Problem 45 - Algebra for College Students

Question

For Problems \(45-60\), set up an algebraic equation and solve each problem. A sum of \(\$ 1750\) is to be divided between two people in the ratio of 3 to 4 . How much does each person receive?

Step-by-Step Solution

Answer

Person one receives $750, person two receives $1000.

Step 1: Assign Variables

Let's assign variables to the amounts each person will receive. Denote the amount the first person receives as \( x \) and the amount the second person receives as \( y \).

Step 2: Understand the Ratio

The amounts are divided in a ratio of 3 to 4. This means that \( \frac{x}{y} = \frac{3}{4} \) or mathematically, it can be expressed as \( 4x = 3y \).

Step 3: Set Up the Equation for Total Amount

Since the total amount is \( 1750 \), we have the equation \( x + y = 1750 \).

Step 4: Solve the System of Equations

We have the system of equations:1. \( 4x = 3y \)2. \( x + y = 1750 \)Substitute \( y = 1750 - x \) into the first equation:\( 4x = 3(1750 - x) \)

Step 5: Simplify and Solve for x

Simplify the substitution: \[ 4x = 5250 - 3x \]Add \( 3x \) to both sides: \[ 7x = 5250 \]Divide both sides by 7:\[ x = 750 \]

Step 6: Solve for y

We know \( y = 1750 - x \), substitute \( x = 750 \) into this:\[ y = 1750 - 750 = 1000 \]

Step 7: Verify the Solution

Verify by checking the ratio \( \frac{x}{y} = \frac{750}{1000} = \frac{3}{4} \), which confirms the ratio requirement. Also, check that the sum is correct: \( 750 + 1000 = 1750 \).

Key Concepts

Ratios and Their Calculation

Ratios are a way to compare quantities in relation to each other. They express how much of one thing there is compared to another. In our exercise, we are given a ratio of 3 to 4, which indicates that for every 3 parts received by the first person, the second person receives 4. When dividing a sum of money, such as $1750, in a given ratio, it ensures that the total is split in the specified portions accordingly. To calculate the amounts each person should receive, assign one part of the ratio to each segment. The total number of parts is 3 + 4 = 7 parts. Each part is worth $1750 divided by 7, which gives you the value of one part. By multiplying the value of one part by the number of parts each person receives, you can find out their amounts.

Understanding Systems of Equations

Systems of equations consist of two or more equations with the same variables. They are used to find values that satisfy all the equations simultaneously. In our exercise, there are two key equations:

• The ratio equation: \(4x = 3y\)
• The total sum equation: \(x + y = 1750\)

By combining these equations, you can solve for the unknowns \(x\) and \(y\). This often involves substitution or elimination methods. Here, we use substitution by expressing \(y\) from the total sum equation as \(y = 1750 - x\) and substituting it into the ratio equation to find the value of \(x\). Once one variable is known, the other can be determined easily.

Decoding Word Problems

Word problems often require careful reading to translate text into mathematical expressions. First, identify the important information and what is required. In this problem, the key details are the total sum of $1750, the division into two parts, and the given ratio of 3 to 4.Next, assign variables to unknown quantities. We use variables, such as \(x\) and \(y\), to represent the amounts received by the two individuals. Recognizing patterns and relationships in the data, such as ratios, can help transform the problem into an algebraic format. Finally, solving these equations will provide the solution to the word problem.

Effective Problem Solving Strategies

Problem-solving requires a systematic approach to tackle the unknowns. One useful strategy is to break down the problem into manageable steps. First, understand the problem by identifying the given information and the relations among them, such as ratios or sums. Writing down all known quantities and what you need to find can provide clarity.

• Use visualization techniques if necessary, like drawing models.
• Next, set up equations. Translate relationships into algebraic expressions or equations.
• Solve equations using algebraic methods like substitution or elimination.
• Check your solution by verifying it satisfies all initial conditions and make logical sense.

This step-by-step problem-solving strategy will ensure that you have systematically addressed all components of the problem.