Problem 70 Raoul received a $\$ 30$ tip f... [FREE SOLUTION]

Chapter 3: Problem 70

Raoul received a $\$ 30$ tip for waiting on a large party. This was $\$ 5$ more than one-fourth of the tip the headwaiter received. How much did the headwaiter receive for a tip?

Short Answer

Expert verified

The headwaiter received a $100 tip.

Step by step solution

Understand the Problem

Raoul received a $30 tip. This was $5 more than one-fourth of the headwaiter's tip. We need to find the total tip the headwaiter received.

Set Up the Equation

Let the headwaiter's tip be $ x $ dollars. According to the problem, Raoul's tip is $5 more than one-fourth of this amount. So, we can write the equation: \[ 30 = \frac{1}{4}x + 5 \]

Solve for x

Start by isolating the fraction. Subtract 5 from both sides of the equation: \[ 30 - 5 = \frac{1}{4}x \] Simplify to obtain $ 25 = \frac{1}{4}x $.

Clear the Fraction

Multiply both sides of the equation by 4 to solve for $ x $: \[ 25 \times 4 = x \] which simplifies to $ x = 100 $.

Verify the Solution

Raoul's tip was $30, and this amounts to $5 more than $ \frac{1}{4} $ of $100. Calculate \( \frac{1}{4} \times 100 = 25 $, and verify that \)25 + $5 = $30. The calculations confirm the problem statement.

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Key Concepts

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

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra that helps you find the value of unknowns. In this exercise, we deal with a linear equation where we have an equation of the form: $ ax + b = c $. Understanding this helps us solve for the variable, which represents a value in the real world.
Here's how to solve a linear equation step-by-step:

First, identify the equation you need to solve. In this case, the problem gives us $ 30 = \frac{1}{4}x + 5 $.
The goal is to isolate $ x $ on one side of the equation. Begin by eliminating constants not associated with $ x $ by performing opposite operations.
Here, subtract 5 from both sides: $ 25 = \frac{1}{4}x $.
To get $ x $ alone, clear the fraction (if present) by multiplying by the reciprocal or the denominator. Thus, multiply both sides by 4: $ x = 100 $.
Always check your answer in the original equation to ensure accuracy.

Following these steps ensures that you methodically tackle any linear equation through a systematic approach.

Mathematical Reasoning

Mathematical reasoning involves thinking logically to solve problems. In this particular word problem, you're tasked with transforming a real-world scenario into a mathematical equation.

Here's how you apply mathematical reasoning in word problems:

Start by carefully reading the problem to understand what is being asked. Identify the quantities involved and their relationships.
Translate the words into mathematical terms. In the exercise, "Raoul's tip being \$5 more than one-fourth of the headwaiter's tip" becomes the equation $ 30 = \frac{1}{4}x + 5 $.
Setting up an equation accurately depends on a clear understanding of how numbers and operations relate to each other in the context of the problem.
Use logic to plan how you'll solve the equation, considering operations that will isolate your unknown variable, $ x $.

With practice, mathematical reasoning becomes second nature, helping you solve similar problems efficiently.

Fraction Operations

Working with fractions can seem challenging, but understanding operations with fractions is key to mastering algebra. In this exercise, fractions play a central role when forming and solving the equation.

Here are the steps to handle fraction operations in equations:

Identify the fraction in the equation. Here, we have $ \frac{1}{4}x $, which shows that the headwaiter鈥檚 tip is being divided by 4.
When solving, start by isolating the fractional term if it's not already by itself. This might involve adding, subtracting, or multiplying both sides of an equation.
To eliminate a fraction, multiply both sides of the equation by the denominator. This clears the fraction, making it easier to find the value of the variable. In our case, after isolating $ \frac{1}{4}x = 25 $, we multiplied by 4 to get $ x = 100 $.
Simplifying fractions where possible also helps keep calculations straightforward and error-free.

Fraction operations are crucial and frequent in algebra. Mastering these allows for more efficient problem-solving across various mathematical contexts.

91影视

Raoul received a \(\$ 30\) tip for waiting on a large party. This was \(\$ 5\) more than one-fourth of the tip the headwaiter received. How much did the headwaiter receive for a tip?

Short Answer

Step by step solution

Understand the Problem

Set Up the Equation

Solve for x

Clear the Fraction

Verify the Solution

Key Concepts

Solving Linear Equations

Mathematical Reasoning

Fraction Operations

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