Fixing the Lemonaid
12) John made some sugarless lemonade (more like lemon-flavored
water) that he liked to sip on. He added 3 ounces of the lemon
juice to 32 ounces of water, but quickly realized he added 1 ounce
too many of the lemon juice. Fortunately, John knew the right
mathematical formulas to help him remedy his problem. How much
water did he have to add to this 35-ounce drink to make the 3
ounces of lemon juice the equivalent (or same percent) as 2 ounces
added to 32 ounces would have been?
First, we'll figure out what percent 3 ounces is to 35 ounces:
(3 divided by 35) times 100 = 8.57%
Next, we'll find out the ratio of 2 ounces to 34 ounces.
(2 divided by 34) times 100 = 5.88%
Now we have some figures to work with. Since 5.88% is the
equivalent of 2 ounces of lemon juice in a 34-ounce mixture, we can
now find out how much water must be added to 3 ounces of lemon juice
(8.57%) in a 35-ounce mixture in order to reduce the lemon juice to 5.88%.
8.57(35) + o(x) = 5.88(35 + x)
The "x" variable represents the water that we're going to dilute
the mixture with.
8.57(35) + o(x) = 5.88(35 + x)
299.95 = 205.8 + 5.88x
94.15 = 5.88x
x = 16 (ounces of water -- actully 16.01, but 1/100 of an ounce
won't make much difference for a lemon-flavored drink.)
Let's check this out:
WHAT JOHN FIRST MADE:
3 ounces of lemon juice 8.57%
32 ounces of water 91.43%
--------------------
35 total ounces
WHAT HE SHOULD HAVE MADE:
2 ounces of lemon juice 5.88%
32 ounces of water 94.12%
---------------------
34 ounces (total)
HOW HE REMEDIED THE PROBLEM WITH A NEW MIXTURE (diluted with
water):
3.00 ounces of lemon juice 5.88%
48.00 ounces of water 94.12%
-------------------------
51.00 (total ounces)
Here's another way to double check our work:
2 3
---- = -----
32 48
Cross multiply: 2 times 48 = 96; And 3 times 32 = 96. So it
checks out okay. It shows that 2 ounces of lemon juice to 32 ounces
of water is the equivalent of 3 ounces of lemon juice to a 48
ounces of water. Also, 2 divided by 32 = 0.0625; And 3 divided
by 48 = 0.0625.
Or...
2 3
----- = -----
34 51
2 X 51 = 102
3 X 34 = 102
2/34 = 5.88235294118E-2
3/51 = 5.88235294118E-2