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]
4.1.4. [tex]$\frac{3}{17}+6$[/tex]
Answer (2)
Divide fractions by multiplying by the reciprocal: 6 4 ÷ 12 3 = 6 4 × 3 12 = 3 8 .
Multiply fractions by multiplying numerators and denominators: 9 7 × 28 4 = 252 28 = 9 1 .
Convert mixed numbers to improper fractions and multiply: 4 6 1 × 6 5 2 = 6 25 × 5 32 = 3 80 .
Add a fraction and a whole number by converting the whole number to a fraction with the same denominator: 17 3 + 6 = 17 3 + 17 102 = 17 105 .
3 8 , 9 1 , 3 80 , 17 105
Explanation
Introduction We have four calculations to perform involving fractions. Let's tackle them one by one!
Dividing Fractions 4.1.1: To divide fractions, we multiply by the reciprocal of the second fraction. So, 6 4 ÷ 12 3 = 6 4 × 3 12 .
Now, let's multiply the numerators and the denominators: 6 × 3 4 × 12 = 18 48 .
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6: 18 ÷ 6 48 ÷ 6 = 3 8 .
So, 6 4 ÷ 12 3 = 3 8 .
Multiplying Fractions 4.1.2: To multiply fractions, we multiply the numerators and the denominators: 9 7 × 28 4 = 9 × 28 7 × 4 .
This gives us 252 28 .
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 28: 252 ÷ 28 28 ÷ 28 = 9 1 .
So, 9 7 × 28 4 = 9 1 .
Multiplying Mixed Numbers 4.1.3: First, we need to convert the mixed numbers to improper fractions.
4 6 1 = 6 4 × 6 + 1 = 6 24 + 1 = 6 25 .
6 5 2 = 5 6 × 5 + 2 = 5 30 + 2 = 5 32 .
Now, we multiply the improper fractions: 6 25 × 5 32 = 6 × 5 25 × 32 = 30 800 .
We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 10: 30 ÷ 10 800 ÷ 10 = 3 80 .
So, 4 6 1 × 6 5 2 = 3 80 .
Adding a Fraction and a Whole Number 4.1.4: To add a fraction and a whole number, we need to convert the whole number to a fraction with the same denominator as the other fraction.
6 = 17 6 × 17 = 17 102 .
Now, we add the two fractions: 17 3 + 17 102 = 17 3 + 102 = 17 105 .
So, 17 3 + 6 = 17 105 .
Final Answers Here are the results of the calculations:
4.1.1: 6 4 ÷ 12 3 = 3 8
4.1.2: 9 7 × 28 4 = 9 1
4.1.3: 4 6 1 × 6 5 2 = 3 80
4.1.4: 17 3 + 6 = 17 105
Examples
Fractions are used in everyday life, such as when cooking, measuring ingredients, or splitting a bill with friends. Understanding how to perform operations with fractions is essential for many practical tasks. For example, if you are baking a cake and need to double the recipe, you will need to multiply the fractions in the recipe by 2. Or, if you are splitting a pizza with friends, you will need to divide the pizza into equal slices, which involves fractions.
We calculated the results for four fraction operations, obtaining: 3 8 for the division, 9 1 for the multiplication of fractions, 3 80 for the multiplication of mixed numbers, and 17 105 for the addition of a fraction and a whole number.
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