.Repeat#4

Number Theory Level 3

2.8 × 4.3 5 ‾ × 1.2 34 ‾ = a b c \large 2.8 \times 4.3\overline{5} \times 1.2\overline{34} = a\frac bc

The expression a b c a\dfrac bc above is a mixed fraction, where a a , b b and c c are pair-wise coprime positive integers. FInd a + b a+b .


The answer is 5974.

Munem Shahriar by Munem Shahriar , 18, Bangladesh

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1 solution

Munem Shahriar
Jan 11, 2018
  • 2.8 = 14 5 2.8 = \dfrac{14}{5}

~~~~~~~~~~~~~~~~~~~~~

  • 4.3 5 ‾ = 435 − 43 90 = 392 90 = 196 45 4.3\overline{5} = \dfrac{435-43}{90} = \dfrac{392}{90} = \dfrac{196}{45}

~~~~~~~~~~~~~~~~~~~~~

  • 1.2 34 ‾ = 1234 − 12 990 = 1222 990 = 611 495 1.2\overline{34} = \dfrac{1234 - 12}{990} = \dfrac{1222}{990} = \dfrac{611}{495}

Now,

14 5 × 196 45 × 611 495 = 15 5959 111375 \dfrac{14}{5} \times \dfrac{196}{45} \times \dfrac{611}{495} = 15\dfrac{5959}{111375}

Hence a + b = 15 + 5959 = 5974 a+b = 15 + 5959 = \boxed{5974}

0 Helpful 0 Interesting 0 Brilliant 0 Confused

a and c (15 and 111375) are not coprime.

Ricardo Moritz Cavalcanti - 10 months, 3 weeks ago

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