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Chapter 6: Problem 10

Write each rate in lowest terms. Travis has 14 blooms on 6 of his plants.

Short Answer

Expert verified
The lowest term for the rate is \(\frac{7}{3}\).

Step by step solution

01

Write the initial rate

Start by expressing the given information as a ratio. Travis has 14 blooms on 6 plants, so write this as \(\frac{14}{6}\).
02

Find the greatest common divisor (GCD)

Determine the greatest common divisor of the numerator (14) and the denominator (6). The factors of 14 are {1, 2, 7, 14} and the factors of 6 are {1, 2, 3, 6}. The greatest common divisor is 2.
03

Divide both terms by the GCD

Divide both the numerator and the denominator by the GCD to simplify the ratio: \(\frac{14 \div 2}{6 \div 2} = \frac{7}{3}\).
04

Verify the simplified rate

Check if the ratio \(\frac{7}{3}\) is in its simplest form by ensuring there are no common factors other than 1 between 7 and 3. Since there are none, the simplified rate is correct.

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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 ratio, it鈥檚 essential to understand the greatest common divisor (GCD). The GCD is the highest number that can completely divide two or more numbers.
In our example, we start with the ratio \(\frac{14}{6}\).
We need to find the GCD of 14 and 6.
Here鈥檚 how to do it:
  • List the factors of each number.
  • The factors of 14 are {1, 2, 7, 14}.
  • The factors of 6 are {1, 2, 3, 6}.
  • Now, identify the greatest common factor that appears in both lists. In this case, it's 2.
With the GCD in hand, you're ready for the next steps toward simplification!
fraction simplification
Simplifying a fraction involves dividing both the numerator and the denominator by their GCD. For our ratio \(\frac{14}{6}\), we鈥檝e determined that the GCD is 2. Now, we simplify the fraction by dividing both terms by this number:
\( \frac{14 \div 2}{6 \div 2} = \frac{7}{3} \).
Here鈥檚 a quick summary:
  • Write the fraction.
  • Find the GCD of the numerator and denominator.
  • Divide both the numerator and the denominator by this GCD.
  • The result is your simplified fraction.
Always remember to check if the new fraction is in its simplest form by ensuring there are no more common factors between the numerator and the denominator.
ratios in lowest terms
A ratio is in its lowest terms when both the numerator and the denominator are as small as possible. This is achieved when the only common factor between them is 1. For example, our simplified ratio \(\frac{7}{3}\) is in its lowest terms because the only common factor between 7 and 3 is 1.
To ensure that a ratio is in its lowest terms:
  • Start with the original ratio, like \(\frac{14}{6}\).
  • Determine the GCD of the numerator and the denominator.
  • Divide both terms by the GCD.
  • Check the resulting ratio to ensure there are no common factors other than 1.
With these steps, you'll always be able to simplify ratios to their lowest terms, making comparisons and calculations much more manageable!

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