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]
Answer (2)
To divide 6 4 by 12 3 , multiply by the reciprocal: 6 4 × 3 12 . Simplify to 3 2 × 3 12 = 9 24 , then simplify the result to 3 8 .
To multiply 9 7 by 28 4 , multiply the fractions: 9 7 × 28 4 . Simplify to 9 7 × 7 1 = 63 7 , then simplify the result to 9 1 .
The result of 6 4 ÷ 12 3 is 3 8 .
The result of 9 7 × 28 4 is 9 1 .
9 1
Explanation
Understanding the Problem We are asked to perform division and multiplication of fractions. We will follow the standard procedures for these operations, simplifying fractions where possible.
Division of Fractions To divide 6 4 by 12 3 , we multiply 6 4 by the reciprocal of 12 3 , which is 3 12 . So, the expression becomes 6 4 × 3 12 .
Simplifying Fractions We can simplify 6 4 to 3 2 . Thus, the expression is now 3 2 × 3 12 .
Multiplying Numerators and Denominators Multiplying the fractions, we get 3 × 3 2 × 12 = 9 24 .
Simplifying the Result Now, we simplify 9 24 by dividing both the numerator and the denominator by their greatest common divisor, which is 3. So, 9 ÷ 3 24 ÷ 3 = 3 8 .
Multiplication of Fractions For the multiplication 9 7 × 28 4 , we multiply the numerators and the denominators: 9 × 28 7 × 4 .
Simplifying Fractions We can simplify 28 4 to 7 1 . So the expression becomes 9 7 × 7 1 .
Multiplying Numerators and Denominators Multiplying the simplified fractions, we have 9 × 7 7 × 1 = 63 7 .
Simplifying the Result Finally, we simplify 63 7 by dividing both the numerator and the denominator by their greatest common divisor, which is 7. Thus, 63 ÷ 7 7 ÷ 7 = 9 1 .
Final Answer Therefore, 6 4 ÷ 12 3 = 3 8 and 9 7 × 28 4 = 9 1 .
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. Similarly, if you have 4 3 of a pizza and you want to divide it among 3 people, you need to divide 4 3 by 3. Understanding how to multiply and divide fractions is essential for accurately adjusting quantities and sharing resources.
The results of the calculations are 6 4 ÷ 12 3 = 3 8 and 9 7 × 28 4 = 9 1 .
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