To solve the problem [tex]\( \frac{10}{3} \div \frac{5}{12} \)[/tex], we need to perform the division of these two fractions and simplify the result to its lowest terms. Here are the steps to do that:
### Step 1: Write the Division as Multiplication by the Reciprocal
Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of [tex]\( \frac{5}{12} \)[/tex] is [tex]\( \frac{12}{5} \)[/tex]. So, we change the division into multiplication:
[tex]\[ \frac{10}{3} \div \frac{5}{12} = \frac{10}{3} \times \frac{12}{5} \][/tex]
### Step 2: Multiply the Fractions
To multiply fractions, we multiply the numerators together and the denominators together:
[tex]\[ \left( \frac{10}{3} \right) \times \left( \frac{12}{5} \right) = \frac{10 \times 12}{3 \times 5} \][/tex]
### Step 3: Calculate the Numerator and the Denominator
Perform the multiplication for the numerator and the denominator:
[tex]\[ 10 \times 12 = 120 \][/tex]
[tex]\[ 3 \times 5 = 15 \][/tex]
So,
[tex]\[ \frac{10}{3} \times \frac{12}{5} = \frac{120}{15} \][/tex]
### Step 4: Simplify the Resulting Fraction
To simplify [tex]\( \frac{120}{15} \)[/tex] to its lowest terms, we need to find the greatest common divisor (GCD) of 120 and 15, and then divide both the numerator and the denominator by this GCD.
- The GCD of 120 and 15 is 15.
[tex]\[ \frac{120 \div 15}{15 \div 15} = \frac{8}{1} \][/tex]
### Step 5: Write the Final Answer
Thus, the quotient of [tex]\( \frac{10}{3} \div \frac{5}{12} \)[/tex] is [tex]\( \frac{8}{1} \)[/tex], which simplifies to 8.
So, the quotient is:
[tex]\[ 8 \][/tex]
Also, the intermediate fraction before simplification was:
[tex]\[ \frac{120}{15} \][/tex]
