Answer :
Let's simplify each fraction one by one to its simplest form. We'll use the greatest common divisor (GCD) to help with the simplification:
a) [tex]\(\frac{30}{12}\)[/tex]
- Find the GCD of 30 and 12, which is 6.
- Divide both the numerator and the denominator by 6.
- [tex]\(\frac{30 ÷ 6}{12 ÷ 6} = \frac{5}{2}\)[/tex]
b) [tex]\(\frac{28}{42}\)[/tex]
- Find the GCD of 28 and 42, which is 14.
- Divide both the numerator and the denominator by 14.
- [tex]\(\frac{28 ÷ 14}{42 ÷ 14} = \frac{2}{3}\)[/tex]
c) [tex]\(\frac{32}{40}\)[/tex]
- Find the GCD of 32 and 40, which is 8.
- Divide both the numerator and the denominator by 8.
- [tex]\(\frac{32 ÷ 8}{40 ÷ 8} = \frac{4}{5}\)[/tex]
d) [tex]\(\frac{45}{27}\)[/tex]
- Find the GCD of 45 and 27, which is 9.
- Divide both the numerator and the denominator by 9.
- [tex]\(\frac{45 ÷ 9}{27 ÷ 9} = \frac{5}{3}\)[/tex]
e) [tex]\(\frac{36}{48}\)[/tex]
- Find the GCD of 36 and 48, which is 12.
- Divide both the numerator and the denominator by 12.
- [tex]\(\frac{36 ÷ 12}{48 ÷ 12} = \frac{3}{4}\)[/tex]
f) [tex]\(\frac{48}{54}\)[/tex]
- Find the GCD of 48 and 54, which is 6.
- Divide both the numerator and the denominator by 6.
- [tex]\(\frac{48 ÷ 6}{54 ÷ 6} = \frac{8}{9}\)[/tex]
g) [tex]\(\frac{75}{50}\)[/tex]
- Find the GCD of 75 and 50, which is 25.
- Divide both the numerator and the denominator by 25.
- [tex]\(\frac{75 ÷ 25}{50 ÷ 25} = \frac{3}{2}\)[/tex]
h) [tex]\(\frac{80}{64}\)[/tex]
- Find the GCD of 80 and 64, which is 16.
- Divide both the numerator and the denominator by 16.
- [tex]\(\frac{80 ÷ 16}{64 ÷ 16} = \frac{5}{4}\)[/tex]
i) [tex]\(\frac{81}{54}\)[/tex]
- Find the GCD of 81 and 54, which is 27.
- Divide both the numerator and the denominator by 27.
- [tex]\(\frac{81 ÷ 27}{54 ÷ 27} = \frac{3}{2}\)[/tex]
j) [tex]\(\frac{72}{90}\)[/tex]
- Find the GCD of 72 and 90, which is 18.
- Divide both the numerator and the denominator by 18.
- [tex]\(\frac{72 ÷ 18}{90 ÷ 18} = \frac{4}{5}\)[/tex]
k) [tex]\(\frac{60}{84}\)[/tex]
- Find the GCD of 60 and 84, which is 12.
- Divide both the numerator and the denominator by 12.
- [tex]\(\frac{60 ÷ 12}{84 ÷ 12} = \frac{5}{7}\)[/tex]
l) [tex]\(\frac{84}{96}\)[/tex]
- Find the GCD of 84 and 96, which is 12.
- Divide both the numerator and the denominator by 12.
- [tex]\(\frac{84 ÷ 12}{96 ÷ 12} = \frac{7}{8}\)[/tex]
So the simplified, irreducible forms of the fractions are:
a) [tex]\(\frac{5}{2}\)[/tex]
b) [tex]\(\frac{2}{3}\)[/tex]
c) [tex]\(\frac{4}{5}\)[/tex]
d) [tex]\(\frac{5}{3}\)[/tex]
e) [tex]\(\frac{3}{4}\)[/tex]
f) [tex]\(\frac{8}{9}\)[/tex]
g) [tex]\(\frac{3}{2}\)[/tex]
h) [tex]\(\frac{5}{4}\)[/tex]
i) [tex]\(\frac{3}{2}\)[/tex]
j) [tex]\(\frac{4}{5}\)[/tex]
k) [tex]\(\frac{5}{7}\)[/tex]
l) [tex]\(\frac{7}{8}\)[/tex]
a) [tex]\(\frac{30}{12}\)[/tex]
- Find the GCD of 30 and 12, which is 6.
- Divide both the numerator and the denominator by 6.
- [tex]\(\frac{30 ÷ 6}{12 ÷ 6} = \frac{5}{2}\)[/tex]
b) [tex]\(\frac{28}{42}\)[/tex]
- Find the GCD of 28 and 42, which is 14.
- Divide both the numerator and the denominator by 14.
- [tex]\(\frac{28 ÷ 14}{42 ÷ 14} = \frac{2}{3}\)[/tex]
c) [tex]\(\frac{32}{40}\)[/tex]
- Find the GCD of 32 and 40, which is 8.
- Divide both the numerator and the denominator by 8.
- [tex]\(\frac{32 ÷ 8}{40 ÷ 8} = \frac{4}{5}\)[/tex]
d) [tex]\(\frac{45}{27}\)[/tex]
- Find the GCD of 45 and 27, which is 9.
- Divide both the numerator and the denominator by 9.
- [tex]\(\frac{45 ÷ 9}{27 ÷ 9} = \frac{5}{3}\)[/tex]
e) [tex]\(\frac{36}{48}\)[/tex]
- Find the GCD of 36 and 48, which is 12.
- Divide both the numerator and the denominator by 12.
- [tex]\(\frac{36 ÷ 12}{48 ÷ 12} = \frac{3}{4}\)[/tex]
f) [tex]\(\frac{48}{54}\)[/tex]
- Find the GCD of 48 and 54, which is 6.
- Divide both the numerator and the denominator by 6.
- [tex]\(\frac{48 ÷ 6}{54 ÷ 6} = \frac{8}{9}\)[/tex]
g) [tex]\(\frac{75}{50}\)[/tex]
- Find the GCD of 75 and 50, which is 25.
- Divide both the numerator and the denominator by 25.
- [tex]\(\frac{75 ÷ 25}{50 ÷ 25} = \frac{3}{2}\)[/tex]
h) [tex]\(\frac{80}{64}\)[/tex]
- Find the GCD of 80 and 64, which is 16.
- Divide both the numerator and the denominator by 16.
- [tex]\(\frac{80 ÷ 16}{64 ÷ 16} = \frac{5}{4}\)[/tex]
i) [tex]\(\frac{81}{54}\)[/tex]
- Find the GCD of 81 and 54, which is 27.
- Divide both the numerator and the denominator by 27.
- [tex]\(\frac{81 ÷ 27}{54 ÷ 27} = \frac{3}{2}\)[/tex]
j) [tex]\(\frac{72}{90}\)[/tex]
- Find the GCD of 72 and 90, which is 18.
- Divide both the numerator and the denominator by 18.
- [tex]\(\frac{72 ÷ 18}{90 ÷ 18} = \frac{4}{5}\)[/tex]
k) [tex]\(\frac{60}{84}\)[/tex]
- Find the GCD of 60 and 84, which is 12.
- Divide both the numerator and the denominator by 12.
- [tex]\(\frac{60 ÷ 12}{84 ÷ 12} = \frac{5}{7}\)[/tex]
l) [tex]\(\frac{84}{96}\)[/tex]
- Find the GCD of 84 and 96, which is 12.
- Divide both the numerator and the denominator by 12.
- [tex]\(\frac{84 ÷ 12}{96 ÷ 12} = \frac{7}{8}\)[/tex]
So the simplified, irreducible forms of the fractions are:
a) [tex]\(\frac{5}{2}\)[/tex]
b) [tex]\(\frac{2}{3}\)[/tex]
c) [tex]\(\frac{4}{5}\)[/tex]
d) [tex]\(\frac{5}{3}\)[/tex]
e) [tex]\(\frac{3}{4}\)[/tex]
f) [tex]\(\frac{8}{9}\)[/tex]
g) [tex]\(\frac{3}{2}\)[/tex]
h) [tex]\(\frac{5}{4}\)[/tex]
i) [tex]\(\frac{3}{2}\)[/tex]
j) [tex]\(\frac{4}{5}\)[/tex]
k) [tex]\(\frac{5}{7}\)[/tex]
l) [tex]\(\frac{7}{8}\)[/tex]
