High School

1. Simplify the following fractions:

(a) \(\frac{36}{48}\)

(b) \(\frac{81}{108}\)

(c) \(\frac{45}{60}\)

Answer :

To simplify a fraction, we need to find the greatest common divisor (GCD) of the numerator and the denominator and then divide both by this number. Let's simplify each fraction step-by-step:

  1. [tex]\frac{36}{48}[/tex]:

    • Find the GCD of 36 and 48.
    • The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, 36.
    • The factors of 48 are: 1, 2, 3, 4, 6, 8, 12, 16, 24, 48.
    • The common factors are: 1, 2, 3, 4, 6, 12. The greatest is 12.
    • Divide both the numerator and the denominator by 12:
    • [tex]\frac{36 \div 12}{48 \div 12} = \frac{3}{4}[/tex].

  2. [tex]\frac{81}{108}[/tex]:

    • Find the GCD of 81 and 108.
    • The factors of 81 are: 1, 3, 9, 27, 81.
    • The factors of 108 are: 1, 2, 3, 4, 6, 9, 12, 18, 27, 36, 54, 108.
    • The common factors are: 1, 3, 9, 27. The greatest is 27.
    • Divide both the numerator and the denominator by 27:
    • [tex]\frac{81 \div 27}{108 \div 27} = \frac{3}{4}[/tex].

  3. [tex]\frac{45}{60}[/tex]:

    • Find the GCD of 45 and 60.
    • The factors of 45 are: 1, 3, 5, 9, 15, 45.
    • The factors of 60 are: 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60.
    • The common factors are: 1, 3, 5, 15. The greatest is 15.
    • Divide both the numerator and the denominator by 15:
    • [tex]\frac{45 \div 15}{60 \div 15} = \frac{3}{4}[/tex].

In summary, the simplified forms of the fractions are as follows:

  • [tex]\frac{36}{48} = \frac{3}{4}[/tex]
  • [tex]\frac{81}{108} = \frac{3}{4}[/tex]
  • [tex]\frac{45}{60} = \frac{3}{4}[/tex]

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