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Chapter 2: Problem 39

Solve each equation. $$ \frac{49}{56}=\frac{z}{8} $$

Short Answer

Expert verified
z = 7

Step by step solution

01

Simplify the Fraction on the Left

Simplify the fraction \(\frac{49}{56}\). Find the greatest common divisor (GCD) of 49 and 56, which is 7. Then, divide both the numerator and the denominator by 7. \(\frac{49 \div 7}{56 \div 7} = \frac{7}{8}\).
02

Set Simplified Fractions Equal

Rewrite the equation with the simplified fraction: \(\frac{7}{8} = \frac{z}{8}\).
03

Solve for \(z\)

Since the two fractions are equal and they have the same denominator, their numerators must also be equal. Therefore, set the numerators equal to each other: \(7 = z\).

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Key Concepts

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

Greatest Common Divisor
The greatest common divisor (GCD) is the largest number that can evenly divide two numbers. In the context of fractions, finding the GCD allows you to simplify fractions easily.
To determine the GCD of two numbers:
  • List the prime factors of each number.
  • Identify the common factors.
  • Select the largest of these common factors.
For example, with 49 and 56, we find that both are divisible by 7, making 7 the GCD. By dividing the numerator and denominator of \( \frac{49}{56} \) by 7, we simplify it to \( \frac{7}{8} \).
Equivalent Fractions
Equivalent fractions are fractions that represent the same value, even though they may have different numerators and denominators. Simplifying a fraction involves finding an equivalent fraction with the smallest possible numerator and denominator. For instance: \( \frac{49}{56} \) and \( \frac{7}{8} \) are equivalent because both represent the same portion of a whole. To check if fractions are equivalent:
  • Cross-multiply the numerators and denominators.
  • Compare the results.
If the results are equal, the fractions are equivalent. Equivalence is crucial for solving equations involving fractions because it helps simplify the problem.
Solving Equations
Solving equations means finding the value of the unknown that makes the equation true. For the equation \[ \frac{49}{56} = \frac{z}{8} \], simplifying the first fraction to \[ \frac{7}{8} \] makes it easier to solve. Since both sides of the equation share the same denominator, the numerators must be equal for the fractions to be identical. Therefore, we set the numerators equal: \[ 7 = z \]. This straightforward step leads us to the solution \[ z = 7 \]. Knowing how to simplify fractions and recognize equivalent fractions makes solving such equations quicker and more intuitive.

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