Problem 30 - Elementary Algebra

Question

Find two numbers in the ratio of -1 to 3 whose sum is 90 .

Step-by-Step Solution

Answer

The numbers are -45 and 135.

Step 1: Understanding the ratio

The problem states that the ratio of the two numbers is -1 to 3. This means if we take one number as -1x, the other number will be 3x.

Step 2: Set up the sum equation

According to the problem, the sum of these two numbers is 90. Therefore, we can write the equation as \(-1x + 3x = 90\).

Step 3: Simplify the equation

Combine like terms in the equation from Step 2 to get \(2x = 90\).

Step 4: Solve for x

To find the value of \(x\), divide both sides of the equation \(2x = 90\) by 2, so \(x = \frac{90}{2} = 45\).

Step 5: Find the numbers

Substitute \(x = 45\) into the expressions for the numbers: the first number is \(-1(45) = -45\) and the second number is \(3(45) = 135\).

Step 6: Verify the solution

Check the sum of the numbers: \(-45 + 135 = 90\), which matches the condition given in the problem.

Key Concepts

Understanding Ratios

Ratios are a way to compare two quantities, showing how many times one value contains or is contained within the other. In this exercise, we dealt with a ratio of -1 to 3. This means for every unit of the first quantity, we have three units of the second. When the ratio involves negative numbers, like -1, it indicates an inverse relationship, where one number is opposite in sign to the other. Understanding this helps set up equations correctly by assigning variables appropriately.

Setting Up Equations

Equations are mathematical statements that express the equality between two expressions. Here, we use equations to solve for unknown values based on given conditions. We represent the two numbers as

• -1 times a variable x, and
• 3 times the same variable x.

So, our equation becomes \[-1x + 3x = 90\]. By representing relationships through equations, you simplify complex information into manageable parts.

Approaches to Problem-Solving

Problem-solving in algebra often involves translating word problems into mathematical statements. Begin by identifying knowns and unknowns. In our case:

• Ratio of numbers: -1 to 3,
• Sum requirement: 90.

By setting up relationships based on these, you can create an equation that leads to the solution. Breaking down the process into steps like simplifying and solving the equation makes finding the answer systematic and straightforward.

Exploring Integer Operations

Integer operations are fundamental in algebra, involving basic arithmetic with whole numbers. In this solution, we performed operations to isolate the variable:

• Add: Combine like terms -1x and 3x to get 2x.
• Divide: Solve 2x = 90 by dividing by 2, which results in x = 45.

These operations help simplify equations and find solutions efficiently. Understanding how to manipulate integers is key for accurately solving algebra problems.