Problem 29 - Elementary Algebra

Question

Find two numbers in the ratio of -3 to 7 whose sum is 80 .

Step-by-Step Solution

Answer

The numbers are -60 and 140.

Step 1: Understand the Ratio

The ratio of the two numbers is given as \(-3:7\). This means if the first number is \(-3x\), then the second number is \(7x\).

Step 2: Set up the Equation Based on the Sum

According to the problem, the sum of the two numbers is 80. Therefore, we can set up the equation \(-3x + 7x = 80\).

Step 3: Simplify the Equation

Combine the terms in the equation: \(-3x + 7x = 4x\). Thus, the equation simplifies to \(4x = 80\).

Step 4: Solve for x

Divide both sides of the equation by 4 to solve for \(x\): \(x = \frac{80}{4} = 20\).

Step 5: Find the First Number

Now that we have \(x\), substitute it into the expression for the first number, \(-3x\): The first number is \(-3(20) = -60\).

Step 6: Find the Second Number

Similarly, substitute \(x\) into the expression for the second number, \(7x\): The second number is \(7(20) = 140\).

Step 7: Verify the Solution

Verify that the sum of the two numbers equals 80: \(-60 + 140 = 80\), which confirms that our solution is correct.

Key Concepts

Understanding Ratios

Ratios express the relationship between two quantities. In the exercise, the ratio is given as \(-3:7\). This tells us how one number compares to the other.
For instance, if the first number was \(-3\) units, the second number would be \(7\) units. Ratios help in identifying the proportionality between different values.

• If the ratio is \(a:b\), the two quantities are \(ax\) and \(bx\) where \(x\) is the scaling factor.
• Negative ratios, like \(-3:7\), work similarly, where one part is negative, indicating a reverse direction or a deficit.

Understanding ratios is essential in dividing quantities based on given conditions.

Solving Equations

Equations are mathematical statements that indicate equality between two expressions. In this problem, the equation is derived from the sum of numbers in a ratio.
Given the numbers as \(-3x\) and \(7x\), we form the equation \(-3x + 7x = 80\). This equation helps us find the unknown scaling factor.
Here are the steps to simplify and solve it:

• Combine like terms: \(-3x + 7x = 4x\).
• Divide both sides by \(4\) to solve for \(x\): so, \(4x = 80\) becomes \(x = 20\).

The solution of the equation gives the value of \(x\), which can then be used to find the actual numbers.

Effective Problem Solving

Problem-solving in mathematics involves understanding the problem, setting up equations, and verifying solutions. This exercise requires finding numbers based on a ratio and a sum.
Here’s a simple guide to solving similar problems:

• Start by clearly understanding what is given and what is needed. Here, you know the ratio and the sum.
• Translate the word problem into mathematical expressions and equations. Use the ratio to define expressions for both numbers.
• Set up an equation that represents the known condition, such as a total sum.
• Solve for the unknown variable and substitute back to find the specific numbers.
• Always verify by substituting the numbers back to confirm they satisfy the initial condition.

Following these steps ensures that you solve problems efficiently and accurately.