Problem 57
Question
Set up an algebraic equation and solve each problem. A 20-foot board is to be cut into two pieces whose lengths are in the ratio of 7 to 3 . Find the lengths of the two pieces.
Step-by-Step Solution
Verified Answer
The lengths of the pieces are 14 feet and 6 feet.
1Step 1: Understand the Ratio
The problem gives a ratio of 7 to 3 for the lengths of the two pieces. This means that if we let the lengths of the two pieces be 7x and 3x respectively, their lengths are in this desired ratio.
2Step 2: Write the Equation
Since the total length of both pieces should equal the entire board, which is 20 feet, we set up the equation based on their lengths. We have:\[ 7x + 3x = 20 \]
3Step 3: Simplify the Equation
Simplify the equation \( 7x + 3x = 20 \) to combine like terms. This results in:\[ 10x = 20 \]
4Step 4: Solve for x
To find the value of \( x \), divide both sides of the equation by 10:\[ x = \frac{20}{10} \]\[ x = 2 \]
5Step 5: Calculate the Lengths of the Pieces
Now that we have \( x = 2 \), calculate the lengths of the two pieces:- The first piece is \( 7x = 7(2) = 14 \) feet.- The second piece is \( 3x = 3(2) = 6 \) feet.
Key Concepts
Ratio and ProportionSolving EquationsLinear Equations
Ratio and Proportion
A ratio is a way of comparing two quantities by division, highlighting their relative sizes. In the exercise given, we encounter a ratio of 7 to 3, which tells us that one piece of board is 7 parts, while the other part is 3 parts, when compared to a whole. Ratios are extremely useful for setting up proportions, which allow us to solve for unknown variables based on relationships between known quantities and their ratios.
Proportion, on the other hand, is an equation stating that two ratios are equal. In our problem, we set up a direct relationship by assigning variables, and later solve this proportion using algebraic equations. By expressing the two pieces' lengths as 7x and 3x, we directly use the given ratio to form our equation in a meaningful way. Understanding these concepts is crucial in solving similar algebraic problems effectively.
Proportion, on the other hand, is an equation stating that two ratios are equal. In our problem, we set up a direct relationship by assigning variables, and later solve this proportion using algebraic equations. By expressing the two pieces' lengths as 7x and 3x, we directly use the given ratio to form our equation in a meaningful way. Understanding these concepts is crucial in solving similar algebraic problems effectively.
Solving Equations
The core of algebra lies in the ability to manipulate and solve equations. In our exercise, the equation to start with was simple: \(7x + 3x = 20\). Solving equations involves finding the value of the variable that makes the equation true.
Breaking down the problem, we combine like terms first. This means adding together terms that look alike, in this case \(7x + 3x\) becomes \(10x\). Once simplified, solving for the variable involves getting x by itself. This is done by dividing both sides of the equation by 10, resulting in \(x = 2\).
Solving equations requires basic arithmetic and an understanding of algebraic manipulation. These skills are valuable not only for math exercises but also for real-world problem-solving.
Breaking down the problem, we combine like terms first. This means adding together terms that look alike, in this case \(7x + 3x\) becomes \(10x\). Once simplified, solving for the variable involves getting x by itself. This is done by dividing both sides of the equation by 10, resulting in \(x = 2\).
Solving equations requires basic arithmetic and an understanding of algebraic manipulation. These skills are valuable not only for math exercises but also for real-world problem-solving.
Linear Equations
Linear equations are a type of algebraic equation where the highest power of the variable is one. They are called 'linear' because they graph as straight lines. In our exercise, the equation \(10x = 20\) is an example of a linear equation.
When solving linear equations, our goal is to simplify the equation to find the value of the unknown variable. This usually involves operations like addition, subtraction, multiplication, or division. We took the equation \(10x = 20\), and by dividing both sides by 10, we determined that \(x = 2\).
Linear equations are foundational in algebra and serve as building blocks for more complex equations and systems. Mastering how to handle these equations ensures a strong understanding of more advanced mathematical concepts.
When solving linear equations, our goal is to simplify the equation to find the value of the unknown variable. This usually involves operations like addition, subtraction, multiplication, or division. We took the equation \(10x = 20\), and by dividing both sides by 10, we determined that \(x = 2\).
Linear equations are foundational in algebra and serve as building blocks for more complex equations and systems. Mastering how to handle these equations ensures a strong understanding of more advanced mathematical concepts.
Other exercises in this chapter
Problem 56
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