yards more red fabric, he now has a total of {\frac{4}{3} + \frac{8}{3} = \frac{12}{3} = 4$}$ yards of red fabric.
This problem may seem simple, but it has real-world applications in various fields, such as:
- Engineering: When designing a system or a structure, engineers need to consider the proportions and relationships between different components.
- Architecture: Architects need to balance the proportions of different elements in a building, such as the ratio of windows to walls.
- Art: Artists often use proportions and ratios to create balanced and aesthetically pleasing compositions.
When solving problems involving fractions and algebra, remember to:
- Simplify fractions: Before solving an equation, simplify any fractions involved to make the problem easier to solve.
- Use algebraic manipulations: Use algebraic manipulations, such as multiplying or dividing both sides of an equation, to solve for the variable.
- Check your work: Always check your work by plugging the solution back into the original equation to ensure that it is true.
By following these tips and tricks, you can become more confident and proficient in solving problems involving fractions and algebra.
Solving the School Spirit Flag Problem: A Math Adventure (Q&A)
In our previous article, we solved the school spirit flag problem by using fractions and algebra. Now, let's dive deeper into the problem and answer some frequently asked questions.
Q: What is the initial ratio of red to blue fabric?
A: The initial ratio of red to blue fabric is {\frac{1}{3}$}$. This means that for every 1 yard of blue fabric, Mateo has {\frac{1}{3}$}$ yards of red fabric.
Q: How much more red fabric does Mateo buy?
A: Mateo buys ${2 \frac{2}{3}\$} yards more red fabric. To represent this as an improper fraction, we can convert the mixed number to an improper fraction:
${$2 \frac{2}{3} = \frac{8}{3}$]
Q: What is the equation that represents the situation after Mateo buys more red fabric?
A: The equation that represents the situation after Mateo buys more red fabric is:
[$\frac{1}{3}x + \frac{8}{3} = x$]
Q: How do we solve the equation?
A: To solve the equation, we can start by multiplying both sides by 3 to eliminate the fraction:
[$x + 8 = 3x$]
Next, we can subtract [x\$} from both sides to get:
${$8 = 2x$]
Finally, we can divide both sides by 2 to solve for [x\$}:
{x = 4$]
Q: What is the value of x?
A: The value of [x\$} is 4. This means that Mateo has 4 yards of blue fabric.
Q: How much red fabric does Mateo have initially?
A: Mateo has {\frac{1}{3} \times 4 = \frac{4}{3}$}$ yards of red fabric initially.
Q: How much red fabric does Mateo have after buying more?
A: Mateo has a total of {\frac{4}{3} + \frac{8}{3} = \frac{12}{3} = 4$}$ yards of red fabric after buying more.
Q: What are some real-world applications of this problem?
A: This problem has real-world applications in various fields, such as:
- Engineering: When designing a system or a structure, engineers need to consider the proportions and relationships between different components.
- Architecture: Architects need to balance the proportions of different elements in a building, such as the ratio of windows to walls.
- Art: Artists often use proportions and ratios to create balanced and aesthetically pleasing compositions.
Q: What are some tips and tricks for solving problems involving fractions and algebra?
A: When solving problems involving fractions and algebra, remember to:
- Simplify fractions: Before solving an equation, simplify any fractions involved to make the problem easier to solve.
- Use algebraic manipulations: Use algebraic manipulations, such as multiplying or dividing both sides of an equation, to solve for the variable.
- Check your work: Always check your work by plugging the solution back into the original equation to ensure that it is true.
By following these tips and tricks, you can become more confident and proficient in solving problems involving fractions and algebra.