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Chapter 4: Problem 1

Express the following fractions as ratios. Reduce to lowest terms. $$\frac{2}{3}$$

Short Answer

Expert verified
The fraction \(\frac{2}{3}\) as a ratio is 2:3.

Step by step solution

01

Understanding Fractions

A fraction like \(\frac{2}{3}\) consists of two parts: the numerator (2) and the denominator (3). It represents the part of a whole. Here, 2 out of 3 parts is what the fraction represents.
02

Expressing as a Ratio

Express the fraction \(\frac{2}{3}\) as a ratio. Ratios are typically expressed as two numbers separated by a colon, such as 2:3. Therefore, \(\frac{2}{3}\) is already expressed as 2:3.
03

Simplifying the Ratio

Determine if the ratio 2:3 can be simplified. A ratio is in its simplest form when the greatest common divisor (GCD) of the two numbers is 1. Here, the GCD of 2 and 3 is 1, so 2:3 is already in its simplest form.

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Key Concepts

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

Numerator and Denominator
The concepts of numerator and denominator are essential to understanding fractions and how they relate to ratios. A fraction consists of two parts: the numerator is the top number, and the denominator is the bottom number. These two numbers work together to show a portion of a whole. For example, in the fraction \(\frac{2}{3}\), 2 is the numerator and 3 is the denominator. This fraction tells us that we have 2 parts out of a total of 3 parts.

  • The numerator indicates how many parts we are considering.
  • The denominator indicates the total number of equal parts that make up the whole.

When converting fractions to ratios, these concepts help maintain the relationship between the two parts, showing how many of one there is compared to the whole.
Ratio Simplification
Ratio simplification is about reducing ratios to their simplest form, similar to reducing fractions. A ratio compares two quantities, showing how many times one number contains another. In our example, the fraction \(\frac{2}{3}\) converts to the ratio 2:3, keeping the relationship between the two numbers but presented in a different format.

To simplify a ratio:
  • Identify any common factors shared by both numbers.
  • Divide both numbers by their greatest common factor until no common factors exist, other than 1.

If the greatest common factor is 1鈥攍ike in our 2:3 ratio鈥攖hen the ratio is already in its simplest form. Ratio simplification ensures the comparison is as straightforward as possible, just like simplifying fractions.
Greatest Common Divisor
The greatest common divisor (GCD), also known as the greatest common factor, is an important concept in simplifying fractions and ratios. It refers to the largest number that can evenly divide two or more numbers without leaving a remainder. Knowing how to find the GCD is key when you want to simplify a fraction or a ratio.

To find the GCD:
  • List the factors of each number.
  • Identify the largest factor common to both numbers.

Example: For the numbers 2 and 3, the factors are:
  • 2: 1, 2
  • 3: 1, 3
The common factor is 1, so the greatest common divisor is 1.

Since 1 is the only common factor, the ratio 2:3 cannot be simplified further. Understanding the GCD helps verify whether a ratio is already in its simplest form.

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Most popular questions from this chapter

Set up the following problems as a proportion and solve. Include labels in the set up and on the answer. If 60 milligrams (mg) of a medication is in 500 milliliters (mL), how many mL of solution contain 36 mg of medication? ________________________

Express the following fractions as ratios. Reduce to lowest terms. $$\frac{6}{8}$$

Set up the following word problems as proportions and solve. Include labels in the set up and on the answer. If 100 grams (g) of ice cream contains 20 g of fat, how many g of fat are in 275 g of ice cream? __________________________

Express the following fractions as ratios. Reduce to lowest terms. $$\frac{1}{5}$$

Solve for x in the following proportions. Carry division two decimal places as necessary. $$125: 0.4=50: x$$
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