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Chapter 4: Problem 11

In Exercises \(11-14,\) write and solve an equation to find the number of coins each friend has. Ken has three more coins than twice the number Javier has. Khalid has five fewer coins than Javier. They have 50 coins altogether.

Short Answer

Expert verified
Javier has 13 coins, Ken has 29 coins, and Khalid has 8 coins.

Step by step solution

01

- Define Variables

Let Javier have \( x \) coins. Define the number of coins Ken and Khalid have in terms of \( x \).
02

- Express Ken's Coins

Ken has three more coins than twice the number Javier has. Therefore, Ken's coins can be represented as \( 2x + 3 \).
03

- Express Khalid's Coins

Khalid has five fewer coins than Javier. Therefore, Khalid's coins can be represented as \( x - 5 \).
04

- Formulate the Equation

Write the equation representing the total number of coins. The equation is \( x + (2x + 3) + (x - 5) = 50 \).
05

- Simplify the Equation

Combine like terms in the equation: \( 4x - 2 = 50 \).
06

- Solve for \( x \)

Add 2 to both sides: \( 4x = 52 \). Then divide both sides by 4 to find \( x = 13 \).
07

- Calculate Coins for Each Friend

Using \( x = 13 \), calculate the number of coins:- Javier: \( x = 13 \)- Ken: \( 2(13) + 3 = 29 \)- Khalid: \( 13 - 5 = 8 \)
08

- Verify the Solution

Ensure the solution is correct by checking the total: \( 13 + 29 + 8 = 50 \). This confirms the solution is correct.

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Key Concepts

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

defining variables
In algebra word problems, the first step is often to define the variables. This helps to translate the words into mathematical symbols and equations.
For example, in the given exercise, we start by defining the variable for the number of coins Javier has. Let's say Javier has \( x \) coins. Then, Ken and Khalid's coins can be defined in relation to \( x \).

Here is how we do it:
  • Ken has three more coins than twice the number Javier has. Thus, Ken's coins can be represented as \( 2x + 3 \).
  • Khalid has five fewer coins than Javier. Thus, Khalid's coins can be represented as \( x - 5 \).
Always start by defining your variables in a way that matches the relationships described in the problem. This step is crucial as it sets the foundation for forming equations.
forming equations
Once you've defined the variables, the next step is to form equations based on the word problem. These equations will help you represent the relationships and constraints given in the problem.
In this example, the total number of coins among the friends is given as 50. To form the equation, we add up all the expressions for Javier, Ken, and Khalid's coins:
  • Javier's coins: \( x \)
  • Ken's coins: \( 2x + 3 \)
  • Khalid's coins: \( x - 5 \)
We then combine these expressions into a single equation representing the total number of coins:\( x + (2x + 3) + (x - 5) = 50 \)

Forming the equations correctly is essential because it allows you to use algebra to solve for the unknown variable.
solving linear equations
After forming the equations, the final step is solving the linear equation. Let's break down how to solve the equation we formulated:
First, simplify the equation by combining like terms:
\( x + 2x + 3 + x - 5 = 50 \) This simplifies to: \( 4x - 2 = 50 \)

Next, isolate the variable by performing operations on both sides of the equation. Add 2 to both sides:
\[ 4x - 2 + 2 = 50 + 2 \] \ 4x = 52 \
Then, divide both sides by 4: \ \frac{4x}{4} = \frac{52}{4} \ This gives: \ x = 13 \
Now, we have determined that Javier has 13 coins. Using this information, calculate the number of coins for Ken and Khalid:
  • Ken: \ 2 \times 13 + 3 = 29 \ coins
  • Khalid: \ 13 - 5 = 8 \ coins
Always check your solution by adding the number of coins each friend has to ensure they total 50: \ 13 + 29 + 8 = 50 \. This confirms your solution is correct.

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