Problem 62
Question
Write each sentence as an equation. Let the variable \(x\) represent the number. The quoticnt of a number and 8 is \(\frac{1}{4}\)
Step-by-Step Solution
Verified Answer
The equation representing the given sentence is: \(\frac{x}{8} = \frac{1}{4}\).
1Step 1: Identify the mathematical operation
The word 'quotient' in the sentence indicates a division operation. So, the sentence 'The quotient of a number and 8' means that the number is divided by 8. This can be written mathematically as \(\frac{x}{8}\).
2Step 2: Write the complete equation
The given sentence says the quotient of a number and 8 is \(\frac{1}{4}\). So, the expression \(\frac{x}{8}\) from step 1 equals \(\frac{1}{4}\). This leads to the equation \(\frac{x}{8} = \frac{1}{4}\).
Key Concepts
Algebraic ExpressionsDivision Operation in AlgebraSolving Equations
Algebraic Expressions
In algebra, an algebraic expression is a combination of numbers, variables, and operations that represent a particular mathematical relationship. Variables, like the letter 'x' in our exercise, serve as placeholders for values we don't yet know, and allow us to express relationships without specifying the exact numbers involved.
For example, when the word problem from the exercise states 'the quotient of a number and 8,' we translate this into an algebraic expression as \(\frac{x}{8}\). Such expressions become building blocks for creating equations that we can solve.
Breaking down word problems into algebraic expressions is an essential skill, as it not only simplifies complex statements but also prepares the ground for solving for the unknowns.
For example, when the word problem from the exercise states 'the quotient of a number and 8,' we translate this into an algebraic expression as \(\frac{x}{8}\). Such expressions become building blocks for creating equations that we can solve.
Breaking down word problems into algebraic expressions is an essential skill, as it not only simplifies complex statements but also prepares the ground for solving for the unknowns.
Division Operation in Algebra
The division operation in algebra is signified by the terms 'quotient,' 'divided by,' or simply the division symbol ÷ or /. When tackling division in algebraic expressions, it's crucial to remember that division is the inverse operation of multiplication. This inverse relationship can be particularly useful when solving equations.
In the context of our exercise, the word 'quotient' directly implies that division is at play. To translate this into algebra, we used the expression \(\frac{x}{8}\), indicating that 'x' is being divided by 8. Remember, in equations like \(\frac{x}{8} = \frac{1}{4}\), the goal is to isolate 'x' by performing the inverse operation—multiplication in this case—on both sides of the equation.
In the context of our exercise, the word 'quotient' directly implies that division is at play. To translate this into algebra, we used the expression \(\frac{x}{8}\), indicating that 'x' is being divided by 8. Remember, in equations like \(\frac{x}{8} = \frac{1}{4}\), the goal is to isolate 'x' by performing the inverse operation—multiplication in this case—on both sides of the equation.
Solving Equations
The process of solving equations involves finding the value(s) of the variable(s) that make the equation true. In algebra, this usually means isolating the variable on one side of the equation. The steps to do this vary depending on the complexity of the equation.
In the given exercise, we have a simple one-step equation: \(\frac{x}{8} = \frac{1}{4}\). To solve this, we would multiply both sides of the equation by 8 to isolate 'x.' This action nullifies the division by 8, leaving us with \(x = 8 \times \frac{1}{4}\), which then simplifies to \(x = 2\). This solution demonstrates the importance of understanding inverse operations and being comfortable with manipulating equations to isolate variables.
In the given exercise, we have a simple one-step equation: \(\frac{x}{8} = \frac{1}{4}\). To solve this, we would multiply both sides of the equation by 8 to isolate 'x.' This action nullifies the division by 8, leaving us with \(x = 8 \times \frac{1}{4}\), which then simplifies to \(x = 2\). This solution demonstrates the importance of understanding inverse operations and being comfortable with manipulating equations to isolate variables.
Other exercises in this chapter
Problem 62
In Exercises \(29-72,\) use the order of operations to simplify each expression. $$4(-15)+|3(-10)|$$
View solution Problem 62
Simplify each algebraic expression. $$14+2(5 x-1)$$
View solution Problem 62
Find each sum. $$|4+(-11)|+|-3+(-4)|$$
View solution Problem 62
Perform the indicated operation. Where possible, reduce the answer to its lowest terms. $$\frac{7}{4} \div \frac{3}{8}$$
View solution