Problem 51 - Prealgebra

Question

Solve each equation. $$\frac{x}{25}=\frac{4}{20}$$

Step-by-Step Solution

Answer

\(x = 5\)

Step 1: Cross-Multiply

Cross-multiply to solve the proportion:
\(20x = 25 \times 4 = 100\)

Step 2: Solve for x

Divide both sides by 20:
\(x = \frac{100}{20} = 5\)

Key Concepts

Cross-Multiplication

Cross-multiplication is a useful method for solving equations that involve fractions or ratios. It helps in eliminating the denominators so you can solve the equation more easily. Essentially, you multiply the numerator of each fraction by the denominator of the other.

• For the equation \( \frac{x}{25} = \frac{4}{20} \), cross-multiplication involves multiplying \( x \) by 20 and 4 by 25.
• After cross-multiplying, you'll rewrite the equation as \( 20x = 100 \).

This process eliminates fractions, making the equation far easier to solve. Cross-multiplication works well because if two fractions are equal, then the product of the extremes (outer terms) equals the product of the means (inner terms). After this simplification step, you can proceed to solve for \( x \) using simpler algebraic methods.

Proportional Relationships

A proportional relationship is a scenario where two ratios are equal. This is often represented using the format \( \frac{a}{b} = \frac{c}{d} \). Pools of ratios are foundational in mathematics and can apply to various real-world contexts, such as scaling recipes or converting currencies.
Understanding proportional relationships is crucial in solving equations because:

• They maintain equality between two distinct ratios or fractions.
• Allow you to find unknown values through cross-multiplication.
• Enable simplifying complex problems by reducing them to comparable smaller parts.

In the given equation, \( \frac{x}{25} = \frac{4}{20} \), we identify a proportional relationship between \( x \), 25, 4, and 20. Recognizing this enables you to apply cross-multiplication and solve the equation efficiently.

Algebraic Manipulation

Algebraic manipulation refers to the various operations and strategies used to simplify or solve equations. After performing cross-multiplication, you often encounter algebraic expressions that require further simplification.

• In the equation \( 20x = 100 \), you'll use algebraic manipulation to isolate \( x \).
• The primary step involves dividing both sides of the equation by 20, yielding \( x = 5 \).

These steps highlight the importance of algebraic manipulation:- **Isolation of Variables**: The technique focuses on isolating the desired variable, simplifying the process of finding its value.- **Ease of Solving**: Through algebraic manipulation, equations become simpler forms which are easier to deal with.Mastering algebraic manipulation techniques is essential for solving all types of equations, as it forms the backbone of algebra.