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Chapter 1: Problem 12

\(\frac{14}{35}\)

Short Answer

Expert verified
The simplified form of the fraction \(\frac{14}{35}\) is \(\frac{2}{5}\)

Step by step solution

01

Identify the Numbers

Identify the numbers involved. Here, the numerator is 14 and the denominator is 35.
02

Find the Greatest Common Divisor (GCD)

Find the GCD of 14 and 35. The GCD of 14 and 35 is 7.
03

Divide the Numerator and the Denominator by the GCD

Divide both the numerator and the denominator by the GCD. So, \(\frac{14}{7}\) gives us 2 and \(\frac{35}{7}\) gives us 5.
04

Write the Simplified Fraction

The simplified form of the fraction is therefore \(\frac{2}{5}\).

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Key Concepts

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

Greatest Common Divisor
To simplify a fraction, it's crucial to find the greatest common divisor (GCD), which is the largest number that can divide both the numerator and the denominator without leaving a remainder. Let's consider how we can do this with
  • List the factors of each number.
  • The GCD is the largest factor that both numbers share.
For the numbers 14 and 35, the factors are:
  • 14: 1, 2, 7, 14
  • 35: 1, 5, 7, 35
By comparing these factors, we can see that the largest number shared by both lists is 7. Therefore, the GCD of 14 and 35 is 7. This method helps us identify the most effective way to simplify the given fraction.
Numerator and Denominator
In every fraction, the numerator is the top number, and the denominator is the bottom number. These two numbers tell us a lot about how we interpret the fraction.
  • The numerator shows how many parts we are considering.
  • The denominator shows the total number of equal parts that make up a whole.
Taking the fraction \( \frac{14}{35} \), here 14 is the numerator, representing the 14 parts out of a whole, and 35 is the denominator, representing the total parts. When simplifying a fraction, you need to focus on these numbers because they define the fraction's value and relationship.
Fraction Simplification
Fraction simplification is the process of reducing a fraction to its simplest form by ensuring that the numerator and the denominator have no common factors other than 1. Once the GCD is identified, the next step is to divide both the numerator and the denominator by this number. This step is what simplifies the fraction.
  • Divide 14 by 7 to get 2.
  • Divide 35 by 7 to get 5.
Thus, the fraction \( \frac{14}{35} \) simplifies to \( \frac{2}{5} \). This means \( \frac{2}{5} \) represents the same value as 14 out of 35, but in a more straightforward, reduced form. Simplifying fractions makes them easier to understand and compare.

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Most popular questions from this chapter

Equivalent Expressions Explain why the expressions \(10 \cdot 16+5\) and \((10 \cdot 16)+5\) are equivalent.

Error Analysis Describe and correct the error. $$ \begin{aligned} -9+\frac{9+20}{3(5)}-(-3) &=-9+\frac{9}{3}+\frac{20}{5}-(-3) \\ &=-9+3+4-(-3) \\ &=1 \end{aligned} $$

Finding Inverses Find (a) the additive inverse and (b) the multiplicative inverse of \(-8\).

\(-2(8)=8(-2)\)

\(-7(5-2) \neq-7(5)-7(2)\)
See all solutions

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