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18 March, 08:03

A tea merchant mixes some black tea costing $42 per pound with 3 lb of Earl Grey tea costing $52 per pound. How many pounds of the black tea should be used if the merchant wants a blend that costs between $45 and $47 per pound?

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  1. 18 March, 08:51
    0
    The amount in pounds of the black tea that should be used if the merchant wants a blend that costs between $45 and $47 per pound should be between 3pounds and 7 pounds. That is,

    3 < x < 7.

    Step-by-step explanation:

    Let the amount of black tea mixed be x pounds

    Total pounds of tea mixed = (x + 3) pounds

    Amount spent on black tea = $42x

    Amount spent on Earl Grey tea = 52 * 3 = $156

    Total Amount spent on the tea = $ (156 + 42x)

    Average amount spent on tea per pound = (156 + 42x) / (3 + x)

    And this average amount is supposed to be between $45 and $47

    Starting with the lower limit, $45,

    [ (156 + 42x) / (3 + x) ] > 45

    Since x is never negative,

    156 + 42x > 45 (x + 3)

    45x + 135 < 156 + 42x

    45x - 42x < 156 - 135

    3x < 21

    x < 7

    Then taking the upper limit,

    [ (156 + 42x) / (3 + x) ] < 47

    Since x is never negative,

    156 + 42x < 47 (x + 3)

    47x + 141 > 156 + 42x

    47x - 42x > 156 - 141

    5x > 15

    x > 3

    This means that x can be between 3 and 7,

    The right range is 3 < x < 7 for the average to fall between $45 and $47.
  2. 18 March, 11:59
    0
    3 ≤ x ≤ 7.

    Step-by-step explanation:

    Let X be the variable that represents in pounds, the amount of black tea the merchant will add; therefore,

    45 ≤ [42x + (52 * 3) ] / (x + 3) ≤ 47

    Solving;

    The left side of the equation:

    45 = [42x + (52 * 3) ] / (x+3)

    45x + 138 = 42x + 156

    3x = 21

    x = 7

    The right side of the equation:

    47 = [42x + (52 * 3) ] / (x+3)

    47x + 141 = 42x + 156

    5x = 15

    x = 3

    The answer is 3 ≤ x ≤ 7
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