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Chapter 11: Problem 7

In Exercises \(4-12,\) solve the proportion. \(\frac{4}{5}=\frac{x}{7}\)

Short Answer

Expert verified
x = 5.6

Step by step solution

01

Set up the Proportion

Write down the given proportion which is \[\frac{4}{5} = \frac{x}{7}.\]
02

Cross Multiply

To solve the proportion, cross multiply the numerators and denominators:\[4 \times 7 = 5 \times x.\]
03

Simplify the Equation

Perform the multiplication on both sides of the equation:\[28 = 5x.\]
04

Solve for x

Isolate the variable \( x \) by dividing both sides of the equation by 5:\[x = \frac{28}{5}.\]
05

Simplify the Result

Simplify the fraction if possible:\[x = 5.6.\]

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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 powerful technique used to solve proportions. A proportion is an equation stating that two ratios or fractions are equal. By cross multiplying, we eliminate the fractions, making it easier to solve the equation. For instance, in the problem \(\frac{4}{5} = \frac{x}{7}\), we cross multiply to create a simple equation: \[4 \times 7 = 5 \times x.\] Here, we multiplied the numerator of one fraction by the denominator of the other fraction. This step transforms a complex fraction problem into something much simpler. It鈥檚 a handy method whenever you're dealing with proportions.
Equation Solving
Once we've used cross multiplication, we move to solving the resulting equation. For \(\frac{4}{5} = \frac{x}{7}\), after cross multiplying, we get \[28 = 5x.\] To find \(x\), we need to isolate it on one side of the equation. This is done by performing the same operation on both sides of the equation. Here, we divide both sides by 5 to isolate \(x\) on its own:
  • Divide 28 by 5.
  • The resulting equation is: \ x = \frac{28}{5}. \
  • Perform the division to simplify the fraction (if needed).
This will give you the value of \(x\), providing the solution to the original proportion.
Mathematical Fractions
Fractions represent parts of a whole and are essential in solving proportions. In the problem \(\frac{4}{5} = \frac{x}{7}\), each number in the fraction is called a term: the top number (numerator) and the bottom number (denominator). Understanding these terms is key as they help identify which values to cross multiply. Additionally, simplification鈥攕uch as turning \( \frac{28}{5}\) into a decimal 5.6鈥攊s crucial for getting the final answer. Simplifying makes the fraction easier to understand and use, especially in real-life applications. When solving mathematical fractions, always strive for the simplest form for clarity.

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