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Chapter 6: Problem 51

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

Short Answer

Expert verified
Cross-multiply to get 20x = 100.

Step by step solution

01

Cross-Multiply

Cross-multiply the fractions to eliminate the denominators. Multiply the numerator of the first fraction by the denominator of the second fraction, and the numerator of the second fraction by the denominator of the first fraction. This gives:\[x imes 20 = 4 imes 25\]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

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.

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Most popular questions from this chapter

There are a number of online calculators that will show what the money in one country is worth in another country. One such converter, the XE Universal Currency Converter@, uses live, up-to-the-minute currency rates. Use the information shown here to determine what the equivalent to one U.S. dollar for each of the following denominations. Round to the nearest thousandth, where necessary. \(\$ 25.00\) U.S. dollars are equivalent to 2704.0125 Japanese yen

Divide. Find the quotient of \(2 \frac{2}{3}\) and \(1 \frac{1}{2}\).

Solve each equation by finding a number to replace \(n\) that will make the equation a true statement. $$6 \cdot n=24$$

The following problems review material from a previous section. Reviewing these problems will help you with the next section. Write as a decimal. $$\frac{120}{3}$$

Solve each equation by finding a number to replace \(n\) that will make the equation a true statement. $$3 \cdot n=27$$
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