Complete the following multiplication and division calculations involving fractions (show all the steps).
4.1.1. [tex]$\frac{4}{6} \div \frac{3}{12}$[/tex]
4.1.2. [tex]$\frac{7}{9} \times \frac{4}{28}$[/tex]
4.1.3. [tex]$4 \frac{1}{6} \times 6 \frac{2}{5}$[/tex]
Answer (2)
To divide fractions, multiply by the reciprocal, resulting in 3 8 . For multiplication, directly multiply the numerators and denominators, yielding 9 1 . When multiplying mixed numbers, convert to improper fractions first, resulting in 3 80 .
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To divide 6 4 by 12 3 , multiply by the reciprocal: 6 4 × 3 12 = 18 48 = 3 8 .
To multiply 9 7 by 28 4 , multiply numerators and denominators: 9 × 28 7 × 4 = 252 28 = 9 1 .
To multiply 4 6 1 by 6 5 2 , convert to improper fractions and multiply: 6 25 × 5 32 = 30 800 = 3 80 .
The final answers are: 3 8 , 9 1 , and 3 80 .
Explanation
Problem Overview We need to perform three calculations involving fractions: division, multiplication, and multiplication of mixed numbers. Let's tackle them one by one!
Dividing Fractions To divide 6 4 by 12 3 , we multiply 6 4 by the reciprocal of 12 3 , which is 3 12 . So, we have: 6 4 ÷ 12 3 = 6 4 × 3 12 = 6 × 3 4 × 12 = 18 48 Now, we simplify the fraction 18 48 by dividing both the numerator and the denominator by their greatest common divisor, which is 6: 18 ÷ 6 48 ÷ 6 = 3 8 We can also write this as a mixed number: 2 3 2
Multiplying Fractions To multiply 9 7 by 28 4 , we multiply the numerators and the denominators: 9 7 × 28 4 = 9 × 28 7 × 4 = 252 28 Now, we simplify the fraction 252 28 by dividing both the numerator and the denominator by their greatest common divisor, which is 28: 252 ÷ 28 28 ÷ 28 = 9 1
Multiplying Mixed Numbers To multiply 4 6 1 by 6 5 2 , we first convert the mixed numbers to improper fractions: 4 6 1 = 6 4 × 6 + 1 = 6 25 6 5 2 = 5 6 × 5 + 2 = 5 32 Now, we multiply the improper fractions: 6 25 × 5 32 = 6 × 5 25 × 32 = 30 800 Now, we simplify the fraction 30 800 by dividing both the numerator and the denominator by their greatest common divisor, which is 10: 30 ÷ 10 800 ÷ 10 = 3 80 We can also write this as a mixed number: 26 3 2
Final Answers Therefore, the results of the calculations are: 4.1.1. 6 4 ÷ 12 3 = 3 8 = 2 3 2 4.1.2. 9 7 × 28 4 = 9 1 4.1.3. 4 6 1 × 6 5 2 = 3 80 = 26 3 2
Examples
Fractions are used in everyday life, such as when cooking, baking, or measuring ingredients. For example, if a recipe calls for 2 1 cup of flour and you want to double the recipe, you need to multiply 2 1 by 2, which equals 1 cup. Similarly, if you want to divide a pizza into 8 equal slices, each slice represents 8 1 of the pizza. Understanding fractions and how to perform operations with them is essential for many practical tasks.
