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Chapter 6: Problem 26

Write the ratio in lowest terms. 3 cups to 9 cups

Short Answer

Expert verified
The ratio is 1 cup to 3 cups.

Step by step solution

01

- Understand the ratio

The ratio given is 3 cups to 9 cups. This can be written as the fraction \(\frac{3}{9}\).
02

- Find the greatest common divisor (GCD)

Identify the greatest common divisor of the numerator (3) and the denominator (9). The GCD of 3 and 9 is 3.
03

- Divide both parts by GCD

Divide both the numerator and the denominator by the GCD. So, \(\frac{3}{9} = \frac{3 \div 3}{9 \div 3} = \frac{1}{3}\).
04

- Write the ratio in lowest terms

The fraction \(\frac{1}{3}\) represents the ratio in its lowest terms. Therefore, 3 cups to 9 cups in the simplest form is 1 cup to 3 cups.

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Key Concepts

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

Fractions
A fraction consists of two numbers: a numerator and a denominator. In our example, 3 cups to 9 cups can be represented as the fraction \(\frac{3}{9}\), where 3 is the numerator and 9 is the denominator. Fractions are used to express a part of a whole. Fractions can also represent ratios, which show the relationship between two quantities.
  • Numerator: The top number in a fraction. It represents the number of parts we have.
  • Denominator: The bottom number in a fraction. It represents the total number of equal parts.
Understanding fractions is crucial because they help us compare different quantities and solve problems involving ratios. Remember, whenever dealing with fractions, it's always good to check if they can be simplified for easier comprehension.
Greatest Common Divisor (GCD)
The greatest common divisor (GCD), also known as the greatest common factor (GCF), is the highest number that divides exactly into two or more numbers. For example, in the fraction \(\frac{3}{9}\), the numbers involved are 3 and 9. We find the GCD to simplify this fraction.
To solve this, we need to:
  • List the factors of 3: 1, 3
  • List the factors of 9: 1, 3, 9
The highest number that appears in both lists of factors is the GCD. Here, it is 3. This means 3 is the greatest number that divides both 3 and 9 without leaving a remainder. Finding the GCD is essential for simplifying fractions to their lowest terms and making calculations easier.
Simplifying Fractions
Simplifying fractions involves reducing them to their lowest terms so that they are easier to work with. To simplify, we use the GCD, dividing both the numerator and the denominator by this number.
For our given ratio of 3 cups to 9 cups, written as \(\frac{3}{9}\), we already found that the GCD is 3. So, we simplify by dividing both the numerator and the denominator by 3:
  • Numerator: 3 梅 3 = 1
  • Denominator: 9 梅 3 = 3
Thus, \(\frac{3}{9}\) simplifies to \(\frac{1}{3}\). This fraction, \(\frac{1}{3}\), indicates that for every 1 cup of the first quantity, there are 3 cups of the second. Simplifying fractions helps us understand and compare ratios more effectively.

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Most popular questions from this chapter

Determine whether the ratios form a proportion. $$\frac{1 \frac{3}{4}}{3}=\frac{7}{12}$$

Solve the proportion. Be sure to check your answers. $$\frac{\frac{3}{8}}{6.75}=\frac{x}{72}$$

Erik bought \(\$ 624\) Canadian with \(\$ 600\) American. At this rate, how many Canadian dollars can he buy with \(\$ 250\) American?

Are the numbers \(1 \frac{2}{3}\) and \(\frac{5}{6}\) proportional to the numbers 5 and \(2 \frac{1}{2} ?\)

Solve the proportion. Be sure to check your answers. $$\frac{16}{12}=\frac{21}{a}$$
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