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Sagot :
Certainly! Let's go through each part of the question step-by-step.
### Part 1: Are the following pairs of fractions equivalent?
To determine if two fractions are equivalent, we consider the property that two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) are equivalent if \( a \cdot d = b \cdot c \).
a. \(\frac{2}{5}\) and \(\frac{34}{85}\)
To find out if these two fractions are equivalent:
Calculate: \(2 \cdot 85\) and \(5 \cdot 34\)
[tex]\[ 2 \cdot 85 = 170 \\ 5 \cdot 34 = 170 \][/tex]
Since both products are equal (\(170 = 170\)), the fractions \(\frac{2}{5}\) and \(\frac{34}{85}\) are equivalent.
So, the answer to this part is True.
b. \(\frac{7}{11}\) and \(\frac{56}{99}\)
Similarly, calculate: \(7 \cdot 99\) and \(11 \cdot 56\)
[tex]\[ 7 \cdot 99 = 693 \\ 11 \cdot 56 = 616 \][/tex]
Since the products are not equal (\(693 \neq 616\)), the fractions \(\frac{7}{11}\) and \(\frac{56}{99}\) are not equivalent.
So, the answer to this part is False.
### Part 2: Write the following fractions in their simplest form
To simplify a fraction, we divide the numerator and the denominator by their greatest common divisor (GCD).
a. \(\frac{16}{14}\)
The GCD of 16 and 14 is 2. Divide both the numerator and the denominator by 2:
[tex]\[ \frac{16 \div 2}{14 \div 2} = \frac{8}{7} \][/tex]
So, \(\frac{16}{14}\) simplified is \(\frac{8}{7}\).
b. \(\frac{15}{25}\)
The GCD of 15 and 25 is 5. Divide both the numerator and the denominator by 5:
[tex]\[ \frac{15 \div 5}{25 \div 5} = \frac{3}{5} \][/tex]
So, \(\frac{15}{25}\) simplified is \(\frac{3}{5}\).
c. \(\frac{13}{18}\)
The GCD of 13 and 18 is 1. Since the GCD is 1, the fraction is already in its simplest form.
So, \(\frac{13}{18}\) remains \(\frac{13}{18}\).
### Part 3: Convert the following improper fractions to mixed numbers
To convert an improper fraction to a mixed number, divide the numerator by the denominator to get the whole number part and use the remainder as the new numerator over the original denominator.
a. \(\frac{11}{2}\)
Divide 11 by 2:
[tex]\[ 11 \div 2 = 5 \text{ with a remainder of } 1 \][/tex]
Thus, \(\frac{11}{2}\) as a mixed number is \(5 \frac{1}{2}\).
b. \(\frac{29}{3}\)
Divide 29 by 3:
[tex]\[ 29 \div 3 = 9 \text{ with a remainder of } 2 \][/tex]
Thus, \(\frac{29}{3}\) as a mixed number is \(9 \frac{2}{3}\).
c. \(\frac{150}{7}\)
Divide 150 by 7:
[tex]\[ 150 \div 7 = 21 \text{ with a remainder of } 3 \][/tex]
Thus, \(\frac{150}{7}\) as a mixed number is \(21 \frac{3}{7}\).
### Conclusion
Based on the detailed steps above, here are the summarized answers:
1. Equivalent Fractions:
- \(\frac{2}{5}\) and \(\frac{34}{85}\) are equivalent: True
- \(\frac{7}{11}\) and \(\frac{56}{99}\) are equivalent: False
2. Simplified Fractions:
- \(\frac{16}{14} = \frac{8}{7}\)
- \(\frac{15}{25} = \frac{3}{5}\)
- \(\frac{13}{18} = \frac{13}{18}\) (Already simplified)
3. Mixed Numbers:
- \(\frac{11}{2} = 5 \frac{1}{2}\)
- \(\frac{29}{3} = 9 \frac{2}{3}\)
- [tex]\(\frac{150}{7} = 21 \frac{3}{7}\)[/tex]
### Part 1: Are the following pairs of fractions equivalent?
To determine if two fractions are equivalent, we consider the property that two fractions \(\frac{a}{b}\) and \(\frac{c}{d}\) are equivalent if \( a \cdot d = b \cdot c \).
a. \(\frac{2}{5}\) and \(\frac{34}{85}\)
To find out if these two fractions are equivalent:
Calculate: \(2 \cdot 85\) and \(5 \cdot 34\)
[tex]\[ 2 \cdot 85 = 170 \\ 5 \cdot 34 = 170 \][/tex]
Since both products are equal (\(170 = 170\)), the fractions \(\frac{2}{5}\) and \(\frac{34}{85}\) are equivalent.
So, the answer to this part is True.
b. \(\frac{7}{11}\) and \(\frac{56}{99}\)
Similarly, calculate: \(7 \cdot 99\) and \(11 \cdot 56\)
[tex]\[ 7 \cdot 99 = 693 \\ 11 \cdot 56 = 616 \][/tex]
Since the products are not equal (\(693 \neq 616\)), the fractions \(\frac{7}{11}\) and \(\frac{56}{99}\) are not equivalent.
So, the answer to this part is False.
### Part 2: Write the following fractions in their simplest form
To simplify a fraction, we divide the numerator and the denominator by their greatest common divisor (GCD).
a. \(\frac{16}{14}\)
The GCD of 16 and 14 is 2. Divide both the numerator and the denominator by 2:
[tex]\[ \frac{16 \div 2}{14 \div 2} = \frac{8}{7} \][/tex]
So, \(\frac{16}{14}\) simplified is \(\frac{8}{7}\).
b. \(\frac{15}{25}\)
The GCD of 15 and 25 is 5. Divide both the numerator and the denominator by 5:
[tex]\[ \frac{15 \div 5}{25 \div 5} = \frac{3}{5} \][/tex]
So, \(\frac{15}{25}\) simplified is \(\frac{3}{5}\).
c. \(\frac{13}{18}\)
The GCD of 13 and 18 is 1. Since the GCD is 1, the fraction is already in its simplest form.
So, \(\frac{13}{18}\) remains \(\frac{13}{18}\).
### Part 3: Convert the following improper fractions to mixed numbers
To convert an improper fraction to a mixed number, divide the numerator by the denominator to get the whole number part and use the remainder as the new numerator over the original denominator.
a. \(\frac{11}{2}\)
Divide 11 by 2:
[tex]\[ 11 \div 2 = 5 \text{ with a remainder of } 1 \][/tex]
Thus, \(\frac{11}{2}\) as a mixed number is \(5 \frac{1}{2}\).
b. \(\frac{29}{3}\)
Divide 29 by 3:
[tex]\[ 29 \div 3 = 9 \text{ with a remainder of } 2 \][/tex]
Thus, \(\frac{29}{3}\) as a mixed number is \(9 \frac{2}{3}\).
c. \(\frac{150}{7}\)
Divide 150 by 7:
[tex]\[ 150 \div 7 = 21 \text{ with a remainder of } 3 \][/tex]
Thus, \(\frac{150}{7}\) as a mixed number is \(21 \frac{3}{7}\).
### Conclusion
Based on the detailed steps above, here are the summarized answers:
1. Equivalent Fractions:
- \(\frac{2}{5}\) and \(\frac{34}{85}\) are equivalent: True
- \(\frac{7}{11}\) and \(\frac{56}{99}\) are equivalent: False
2. Simplified Fractions:
- \(\frac{16}{14} = \frac{8}{7}\)
- \(\frac{15}{25} = \frac{3}{5}\)
- \(\frac{13}{18} = \frac{13}{18}\) (Already simplified)
3. Mixed Numbers:
- \(\frac{11}{2} = 5 \frac{1}{2}\)
- \(\frac{29}{3} = 9 \frac{2}{3}\)
- [tex]\(\frac{150}{7} = 21 \frac{3}{7}\)[/tex]
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