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Chapter 6: Problem 398

In the following exercises, write each sentence as a proportion. 3 is to 8 as 12 is to 32

Short Answer

Expert verified
\(\frac{3}{8} = \frac{12}{32}\)

Step by step solution

01

Identify the quantities

In the given sentence, identify the two pairs of corresponding numbers. Here, 3 and 8 are one pair, and 12 and 32 are the other pair.
02

Set up the proportion

Write the identified pairs as a fraction. For the first pair, the fraction is \(\frac{3}{8}\). For the second pair, the fraction is \(\frac{12}{32}\).
03

Form the proportion statement

Combine the fractions into a proportion statement: \(\frac{3}{8} = \frac{12}{32}\). This shows that 3 is to 8 as 12 is to 32.

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Key Concepts

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

ratios
Ratios are a way to compare two quantities. They can tell us how much of one thing there is compared to another. In this exercise, we are comparing numbers using ratios.
A ratio can be written in different forms:
  • Using a colon: 3:8
  • As a fraction: \(\frac{3}{8}\)
  • Using the word 'to': 3 is to 8
Understanding how to switch between these forms helps you solve problems more easily.
In our exercise, we started with a sentence: '3 is to 8 as 12 is to 32.' This tells us two pairs of numbers we need to compare.
Next, we can convert these ratios into fractions to see if they are equal. This makes it easier to understand if the two ratios are indeed the same.
fractions
Fractions represent a part of a whole. They have a numerator (top number) and a denominator (bottom number). For example, \(\frac{3}{8}\) is a fraction.
To solve the proportion exercise, we first write the pairs of numbers as fractions:
  • The first pair, 3 and 8, can be written as \(\frac{3}{8}\)
  • The second pair, 12 and 32, can be written as \(\frac{12}{32}\)
Comparing these fractions helps us see if they represent the same ratio. When two fractions are equal, it means the ratios are proportionate. This concept is key to understanding proportions and solving these types of exercises.
We can simplify fractions to check if they are equal more easily. For example, \(\frac{12}{32}\) simplifies to \(\frac{3}{8}\), confirming that \(\frac{3}{8} = \frac{12}{32}\).
basic algebra
Basic algebra involves solving problems by finding relationships between numbers.
In this exercise, we use algebraic thinking to set up and solve proportions. Here鈥檚 how:
  • Identify the relationships: Find the pairs of numbers. Example: 3 is to 8 and 12 is to 32.
  • Convert into fractions: Write these pairs as fractions. Example: \(\frac{3}{8}\) and \(\frac{12}{32}\).
  • Set up the proportion: Formulate the equation. Example: \(\frac{3}{8} = \frac{12}{32}\).
Using algebraic methods, like cross-multiplying, can help confirm proportions are correct. Cross-multiplying means you multiply the numerator of one fraction by the denominator of the other. If the products are equal, the ratios are proportional:
  • For \(\frac{3}{8}\ and \frac{12}{32}\), cross multiplying gives us \(3 \times 32 = 96\) and \(8 \times 12 = 96\).
  • Since both products are equal, the ratios are proportional.
This technique makes it easy to verify our solutions and deepen our understanding of algebraic relationships.

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