Problem 11 - Prealgebra

Question

Translate each sentence into an equation. Then find each number. The quotient of a number and \(-4,\) less \(8,\) is \(-42\)

Step-by-Step Solution

Answer

The number is 136.

Step 1: Determine the Total Structure of the Problem

The problem involves finding a number that satisfies a certain condition when divided and subtracted. We need to translate the sentence into a mathematical equation.

Step 2: Translate the Sentence into an Equation

The sentence "The quotient of a number and \(-4\), less \(8\), is \(-42\)" can be broken down: - "The quotient of a number and \(-4\)" translates to \(\frac{x}{-4}\).- "less \(8\)" means subtracting \(8\) from this quotient.- "is \(-42\)" means this entire expression is equal to \(-42\).Thus, the equation is: \[ \frac{x}{-4} - 8 = -42 \]

Step 3: Solve the Equation for the Number

We have the equation \( \frac{x}{-4} - 8 = -42 \). We need to isolate \(x\): 1. Add \(8\) to both sides to get: \[ \frac{x}{-4} = -42 + 8 \] which simplifies to \( \frac{x}{-4} = -34 \). 2. Multiply both sides by \(-4\) to solve for \(x\): \[ x = -34 \times (-4) = 136 \].

Step 4: Verify the Solution

To check if our solution is correct, substitute \(x = 136\) back into the original setup: 1. Calculate the quotient: \( \frac{136}{-4} = -34 \). 2. Subtract \(8\) from this: \(-34 - 8 = -42\).This confirms the original statement, so \(x = 136\) is correct.

Key Concepts

Translation from Words to Math

Reading mathematical problems in words and translating them into equations is a fundamental skill in algebra. It involves understanding the language of mathematics and seeing how everyday phrases can represent mathematical operations. In the given exercise, we translate the sentence "The quotient of a number and \(-4\), less \(8\), is \(-42\)" into an equation.

Here's how we break it down:

• "The quotient of a number and \(-4\)" indicates that a number, let's call it \(x\), is divided by \(-4\). This can be written as \(\frac{x}{-4}\).
  
• "Less \(8\)" means we are subtracting \(8\) from this quotient, leading to the expression \(\frac{x}{-4} - 8\).
  
• "Is \(-42\)" indicates that the previous expressions equals \(-42\), forming the complete equation \(\frac{x}{-4} - 8 = -42\).
  

By carefully translating from words to math, we convert verbal descriptions into algebraic expressions, which we can solve using algebraic principles.

Solving Equations

Once we have translated the word problem into a mathematical equation, the next step is solving the equation to find the value of the unknown variable.
The equation provided is \(\frac{x}{-4} - 8 = -42\).
To solve it, we need to isolate \(x\):

• Add \(8\) to both sides of the equation: \(\frac{x}{-4} = -42 + 8\), which simplifies to \(\frac{x}{-4} = -34\).
  
• Now, to get rid of the division by \(-4\), multiply both sides of the equation by \(-4\): \(x = -34 \times (-4)\), resulting in \(x = 136\).
  

The process of solving equations often involves reversing the operations listed to isolate the variable, ensuring each step logically follows the next, leading to the correct solution.

Math Verification Process

Verifying your solution is just as important as finding it, as it confirms the accuracy of your answer. In algebra, this means substituting your answer back into the original equation to see if it works.
For the solution \(x = 136\), here's how verification works:

• Substitute \(x = 136\) back into the original problem's setup: Calculate the quotient \(\frac{136}{-4} = -34\).
  
• Then subtract \(8\) from this result: \(-34 - 8 = -42\), which matches the original condition of the problem.
  

The result checks out because the left side of the equation equals the right side, confirming that \(x = 136\) satisfies the equation. This verification step helps build confidence in algebraic solutions and ensures no errors were made during calculation.