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Chapter 5: Problem 199

The age of Noelle's dad is six less than three times Noelle's age. The sum of their ages is seventyfour. Find their ages.

Short Answer

Expert verified
Noelle is 20 years old, and her dad is 54 years old.

Step by step solution

01

Define the Variables

Let Noelle's age be denoted by \( n \). Therefore, Noelle's dad's age would be \( 3n - 6 \) as per the problem statement.
02

Set Up the Equation

The problem states the sum of their ages is 74. Therefore, we can write the equation: \[ n + (3n - 6) = 74 \]
03

Simplify the Equation

Combine the terms involving \( n \) and simplify: \[ n + 3n - 6 = 74 \ 4n - 6 = 74 \]
04

Solve for \( n \)

Add 6 to both sides of the equation to isolate the term with \( n \): \[ 4n - 6 + 6 = 74 + 6 \ 4n = 80 \]
05

Find Noelle's Age

Divide both sides by 4 to solve for \( n \): \[ n = \frac{80}{4} \ n = 20 \] Therefore, Noelle is 20 years old.
06

Find Noelle's Dad's Age

Use the relationship given in the problem to find Noelle's dad's age: \[ \text{Dad's Age} = 3n - 6 \ = 3 \times 20 - 6 \ = 60 - 6 \ = 54 \] Therefore, Noelle's dad is 54 years old.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

algebraic equations
Algebraic equations are a fundamental part of solving many word problems, including age problems. An algebraic equation is a mathematical statement that shows the equality between two expressions. These equations can include variables, coefficients, and constants. In the exercise, the equation represents the sum of Noelle's and her dad's ages. The goal is to manipulate and solve these equations to find the value of the unknown variables.
variable definition
Defining variables is one of the first steps in solving word problems. A variable is a symbol, often an alphabetical letter, that stands in for an unknown value. In our problem, we defined Noelle's age as Then, using information from the problem, we expressed her dad's age in terms of Noelle's age as This approach helps in breaking down the problem into manageable parts and setting the stage for creating the algebraic equation.
age problems
Age problems are common in algebra and typically involve the ages of two or more people at different points in time. The key is to create expressions and equations that accurately represent the relationships given in the problem. For instance, in the exercise, we know that Noelle's dad's age is six less than three times Noelle's age. The steps involve:
  • Defining the ages using variables
  • Creating an equation based on the sum of their ages
  • Solving the equation step-by-step
These problems require careful reading and translation of the word problem into algebraic language.
equation solving
Equation solving involves determining the value of the variable that makes the equation true. After setting up the equation , we combined like terms to simplify it. The next steps involved isolating the variable by performing inverse operations:
  • Adding 6 to both sides
  • Dividing by 4
Each step simplifies the equation until we find that Noelle is 20 years old. Solving equations requires patience and attention to detail.
simplification
Simplification is the process of reducing an equation or expression to its simplest form. It often involves combining like terms or eliminating constants from both sides of an equation. In this exercise:
After that, we:
  • Added 6 to both sides to get
  • Divided both sides by 4 to isolate
Simplifying equations makes it much easier to find the values of unknown variables and ultimately solve the problem.

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Most popular questions from this chapter

In the following exercises, translate to a system of equations and solve. Two angles are supplementary. The measure of the larger angle is five less than four times the measure of the smaller angle. Find the measures of both angles.

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