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Chapter 9: Problem 27

Solve the proportion: \(\frac{9}{x}=\frac{27}{60}\).

Short Answer

Expert verified
The value of \(x\) is 20.

Step by step solution

01

Identify the Proportion

We start with the given proportion: \( \frac{9}{x} = \frac{27}{60} \). The goal is to find the value of \(x\) that makes this proportion true.
02

Cross-Multiply

In a proportion, the cross-products are equal. Thus, multiply 9 by 60 and \(x\) by 27 to get the equation: \( 9 \times 60 = 27 \times x \).
03

Calculate the Cross-Products

Calculate \(9 \times 60\), which results in 540. The equation is now \(540 = 27x\).
04

Solve for \(x\)

To find \(x\), isolate it by dividing both sides of the equation \(540 = 27x\) by 27. This gives \(x = \frac{540}{27}\).
05

Simplify the Fraction

Simplify \(\frac{540}{27}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 27. So, \( \frac{540}{27} = 20 \).

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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 very effective tool for solving proportions, such as \( \frac{a}{b} = \frac{c}{d} \). This method is based on an important rule of equivalent fractions that states: "In a proportion, the cross-products are always equal." Thus, when you multiply the numerator of one ratio by the denominator of the other ratio, both products are equal.
  • Start by identifying the two fractions involved in the proportion.
  • Then, multiply the numerator of the first fraction by the denominator of the second fraction.
  • Do the same with the denominator of the first fraction and the numerator of the second.
  • Equate the two results to form an equation.
Doing cross-multiplication allows you to solve for unknowns easily. In our exercise, cross-multiplying 9 by 60 and \( x \) by 27 formed the equation \( 9 \times 60 = 27 \times x \). This essential step allowed us to transition from the proportion to a simple algebraic equation.
Simplifying Fractions
Fraction simplification involves reducing a fraction to its simplest form. This is done by dividing the numerator and the denominator by their greatest common factor (GCF). When a fraction is fully simplified, the numerator and the denominator have no common factors other than 1.
  • First, find the GCF of the numerator and the denominator.
  • Divide the numerator and denominator by their GCF.
  • The result is the simplified fraction.
In our scenario, we simplified \( \frac{540}{27} \). By identifying and dividing by the GCF, which is 27, the fraction simplified to 20, thus helping us find the solution to the equation.
Greatest Common Divisor
The greatest common divisor (GCD) is a key mathematical concept used to simplify fractions. It is the largest positive integer that divides both numbers without leaving a remainder. To find the GCD, you can use several methods like listing the factors, using prime factorization, or employing the Euclidean algorithm.
  • Listing Factors: Write down all factors of each number, then find the largest factor that both lists share.
  • Prime Factorization: Break down both numbers into their prime factors, then multiply the common prime factors.
  • Euclidean Algorithm: A faster method which uses the remainder of the division process to reduce the problem step-by-step.
In our example, the GCD of 540 and 27 was 27, which greatly simplified our final step, allowing us to easily simplify \( \frac{540}{27} \) to 20.

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