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Chapter 6: Problem 277

In the following exercises, solve each proportion. $$ \frac{2.8}{k}=\frac{2.1}{1.5} $$

Short Answer

Expert verified
k = 2

Step by step solution

01

- Set Up the Proportion

Start with the given proportion: \( \frac{2.8}{k}=\frac{2.1}{1.5} \).
02

- Cross Multiply

To solve for \( k \), use cross multiplication. Multiply both sides so that: \[ 2.8 \times 1.5 = 2.1 \times k \]
03

- Perform the Multiplication

Calculate the product of the numbers on both sides. On the left side, \( 2.8 \times 1.5 = 4.2 \). So we have: \( 4.2 = 2.1k \)
04

- Solve for k

Isolate \( k \) by dividing both sides by 2.1: \[ k = \frac{4.2}{2.1} \]
05

- Simplify

Simplify the fraction: \( \frac{4.2}{2.1} = 2 \). Thus, \( k = 2 \).

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Key Concepts

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

cross multiplication
Understanding cross multiplication is key to solving proportions like the given exercise. Start with the proportion: \( \frac{2.8}{k} = \frac{2.1}{1.5} \). To solve for 魏, we use cross multiplication, which involves multiplying the numerator of one fraction by the denominator of the other fraction. Multiply both sides of the equation to eliminate the denominators: \[ 2.8 \times 1.5 = 2.1 \times k \]. This transforms the proportion into a simpler equation: \[4.2 = 2.1k \].
simplifying fractions
To find the value of k, we need to simplify the resulting fraction. After cross multiplication, we have: \[ 4.2 = 2.1k \]. To isolate k, divide both sides of the equation by 2.1: \[ k = \frac{4.2}{2.1} \]. Next, simplify this fraction. Since 4.2 and 2.1 can be easily divided, you get: \[ \frac{4.2}{2.1} = 2 \]. Hence, 魏 equals 2. Simplifying fractions helps in reducing complex calculations to simpler, manageable terms.
prealgebra problem-solving
The solution involves key prealgebra problem-solving skills. The first step is setting up the proportion from the problem statement. Then use cross multiplication to transform the equation into a simpler form. Perform the necessary arithmetic to simplify and solve for the variable. Key steps include:
  • Recognizing the proportion
  • Using cross multiplication
  • Simplifying fractions
  • Isolating the variable
. Each step builds upon basic math principles, leading to the final solution efficiently.
isolation of variables
Isolating the variable is one of the final steps in solving the proportion. After setting up the equation \[ 4.2 = 2.1k \], we isolate k by dividing both sides by 2.1: \[ k = \frac{4.2}{2.1} \]. This isolates k on one side of the equation, making it easier to simplify. In this case, it simplifies to \[ k = 2 \]. Isolating variables involves the following steps:
  • Performing inverse operations
  • Maintaining the balance of the equation
  • Simplifying the fraction if necessary
. This method is widely used in algebra to find the values of unknowns.

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Most popular questions from this chapter

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