2014 AMC 12B Problem 6

Question

Ed and Ann both have lemonade with their lunch. Ed orders the regular size. Ann gets the large lemonade, which is

50\%

more than the regular. After both consume

\tfrac34

of their drinks, Ann gives Ed a third of what she has left, and

2

additional ounces. When they finish their lemonades they realize that they both drank the same amount. How many ounces of lemonade did they drink together?

30

32

36

40

50

Answer: D

Concepts:fraction linear equation

Difficulty rating: 1460

Small hint:

Let a regular lemonade be

a

ounces, so the large is

\tfrac32 a

Big hint:

After drinking

\tfrac34,

Ann has

\tfrac38 a

left and hands Ed

\tfrac13

of that plus

2

Solution:

Let a regular lemonade hold

a

ounces, so Ann’s large holds

\tfrac32 a.

After each drinks

\tfrac34,

Ann has

\tfrac14\cdot\tfrac32 a = \tfrac38 a

left, and she gives Ed

\tfrac13\cdot\tfrac38 a + 2 = \tfrac18 a + 2

ounces.

Ed drinks his full

a

ounces plus that gift, and Ann drinks her

\tfrac32 a

minus the gift. Setting these equal,

a + \left(\tfrac18 a + 2\right) = \tfrac32 a - \left(\tfrac18 a + 2\right),

which gives

4 = \tfrac14 a,

so

a = 16.

Then Ed drank

16

ounces and Ann drank

24

ounces, for a total of

40