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Chapter 2: Problem 23

Decide whether each proportion is true or false. $$ \frac{5}{35}=\frac{8}{56} $$

Short Answer

Expert verified
True. Both fractions simplify to \( \frac{1}{7} \).

Step by step solution

01

- Identify the Proportions

The given proportion is \( \frac{5}{35} = \frac{8}{56} \).
02

- Simplify the First Fraction

Simplify \( \frac{5}{35} \). Notice that 5 is a common factor of both the numerator and the denominator. \( \frac{5 \div 5}{35 \div 5} = \frac{1}{7} \).
03

- Simplify the Second Fraction

Simplify \( \frac{8}{56} \). Notice that 8 is a common factor of both the numerator and the denominator. \( \frac{8 \div 8}{56 \div 8} = \frac{1}{7} \).
04

- Compare the Simplified Fractions

Both simplified fractions are \( \frac{1}{7} \). Since they are equal, the original proportion \( \frac{5}{35} = \frac{8}{56} \) is true.

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Key Concepts

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

proportion
A proportion is an equation that states that two ratios are equal. In simpler terms, it shows that two fractions represent the same relationship between their numerators and denominators. For example, if we have the proportion \( \frac{5}{35} = \frac{8}{56} \), we are asserting that the two fractions are equivalent.

To decide whether a proportion is true or false, we typically simplify both fractions to their simplest forms and compare them.
This makes it easy to see if the ratio presented by each fraction is indeed identical.
simplifying fractions
Simplifying fractions means reducing them to their simplest form, where the numerator and the denominator share no common factors other than 1.

To simplify a fraction:
  • Find the greatest common divisor (GCD) of both the numerator and the denominator.
  • Divide both the numerator and the denominator by their GCD.

Let's apply this to our fractions in the example: \( \frac{5}{35} \) and \( \frac{8}{56} \).

For \( \frac{5}{35} \):
  • The GCD of 5 and 35 is 5.
  • Divide both by 5: \( \frac{5 \div 5}{35 \div 5} = \frac{1}{7} \).

For \( \frac{8}{56} \):
  • The GCD of 8 and 56 is 8.
  • Divide both by 8: \( \frac{8 \div 8}{56 \div 8} = \frac{1}{7} \).

Both fractions simplify to \( \frac{1}{7} \), so the original fractions are in proportion.
numerator and denominator
The terms numerator and denominator are key parts of any fraction.

  • The numerator is the number above the fraction bar and represents how many parts of the whole are being considered.
  • The denominator is the number below the fraction bar and represents the total number of equal parts in the whole.

In our example, the numerator and denominator for \( \frac{5}{35} \) are 5 and 35, respectively.
Similarly, for \( \frac{8}{56} \), the numerator is 8 and the denominator is 56.

Understanding the roles of numerator and denominator helps in simplifying fractions and determining the equality of proportions.

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