To solve the problem of dividing [tex]\(1 \frac{1}{3}\)[/tex] by [tex]\(1 \frac{3}{4}\)[/tex], we should follow these step-by-step instructions for converting and calculating the quotient in its lowest terms:
### Step 1: Convert the Mixed Numbers to Improper Fractions
1. Convert [tex]\(1 \frac{1}{3}\)[/tex]:
- The whole number is [tex]\(1\)[/tex].
- The fraction part is [tex]\(\frac{1}{3}\)[/tex].
- To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator.
- [tex]\[1 \frac{1}{3} = \frac{1 \cdot 3 + 1}{3} = \frac{4}{3}\][/tex]
2. Convert [tex]\(1 \frac{3}{4}\)[/tex]:
- The whole number is [tex]\(1\)[/tex].
- The fraction part is [tex]\(\frac{3}{4}\)[/tex].
- Similarly, convert the mixed number to an improper fraction:
- [tex]\[1 \frac{3}{4} = \frac{1 \cdot 4 + 3}{4} = \frac{7}{4}\][/tex]
### Step 2: Perform the Division of Fractions
Dividing by a fraction is the same as multiplying by its reciprocal. Thus, to divide [tex]\(\frac{4}{3}\)[/tex] by [tex]\(\frac{7}{4}\)[/tex], we multiply [tex]\(\frac{4}{3}\)[/tex] by the reciprocal of [tex]\(\frac{7}{4}\)[/tex], which is [tex]\(\frac{4}{7}\)[/tex]:
[tex]\[ \frac{4}{3} \div \frac{7}{4} = \frac{4}{3} \times \frac{4}{7} \][/tex]
### Step 3: Multiply the Fractions
To multiply two fractions, multiply their numerators and their denominators:
[tex]\[ \frac{4 \times 4}{3 \times 7} = \frac{16}{21} \][/tex]
### Step 4: Simplify the Resulting Fraction
We need to ensure that the resulting fraction [tex]\(\frac{16}{21}\)[/tex] is in its simplest form. To do this, we check if the numerator and denominator have any common divisors other than 1:
- The greatest common divisor (GCD) of 16 and 21 is 1.
Since the GCD is 1, the fraction [tex]\(\frac{16}{21}\)[/tex] is already in its simplest form.
### Final Answer:
[tex]\[ 1 \frac{1}{3} \div 1 \frac{3}{4} = \frac{16}{21} \][/tex]
Thus, the quotient of [tex]\(1 \frac{1}{3}\)[/tex] divided by [tex]\(1 \frac{3}{4}\)[/tex] in its lowest terms is [tex]\(\frac{16}{21}\)[/tex].
