The mean of 10 numbers is 12.5; the mean of the first six is 15 and the last five is 10. The sixth number is

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none of these

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Solution

The correct option is A 15
$L e t t h e m e a n o f t h e l a s t f o u r b e A_{2} . T h e n b y t h e f o r m u l a f o r c o m b i n e d m e a n, 12.5 = \frac{6 \times 15 + 4 \times A_{2}}{6 + 4}; o r 125 = 90 + 4 A_{2}; ∴ A_{2} = \frac{35}{4} L e t t h e s i x t h n u m b e r = x; t h e n t a k i n g t h e s i x t h n u m b e r a s a c o l l e c t i o n, t h e c o m b i n e d m e a n o f t h i s c o l l e c t i o n a n d t h e c o l l e c t i o n o f t h e l a s t f i v e i s 10, b y q u e s t i o n . ∴ B y d e f i n i t i o n o f c o m b i n e d m e a n 10 = \frac{1 \times x + 4 \times \frac{35}{4}}{1 + 4}; 50 = x + 35; ∴ x = 15 ∴ s i x t h n u m b e r = 15$

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