Problem 54 - Elementary Algebra Within Reach

Question

Solve the proportion. $$\frac{5}{t}=\frac{75}{165}$$

Step-by-Step Solution

Answer

The value of \(t\) is 11.

Step 1: Set up the proportion

The given proportion is \(\frac{5}{t}=\frac{75}{165}\). The goal is to solve for the variable \(t\). This is a ratio equated to another ratio, forming a proportion.

Step 2: Cross multiply

Cross multiplication is the process of cancelling out fractions present in equation. In this case, it would look like this: \(5*165 = 75*t\).

Step 3: Solve for the variable

Now simplify the expression: \(825 = 75t\). To isolate \(t\), divide both sides of the equation by 75.

Step 4: Calculate the value of t

Perform the division operation: \(\frac{825}{75} = t\).

Key Concepts

Solving Proportions

A proportion is an equation that states two ratios are equal. In algebra, solving proportions involves finding the unknown variable in a proportion. Consider the equation given in the exercise: \[ \frac{5}{t} = \frac{75}{165} \] Our task is to determine the value of the variable \( t \) that makes this statement true.

• To do this, we use algebraic manipulations. It usually involves the concept of cross-multiplication to eliminate the fractions, making it easier to solve for the unknown variable.
• After setting up the equation, we perform operations to isolate the variable.

It's important to ensure that both sides of the proportion are balanced, meaning they are the same value when computed. Properly solving proportions helps us understand relationships between different quantities.

Cross Multiplication

Cross multiplication is a crucial tool in solving proportions. It is a method of eliminating fractions by multiplying the numerator of one fraction by the denominator of the other fraction, and doing the same for the remaining terms. This transforms the equation into a more solvable form. Let's look at how it works:Given the equation: \[ \frac{5}{t} = \frac{75}{165} \]

• Cross multiply by multiplying the numerator on one side by the denominator on the other side: \(5 \times 165\) equals \(75 \times t\).
• This gives us the expression: \(825 = 75t\).

This algebraic manipulation effectively helps us remove the fractions, making it straightforward to solve for \( t \). Proper use of cross multiplication ensures that the original relationship between the two ratios is maintained while simplifying the equation.

Algebraic Fractions

Algebraic fractions are fractions where the numerator, denominator, or both contain algebraic expressions. Solving equations involving algebraic fractions, like in our exercise, involves understanding how we can manipulate these expressions efficiently.With the example equation \( \frac{5}{t} = \frac{75}{165} \), both sides of the equation are set up as fractions:

• The ratio \(\frac{5}{t}\) implies that 5 is part of the whole represented by \(t\).
• The other fraction \(\frac{75}{165}\) represents a known ratio or relationship.

The challenge is often to isolate the variable in the algebraic fraction. By cross-multiplying, as we described, we can clear the fractions, simplify the equation, and ultimately solve for \(t\). Working with algebraic fractions requires practice, but using strategies like cross multiplication can make it much simpler.