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Chapter 6: Problem 58

Write each fraction in simplest form. See Section 4.2. $$ \frac{10}{15} $$

Short Answer

Expert verified
The fraction \( \frac{10}{15} \) simplifies to \( \frac{2}{3} \).

Step by step solution

01

Identify the Greatest Common Divisor (GCD)

To simplify a fraction, we first need to find the greatest common divisor (GCD) of the numerator and the denominator. The numerator is 10 and the denominator is 15. We list the factors of both numbers: - Factors of 10: 1, 2, 5, 10 - Factors of 15: 1, 3, 5, 15 The greatest common factor shared by both 10 and 15 is 5.
02

Divide the Numerator and Denominator by the GCD

Now, divide both the numerator and the denominator by their greatest common divisor, which is 5, to simplify the fraction:\[ \frac{10}{15} = \frac{10 \div 5}{15 \div 5} = \frac{2}{3} \]
03

Confirm the Fraction is in Simplest Form

Check to ensure that the fraction \( \frac{2}{3} \) cannot be reduced further. Since 2 and 3 have no common factors other than 1, \( \frac{2}{3} \) is in its simplest form.

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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 any fraction, we need to find the greatest common divisor (GCD) of both the numerator and the denominator. The GCD is the largest number that can divide both numbers without leaving a remainder.

Here's how to find the GCD:
  • List the factors of both the numerator and denominator.
  • Identify the largest factor that appears in both lists.
For example, in the fraction \( \frac{10}{15} \), we have:
  • Factors of 10: 1, 2, 5, 10
  • Factors of 15: 1, 3, 5, 15
The greatest number that appears in both lists is 5. Hence, 5 is the GCD. Finding the GCD is crucial as it shows how much we can "shrink" the fraction by dividing both components by this number.
Numerator
The numerator is the number on the top part of the fraction. It represents how many parts of the whole we are considering. In the fraction \( \frac{10}{15} \), 10 is the numerator.

When simplifying a fraction, the numerator is divided by the GCD. By doing so:
  • We reduce the size of the number while maintaining the same ratio.
  • The fraction becomes easier to read and more manageable for calculations.
After dividing the numerator by the GCD (which is 5 in this case), we get \( 10 \div 5 = 2 \). This gives us the simplified numerator, which, along with the denominator, makes up the simplest form of the original fraction.
Denominator
The denominator is the bottom number of a fraction and it tells us into how many equal parts the whole is divided. For \( \frac{10}{15} \), the denominator is 15.

In simplification:
  • The denominator is also divided by the GCD.
  • We aim to maintain the equivalent value while altering the form.
When we divide the denominator 15 by the GCD 5, we get \( 15 \div 5 = 3 \).
This step gives us the simplified denominator, completing the formation of the simplest fraction \( \frac{2}{3} \). The simplest form has the same overall value as the original but is more concise and user-friendly for further computations.

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Most popular questions from this chapter

One pound of firmly packed brown sugar yields \(2 \frac{1}{4}\) cups. How many pounds of brown sugar will be required in a recipe that calls for 6 cups of firmly packed brown sugar?

Suppose that the amount of a product decreases, say from an 80 -ounce container to a 70 -ounce container, but the price of the container remains the same. Does the unit price increase or decrease? Explain why.

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Insert \(<\) or \(>\) to form a true statement. $$ 2 \frac{1}{2} \quad 2 \frac{1}{3} $$

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