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Chapter 5: Problem 405

In the following exercises, write each ratio as a fraction. $$42 \text { to } 48$$

Short Answer

Expert verified
The ratio 42 to 48 as a fraction is \( \frac{7}{8} \).

Step by step solution

01

- Write the ratio as a fraction

Start by writing the given ratio 42 to 48 as a fraction. This can be done by using the format \( \frac{a}{b} \), where \( a \) is 42 and \( b \) is 48. Therefore, the ratio can be written as \( \frac{42}{48} \).
02

- Simplify the fraction

Both the numerator (42) and the denominator (48) can be divided by their greatest common divisor (GCD) to simplify the fraction. Find the GCD of 42 and 48.
03

- Find the GCD

The prime factorization of 42 is \( 2 \times 3 \times 7 \), and the prime factorization of 48 is \( 2^4 \times 3 \). The GCD is the product of the smallest power of common prime factors: \( 2^1 \times 3^1 = 6 \).
04

- Divide both terms by the GCD

Divide both the numerator and the denominator by the GCD (6). \( \frac{42 \div 6}{48 \div 6} = \frac{7}{8} \).

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Key Concepts

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

simplifying fractions
Simplifying fractions means making the fraction as simple as possible. In other words, you reduce the fraction to its simplest form.
A fraction is simplified when the numerator (top number) and the denominator (bottom number) have no common factors other than 1. For instance, in the ratio problem provided, we had to simplify the fraction \( \frac{42}{48} \).
Simplifying involves finding a number that both 42 and 48 can be divided by, which in this case is their Greatest Common Divisor (GCD). This allows us to reduce \( \frac{42}{48} \) to \( \frac{7}{8} \). Understanding the process of simplifying fractions will make other math concepts easier in the future.
greatest common divisor
The Greatest Common Divisor (GCD), also known as the greatest common factor, is the largest number that divides two or more integers without leaving a remainder. To find the GCD of two numbers, you can either list out all their factors and find the largest one they have in common, or use prime factorization.
For example, to find the GCD of 42 and 48:
  • List the factors of 42: 1, 2, 3, 6, 7, 14, 21, 42
  • List the factors of 48: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
  • The common factors are 1, 2, 3, and 6, with 6 being the greatest.
Knowing the GCD helps us in reducing the fraction by dividing both the numerator and the denominator by that GCD.
prime factorization
Prime factorization is breaking down a number into its prime number factors. Prime numbers are numbers greater than 1 that have no other factors besides 1 and themselves. To find the prime factorization of a number, repeatedly divide by the smallest prime number until you reach 1.
For instance, to find the prime factorization of 42 and 48:

  • The prime factorization of 42 is \( 2 \times 3 \times 7 \).
  • The prime factorization of 48 is \( 2^4 \times 3 \).
We use these factors to determine the GCD. In this case, the common prime factors are 2 and 3. The smallest power is \( 2^1 \times 3^1 = 6 \), which is the GCD. This understanding allows us to reduce fractions by dividing both the numerator and denominator by the GCD, simplifying the fraction to its simplest form.

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