Problem 6 - Understanding Elementary Algebra with Geometry

Question

Solve each of the following verbal problems algebraically. You may use either a one or a two-variable approach. The ratio of two positive numbers is 3 to \(4 .\) If one of the numbers is 5 more than the other, what are the two numbers?

Step-by-Step Solution

Answer

The two numbers are 25 and 20.

Step 1: Define variables

Let the two positive numbers be denoted as x and y. Assume that the ratio of x to y is given by 3:4.

Step 2: Set up ratio equation

According to the ratio, we have \[ \frac{x}{y} = \frac{3}{4} \] or equivalently \[ 4x = 3y \].

Step 3: Express one number in terms of the other

Given that one number is 5 more than the other, we can set up the equation \[ x = y + 5 \].

Step 4: Substitute and solve for y

Substitute the expression for x into the ratio equation: \[ 4(y + 5) = 3y \]. Simplify and solve for y: \[ 4y + 20 = 3y \] \[ y = 20 \].

Step 5: Solve for x

Now that we know y, use the equation \[ x = y + 5 \] to find x: \[ x = 20 + 5 \] \[ x = 25 \].

Step 6: Conclusion

The two numbers that satisfy the given conditions are 25 and 20.

Key Concepts

ratio and proportion

Understanding ratio and proportion is crucial when solving problems involving the relationship between two quantities. A ratio indicates how many times one number contains another. For instance, in this exercise, the ratio of two numbers is given as 3 to 4, written mathematically as \(\frac{3}{4}\). This means if you have 3 parts of one number, you must have 4 parts of the second number.
In the given problem, if we let the two numbers be denoted as x and y, we can write their ratio as \(\frac{x}{y} = \frac{3}{4}\), leading to the equation \(4x = 3y\). Using these relationships helps us create algebraic equations that we can solve step-by-step.

linear equations

Linear equations are equations that represent straight lines and involve variables raised to the power of one. This problem provides a perfect example of forming and solving linear equations. We first get a linear equation from the ratio given \( 4x = 3y \). Additionally, the problem states that one number is 5 more than the other, leading to another linear equation \( x = y + 5 \).
In this exercise, by combining these two linear equations, we can substitute one into the other to solve for the unknown variables. Simplifying and solving these equations step-by-step is essential for finding the values of the variables.

variable substitution

Variable substitution is a method used in algebra to replace a variable with an equivalent expression. In this problem, we use substitution to simplify solving our equations. First, we express one variable in terms of the other using the provided relationship \( x = y + 5 \).
Next, we substitute this expression for x into our initial ratio equation \(4(y + 5) = 3y \). This allows us to simplify and solve a single linear equation rather than dealing with two variables at once. Substitution helps streamline problems and is a powerful tool in algebra.

algebraic expressions

Algebraic expressions are combinations of variables, numbers, and operations (like addition and multiplication) that represent specific values. In the given exercise, we see algebraic expressions at work in the terms \( 4x = 3y \) and \( x = y + 5 \). These expressions define relationships between the variables x and y.
By understanding how to manipulate these expressions—such as combining like terms or isolating variables—students can uncover solutions to the problem. Practicing the manipulation of algebraic expressions builds a strong foundation for more complex mathematical concepts.