SOLUTION: A bakery baked 1200 cupcakes and packed them into boxes. There were 5/8 as many big boxes as small boxes. Each big box contained 7 more cupcakes than each small box. There were 240

Algebra ->  Percentage-and-ratio-word-problems -> SOLUTION: A bakery baked 1200 cupcakes and packed them into boxes. There were 5/8 as many big boxes as small boxes. Each big box contained 7 more cupcakes than each small box. There were 240      Log On


   

Question 1203188: A bakery baked 1200 cupcakes and packed them into boxes. There were 5/8 as many big boxes as small boxes. Each big box contained 7 more cupcakes than each small box. There were 240 more cupcakes packed in big boxes than in the small boxes. How many cupcakes were there in each small box?
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39616) About Me  (Show Source):
You can put this solution on YOUR website!
This may NOT be the most effective or neatest way to handle the problem:
                 Boxes            CupcakesInBox   CountCupcakes

BIG BOX         (5/8)b               c+7         (5/8)b(c+7)

SMALL BOX            b                 c           bc

Total                                             1200

system%28%285%2F8%29b%28c%2B7%29-bc=240%2Cand%2C%285%2F8%29b%28c%2B7%29%2Bbc=1200%29

Simplifiable and and is system of two unknowns in two equations.
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NOTICE THAT THIS SYSTEM APPEARS TO BE IN A FORMAT OF
system%28X-Y=240%2CX%2BY=1200%29
which might be helpful.

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Solving this

2X=1440
X=720

2Y=960
Y=480

and then back-substituting

system%28bc=480%2Cand%2C%285%2F8%29b%28c%2B7%29=720%29

system%28bc=480%2Cand%2Cb%28c%2B7%29=1152%29

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bc%2B7b=1152
480%2B7b=1152
7b=1152-480
7b=672
highlight%28b=96%29----which only means that 96 big boxes were used.

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Would indicate 5 cupcakes in each small box.

Answer by ikleyn(52775) About Me  (Show Source):
You can put this solution on YOUR website!
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A bakery baked 1200 cupcakes and packed them into boxes.
There were 5/8 as many big boxes as small boxes.
Each big box contained 7 more cupcakes than each small box.
There were 240 more cupcakes packed in big boxes than in the small boxes.
How many cupcakes were there in each small box?
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The problem is to solve in two steps. 

        First step


Total x cupcakes packed in big   boxes
total y cupcakes packed in small boxes

From the problem

    x + y = 1200    (1)
    x - y =  240    (2)

Add equations (1) and (2).  You will get

    2x = 1440  -->  x = 1440/2 = 720              (the number of cupcakes packed in big boxes)

Then from eq(1)  y = 1200 - x = 1200 - 720 = 480  (the number of cupcakes packed in small boxes).
     

        Second step


Let z be the number of cupcakes in each small box (= the value under the problem's question)
Then the number of cupcases in each big box is (z+7).


The number of small boxes is 480%2Fz        (3)

The number of big   boxes is 720%2F%28z%2B7%29   (4)


According to 5the problem, the ratio of the number of big boxes (4)
to the number of small boxes (3) is 5/8

   %28%28720%2F%28z%2B7%29%29%29%2F%28%28480%2Fz%29%29 = 5%2F8.


Simplify and find z

    %283%2Az%29%2F%282%2A%28z%2B7%29%29 = 5%2F8

    3z*8 = 5*2*(z+7)

    24z  = 10z + 70

    24z - 10z = 70

        14z   = 70

          z   = 70/14 = 5.


ANSWER.  The number of cupcakes in each small box is 5 (five).

Solved.