Problem 36 - Elementary Algebra Within Reach

Question

Solve the proportion. $$\frac{4}{t}=\frac{2}{25}$$

Step-by-Step Solution

Answer

The solution to the proportion is \( t = 50 \).

Step 1: Cross Multiply

Cross multiply the given fraction. That is, multiply the numerator of the first fraction (4) by the denominator of the second fraction (25) and set this equal to the product of the denominator of the first fraction (t) and the numerator of the second fraction (2). Giving the equation \( 4*25 = t*2 \).

Step 2: Simplify the equation

Simplify both sides of the equation to get \(100 = 2t\).

Step 3: Solve for t

Solve this equation for \( t \) by dividing both sides of the equation by 2. This yields \( t = \frac{100}{2} \).

Key Concepts

Cross Multiplication

Cross multiplication is a classic technique used to solve proportions, which are equations that set two fractions equal to each other. When you have a proportion like \( \frac{a}{b} = \frac{c}{d} \), you can use cross multiplication to find the unknown variable. To cross multiply, you multiply the numerator of one fraction by the denominator of the other fraction and set them equal to each other. For example, in the proportion \( \frac{4}{t} = \frac{2}{25} \), we multiply 4 (the numerator of the first fraction) and 25 (the denominator of the second fraction) to get one side of the equation. Similarly, we multiply t (the denominator of the first fraction) and 2 (the numerator of the second fraction) to get the other side. This process helps us transform the proportion into an algebraic equation that we can then solve for the unknown variable.

Algebraic Equations

An algebraic equation is a mathematic statement where two expressions are set equal to each other, containing one or more variables. Algebraic equations are powerful tools that allow us to find unknown values by applying various algebraic techniques. In the context of the exercise, the cross multiplication process gives us an algebraic equation \( 4 \times 25 = t \times 2 \). Here, the goal is to manipulate this equation using algebraic operations such as simplification and variable isolation to find the value of t.

Simplifying Equations

Simplifying equations is all about reducing them to their simplest form to make them easier to solve. It involves performing arithmetic operations and combining like terms. After cross multiplying our given proportion, we get the equation \( 4 \times 25 = t \times 2 \), which simplifies down to \( 100 = 2t \). This is a much cleaner and more solvable form compared to the result of cross multiplication. Through simplification, we reduce complexity, making it easier to see the path forward towards isolating the variable and solving for it.

Variable Isolation

Variable isolation is a crucial step in solving algebraic equations. It involves rearranging the equation so that the variable we're solving for is by itself on one side of the equation. The purpose is to find the value of this variable. To isolate the variable t in the simplified equation \( 100 = 2t \), we divide both sides of the equation by 2, which is the coefficient of t. This operation gives us \( t = \frac{100}{2} \), and simplifying the right side of the equation gives us the value of t. By isolating t, we've successfully solved the proportion.