What is the fraction of x's?

Red letters and blue letters are used in a game. Some of the letters are X. All the other letters are Y. The ratio of the number of red letters to blue letters is 2:5. The ratio of the number of red x's to red y's is 1:3. The ratio of blue x's to blue y's is 3:1. Work out what fractions of all the letters are x's.

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  • atsuo
    Lv 6
    4 weeks ago

    Let the number of red Xs be RX and the number of red Ys be RY . 

    RX:RY = 1:3 , so RY = 3RX and RX+RY = 4RX . 

    Let the number of blue Xs be BX and the number of blue Ys be BY . 

    (RX+RY):(BX+BY) = 2:5 , so BX+BY = (5/2)(RX+RY) = (5/2)4RX = 10RX . 

    BX:BY = 3:1 , so BX = (3/4)(BX+BY) = (3/4)10RX = (15/2)RX and 

    BY = (1/4)(BX+BY) = (1/4)10RX = (5/2)RX . 

    The answer is 

    (RX+BX)/((RX+RY)+(BX+BY)) 

    = (RX+(15/2)RX)/(4RX+10RX) 

    = (17/2)RX/(14RX) 

    = 17/28

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