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Answer :
Let's simplify each of these fractions to their irreducible forms by finding their greatest common divisor (GCD) and then dividing both the numerator and the denominator by it.
a) For the fraction [tex]\(\frac{24}{36}\)[/tex]:
1. Find the GCD of 24 and 36:
- The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
- The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.
- The common divisors are 1, 2, 3, 4, 6, 12, and the greatest is 12.
2. Divide both the numerator and the denominator by their GCD:
- [tex]\(\frac{24}{36} = \frac{24 \div 12}{36 \div 12} = \frac{2}{3}\)[/tex]
b) For the fraction [tex]\(\frac{60}{25}\)[/tex]:
1. Find the GCD of 60 and 25:
- The divisors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
- The divisors of 25 are 1, 5, 25.
- The common divisors are 1 and 5, and the greatest is 5.
2. Divide both the numerator and the denominator by their GCD:
- [tex]\(\frac{60}{25} = \frac{60 \div 5}{25 \div 5} = \frac{12}{5}\)[/tex]
c) For the fraction [tex]\(\frac{540}{320}\)[/tex]:
1. Find the GCD of 540 and 320:
- The common divisor calculations can be done using a method such as listing out factors, but we'll directly note the GCD here, which is 20.
2. Divide both the numerator and the denominator by their GCD:
- [tex]\(\frac{540}{320} = \frac{540 \div 20}{320 \div 20} = \frac{27}{16}\)[/tex]
d) For the fraction [tex]\(\frac{120}{90}\)[/tex]:
1. Find the GCD of 120 and 90:
- The divisors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.
- The divisors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.
- The common divisors are 1, 2, 3, 5, 6, 10, 15, 30, and the greatest is 30.
2. Divide both the numerator and the denominator by their GCD:
- [tex]\(\frac{120}{90} = \frac{120 \div 30}{90 \div 30} = \frac{4}{3}\)[/tex]
So, the irreducible forms of the fractions are:
- a) [tex]\(\frac{2}{3}\)[/tex]
- b) [tex]\(\frac{12}{5}\)[/tex]
- c) [tex]\(\frac{27}{16}\)[/tex]
- d) [tex]\(\frac{4}{3}\)[/tex]
a) For the fraction [tex]\(\frac{24}{36}\)[/tex]:
1. Find the GCD of 24 and 36:
- The divisors of 24 are 1, 2, 3, 4, 6, 8, 12, 24.
- The divisors of 36 are 1, 2, 3, 4, 6, 9, 12, 18, 36.
- The common divisors are 1, 2, 3, 4, 6, 12, and the greatest is 12.
2. Divide both the numerator and the denominator by their GCD:
- [tex]\(\frac{24}{36} = \frac{24 \div 12}{36 \div 12} = \frac{2}{3}\)[/tex]
b) For the fraction [tex]\(\frac{60}{25}\)[/tex]:
1. Find the GCD of 60 and 25:
- The divisors of 60 are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
- The divisors of 25 are 1, 5, 25.
- The common divisors are 1 and 5, and the greatest is 5.
2. Divide both the numerator and the denominator by their GCD:
- [tex]\(\frac{60}{25} = \frac{60 \div 5}{25 \div 5} = \frac{12}{5}\)[/tex]
c) For the fraction [tex]\(\frac{540}{320}\)[/tex]:
1. Find the GCD of 540 and 320:
- The common divisor calculations can be done using a method such as listing out factors, but we'll directly note the GCD here, which is 20.
2. Divide both the numerator and the denominator by their GCD:
- [tex]\(\frac{540}{320} = \frac{540 \div 20}{320 \div 20} = \frac{27}{16}\)[/tex]
d) For the fraction [tex]\(\frac{120}{90}\)[/tex]:
1. Find the GCD of 120 and 90:
- The divisors of 120 are 1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120.
- The divisors of 90 are 1, 2, 3, 5, 6, 9, 10, 15, 18, 30, 45, 90.
- The common divisors are 1, 2, 3, 5, 6, 10, 15, 30, and the greatest is 30.
2. Divide both the numerator and the denominator by their GCD:
- [tex]\(\frac{120}{90} = \frac{120 \div 30}{90 \div 30} = \frac{4}{3}\)[/tex]
So, the irreducible forms of the fractions are:
- a) [tex]\(\frac{2}{3}\)[/tex]
- b) [tex]\(\frac{12}{5}\)[/tex]
- c) [tex]\(\frac{27}{16}\)[/tex]
- d) [tex]\(\frac{4}{3}\)[/tex]
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