No 5. Calculate:

Calculate:

1. K) \( \frac{1}{36} + \frac{1}{54} + \frac{1}{36} + \frac{1}{40} \)
   • Find a common denominator for 36 and 54. The least common multiple is 108.
   • \( \frac{3}{108} + \frac{2}{108} + \frac{3}{108} + \frac{1}{40} = \frac{8}{108} + \frac{1}{40} = \frac{2}{27} + \frac{1}{40} \)
   • Find a common denominator for 27 and 40. The least common multiple is 1080.
   • \( \frac{2 \cdot 40}{27 \cdot 40} + \frac{1 \cdot 27}{40 \cdot 27} = \frac{80}{1080} + \frac{27}{1080} = \frac{107}{1080} \)
2. M) \( \frac{4}{15} + \frac{5}{12} \)
   • Find a common denominator for 15 and 12. The least common multiple is 60.
   • \( \frac{4 \cdot 4}{15 \cdot 4} + \frac{5 \cdot 5}{12 \cdot 5} = \frac{16}{60} + \frac{25}{60} = \frac{41}{60} \)
3. H) \( \frac{11}{12} + \frac{3}{18} \)
   • Find a common denominator for 12 and 18. The least common multiple is 36.
   • \( \frac{11 \cdot 3}{12 \cdot 3} + \frac{3 \cdot 2}{18 \cdot 2} = \frac{33}{36} + \frac{6}{36} = \frac{39}{36} = \frac{13}{12} \)
4. O) \( \frac{23}{30} + \frac{2}{45} + \frac{3}{56} + \frac{7}{126} \)
   • First pair: \( \frac{23}{30} + \frac{2}{45} \)
     • Find a common denominator for 30 and 45. The least common multiple is 90.
     • \( \frac{23 \cdot 3}{30 \cdot 3} + \frac{2 \cdot 2}{45 \cdot 2} = \frac{69}{90} + \frac{4}{90} = \frac{73}{90} \)
   • Second pair: \( \frac{3}{56} + \frac{7}{126} \)
     • Find a common denominator for 56 and 126. The least common multiple is 504.
     • \( \frac{3 \cdot 9}{56 \cdot 9} + \frac{7 \cdot 4}{126 \cdot 4} = \frac{27}{504} + \frac{28}{504} = \frac{55}{504} \)
   • Now add the results of the pairs: \( \frac{73}{90} + \frac{55}{504} \)
     • Find a common denominator for 90 and 504. The least common multiple is 2520.
     • \( \frac{73 \cdot 28}{90 \cdot 28} + \frac{55 \cdot 5}{504 \cdot 5} = \frac{2044}{2520} + \frac{275}{2520} = \frac{2319}{2520} \)
     • Simplify the fraction by dividing by 3: \( \frac{2319 \div 3}{2520 \div 3} = \frac{773}{840} \)

Ответ: K) $\(\frac{107}{1080}\)$; M) $\(\frac{41}{60}\)$; H) $\(\frac{13}{12}\)$; O) $\(\frac{773}{840}\)$