Complete the following multiplication and division calculations involving fractions (show all the steps).
1. [tex]$\frac{4}{6} \div \frac{3}{12}$[/tex]
2. [tex]$\frac{7}{9} \times \frac{4}{28}$[/tex]
3. [tex]$4 \frac{1}{6} \times 6 \frac{2}{5}$[/tex]
4. [tex]$\frac{3}{17}+6$[/tex]
Answer (2)
Divide 6 4 by 12 3 by multiplying 6 4 by 3 12 and simplifying to get 3 8 .
Multiply 9 7 by 28 4 by multiplying numerators and denominators and simplifying to get 9 1 .
Multiply 4 6 1 by 6 5 2 by converting to improper fractions, multiplying, and simplifying to get 3 80 .
Add 17 3 to 6 by converting 6 to a fraction with denominator 17 and adding, resulting in 17 105 .
3 8 , 9 1 , 3 80 , 17 105
Explanation
Problem Overview We are asked to perform four arithmetic operations involving fractions. Let's tackle them one by one!
Dividing Fractions 4.1.1. 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 by dividing both the numerator and the denominator by their greatest common divisor, which is 6: 18 48 = 18 ÷ 6 48 ÷ 6 = 3 8 We can also express this as a mixed number: 2 3 2
Multiplying Fractions 4.1.2. 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. Notice that both 28 and 252 are divisible by 28: 252 28 = 252 ÷ 28 28 ÷ 28 = 9 1
Multiplying Mixed Numbers 4.1.3. 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 We simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 10: 30 800 = 30 ÷ 10 800 ÷ 10 = 3 80 We can also express this as a mixed number: 26 3 2
Adding a Fraction and a Whole Number 4.1.4. To add 17 3 and 6, we express 6 as a fraction with a denominator of 17: 6 = 17 6 × 17 = 17 102 Now, we add the two fractions: 17 3 + 6 = 17 3 + 17 102 = 17 3 + 102 = 17 105 We can also express this as a mixed number: 6 17 3
Final Answers Therefore, the results of the calculations are: 4.1.1. 3 8 or 2 3 2 4.1.2. 9 1 4.1.3. 3 80 or 26 3 2 4.1.4. 17 105 or 6 17 3
Examples
Fractions are used in everyday life, such as when cooking, measuring ingredients, or splitting a bill with friends. Understanding how to perform arithmetic operations with fractions is essential for accurately calculating quantities and proportions in various real-world scenarios. For example, if you need to double a recipe that calls for 3 2 cup of flour, you would multiply 3 2 by 2 to find the new amount of flour needed, which is 3 4 or 1 3 1 cups.
We completed four operations involving fractions, finding results of 3 8 , 9 1 , 3 80 , and 17 105 . Each problem was solved step-by-step, simplifying where necessary. This covers dividing, multiplying fractions and mixed numbers, and adding fractions with whole numbers.
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