Question

Finance A total of 10,000 dollar is to be divided between Sean and George, with George to receive 3000 dollar less than Sean. How much will each receive?

Short answer

Sean receives 6500 dollars and George receives 3500 dollars.

Step-by-step solution

Define Variables

Let’s define Sean’s amount as \( S \). Since George receives 3000 dollars less than Sean, George's amount can be defined as \( S - 3000 \).

Set Up the Equation

Combine Sean’s and George’s amounts to set up the equation: \( S + (S - 3000) = 10000 \).

Simplify the Equation

Simplify the equation from Step 2 to find Sean’s amount: \[ S + S - 3000 = 10000 \] \[ 2S - 3000 = 10000 \] \[ 2S = 13000 \] \[ S = 6500 \]

Calculate George's Amount

Subtract 3000 dollars from Sean’s amount to find George’s amount: \[ S - 3000 = 6500 - 3000 = 3500 \]

Key concepts

Defining Variables

To solve any linear equation problem, we start by defining variables. This means deciding what each unknown value will be represented by. In our exercise, we need to find out how much money Sean and George each get. Let's denote Sean's amount by the variable \( S \). Since George gets 3000 dollars less than Sean, we can represent George's amount as \( S - 3000 \). By defining these variables, we lay the foundation for setting up our equation.

Setting Up Equations

Next, we need to set up an equation based on the information given. We know that the total amount of money is 10,000 dollars. This total is split between Sean and George. Therefore, we add Sean’s and George’s amounts and set this equal to 10,000. The equation looks like this: \( S + (S - 3000) = 10000 \). By carefully combining Sean and George's amounts, we create a linear equation that will help us solve for \( S \).

Simplifying Equations

Once we have our equation, we need to simplify it to solve for the variable. Start by expanding the equation: \[ S + S - 3000 = 10000 \]. Combine the \( S \) terms: \[ 2S - 3000 = 10000 \]. Next, make the equation easier to solve by isolating the \( 2S \) term. Add 3000 to both sides: \[ 2S = 13000 \]. Finally, divide both sides by 2 to isolate \( S \): \[ S = 6500 \]. This tells us that Sean receives 6500 dollars.

Basic Arithmetic Operations

Basic arithmetic operations like addition, subtraction, multiplication, and division are crucial in solving linear equations. In the previous steps, we used these operations to simplify the equation. We added and combined like terms, subtracted to isolate terms, and divided to solve for the variable. Finally, to find George's amount, we use subtraction: \( S - 3000 = 6500 - 3000 = 3500 \). Thus, George receives 3500 dollars. These arithmetic operations help us manipulate equations to find the values of our variables.