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Chapter 6: Problem 32

Write the ratio in lowest terms with whole numbers in the numerator and denominator. \(1 \frac{1}{4}\) in. to \(1 \frac{3}{8}\) in.

Short Answer

Expert verified
The ratio in lowest terms is \( \frac{10}{11} \).

Step by step solution

01

Convert Mixed Numbers to Improper Fractions

Convert the mixed numbers to improper fractions. For the first mixed number, \ 1 \frac{1}{4}, convert it: \[ 1 \frac{1}{4} = \frac{4 \times 1 + 1}{4} = \frac{5}{4} \]. For the second mixed number, \ 1 \frac{3}{8}, convert it: \[ 1 \frac{3}{8} = \frac{8 \times 1 + 3}{8} = \frac{11}{8} \].
02

Write the Ratio as a Fraction

Write the given ratio as a fraction using the improper fractions from Step 1. The ratio of \(1 \frac{1}{4}\) to \(1 \frac{3}{8}\) is \[ \frac{\frac{5}{4}}{\frac{11}{8}} \].
03

Simplify the Fraction

Simplify the fraction by multiplying by the reciprocal of the denominator: \[ \frac{\frac{5}{4}}{\frac{11}{8}} = \frac{5}{4} \times \frac{8}{11} \]. This simplifies to: \[ \frac{5 \times 8}{4 \times 11} = \frac{40}{44} \].
04

Reduce to Lowest Terms

Divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 40 and 44 is 4. Therefore, \[ \frac{40 \text{ 梅 } 4}{44 \text{ 梅 } 4} = \frac{10}{11} \].

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Key Concepts

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

Mixed Numbers
A mixed number combines a whole number and a fraction. For example, in the mixed number \(1 \frac{1}{4}\), 1 is the whole number and \(\frac{1}{4}\) is the fraction. Mixed numbers are useful because they express quantities greater than one in an easy-to-understand format. To work with mixed numbers in calculations, they are often converted into improper fractions.
Here鈥檚 how to do it:
  • Multiply the whole number by the denominator of the fraction.
  • Add the result to the numerator of the fraction.
  • Place this sum over the original denominator.
For example, to convert \(1 \frac{1}{4}\) to an improper fraction:
  • Multiply 1 (whole number) by 4 (denominator): \(1 \times 4 = 4\).
  • Add the numerator, 1: \(4 + 1 = 5\).
  • Place this over the original denominator: \(\frac{5}{4}\).
Improper Fractions
Improper fractions have a numerator larger than the denominator. They represent values greater than or equal to one. For example, \(\frac{5}{4}\) means 5 parts of 4, which is more than 1 whole part.
When working with mixed numbers and needing to perform calculations, converting them to improper fractions makes multiplication and division easier.
For instance, converting the mixed number \(1 \frac{3}{8}\):
  • Multiply 1 (whole number) by 8 (denominator): \(1 \times 8 = 8\).
  • Add the numerator, 3: \(8 + 3 = 11\).
  • Place this over the original denominator: \(\frac{11}{8}\).
This improper fraction \(\frac{11}{8}\) can now be easily used in other math operations.
Reciprocal
The reciprocal of a fraction is created by swapping the numerator and the denominator. For example, the reciprocal of \(\frac{5}{4}\) is \(\frac{4}{5}\). Reciprocal fractions are crucial when dividing fractions.
Instead of dividing directly, you multiply by the reciprocal. For example:
  • To divide \(\frac{5}{4}\) by \(\frac{11}{8}\), find the reciprocal of \(\frac{11}{8}\), which is \(\frac{8}{11}\).
  • Then, multiply: \(\frac{5}{4} \times \frac{8}{11}\) results in \(\frac{40}{44}\).
This technique simplifies the division of fractions.
Greatest Common Divisor (GCD)
The greatest common divisor (GCD) of two numbers is the largest number that divides both of them without leaving a remainder. Finding the GCD is essential for simplifying fractions.
For instance, to reduce \(\frac{40}{44}\) to its lowest terms:
  • List the factors of 40: 1, 2, 4, 5, 8, 10, 20, 40.
  • List the factors of 44: 1, 2, 4, 11, 22, 44.
  • The common factors are 1, 2, and 4. The greatest is 4.
  • Divide both the numerator and the denominator by the GCD: \(\frac{40 \div 4}{44 \div 4} = \frac{10}{11}\).
This gives the simplified fraction \(\frac{10}{11}\).

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