Problem #1

Eleven students have a math test with the average score 65. The difference between the highest and the lowest scores is 20. If the highest and the lowest scores are ignored then average is 63. The highest score is …

Solusi
Nilai 11¬†siswa, misalkan¬†x1, x2, …, x11. Reratanya 65.
$latex \frac {x_{1} + x_{2} + … + x_{11}}{11} = 65 $
atau
$latex x_{1} + x_{2} + … + x_{11} = 65 \times 11 = 715 $ (pers. i)

Misalkan x1 nilai tertinggi dan x11 nilai terendah, selisihnya 20.
$latex x_{1} – x_{11} = 20 $ (pers. ii)

Rerata nilai tanpa menyertakan nilai tertinggi dan nilai terendah adalah 63.
$latex \frac {x_{2} + x_{3} + … + x_{10}}{9} = 63 $
atau
$latex x_{2} + x_{3} + … + x_{10} = 63 \times 9 = 567 $ (pers. iii)

Substitusi pers. iii ke pers. i
$latex x_{1} + x_{2} + … + x_{10} + x_{11} = 715 $
$latex x_{1} + x_{11} + (x_{2} + … + x_{10}) = 715 $
$latex x_{1} + x_{11} + 567 = 715 $
$latex x_{1} + x_{11} = 148 $ (pers. iv)

Berdasarkan pers. ii dan pers. iv
$latex x_{1} + x_{11} = 148 $
$latex x_{1} – x_{11} = 20 $

Eliminasi $latex x_{11} $, maka diperoleh:

$latex 2x_{1} = 168 $
$latex x_{1} = 84 $

Jadi, nilai tertinggi = 84.
 

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