Two-Variable (Simultaneous) Equations: Substitution

Paul's statue collection consisted entirely of statues made either of bronze, silver, or gold. He had exactly twice as many silver statues as bronze statues, and exactly twice as many gold statues as silver statues. Also, the total number of statues in Paul's collection was greater than 46 but less than 55. How many statues were in Paul's collection?

Incorrect. [[snippet]] 54 is incorrect. Consider the numbers that divide evenly into 54.

Incorrect. [[snippet]] 50 is incorrect. Consider the numbers that divide evenly into 50.

Correct! Let _B_ = the number of bronze statues, _S_ = the number of silver statues, _G_ = the number of gold statues, and _T_ = the total number of statues in the collection. _B_ + _S_ + _G_ = _T_ , for which _S_ = 2_B_ and _G_ = 2_S_. Also, _G_ = 2_S_ = 2(2_B_) = 4_B_. You can then substitute to express _T_ in terms of _B_ : _B_ + _S_ + _G_ = _T_ _B_ + 2_B_ + 4_B_ = _T_ 7_B_ = _T_. Therefore, _T_ must be divisible by seven. With this in mind, 49 is the only number greater than 46 and less than 55 that is divisible by seven, so there must be 49 statues in Paul's collection.

Default step content.

Incorrect. [[snippet]] 47 is incorrect. Consider the numbers that divide evenly into 47.

Incorrect. There cannot be 41 statues in Paul's collection because the problem specified the number of statues is greater than 46. [[snippet]]

54

50

49

47

41