HARD MATH PROBLEM PLEASE HELP?

Question

A manufacturer of soft drinks advertises their orange soda as "naturally flavored," although it contains only 5% orange juice. A new federal regulation mandates that to be called "naturally flavored" a drink must contain at least 10% fruit juice. How much pure orange juice must this manufacturer add to 900 gallons of orange soda to conform to this new regulation? The answer is 50 gallons but wouldn't it be 45 gallons. I just don't know how to get the answer.

Answer 1

This is definitely a tricky problem, but I'll show you how to solve it easily step by step:

Let x = the initial 900 gallons = 95% of which is unnatural of total juice = 0.95T (T is defined below).

Let y = the number of additional gallons you must add to x to conform to 10% natural juice.

Let T = total number of gallons at the end = 90%unnatural juice + 10% natural juice = 100% = x + y

Setup your equation and use substitutions from ablove to solve for y:

Begin with T = x + y

T = 900 + y = 0.95T + y = 0.95(x + y) + y

900 + y = 0.95x + 0.95y + y

900 = 0.95x + 0.95y

900 = 947.37 + 0.95y

-47.37 = 0.95y

y = -49.86 gallons

Negative gallons does not make sense, so you must take the absolute value:

|y| = |-49.86| gallons = 49.86 gallons = 50 gallons

So your answer is 50 gallons.

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Check your answer:

Remember, the initial x = 900 gallons which is 95%T,  where

T = x + y = 900 + 50 = 950.

Therefore x/T = 900/950 = 95%.

I hope this has helped.

Answer 2

5%(900) = 45 gallons

add additional  5% = 10% = 45 + 45 = 90 gallons to be 10 % fruit juice

so you add additional 5% to be 10% = 45 gallons added more to the 5 %

therefore 900 gallons = 10 % fruit juice + 90 % other

Answer 3

you already have 45 gal of orange juice plus 855 gals of liquid

(900+X)*.9=855

       900+X= 855/.9                                                

       900+X=950

                X=950-900

                X=50 gallons of Orange juice you must add

Answer 4

Let x = gallons of pure orange juice to be added to 900 gallons of orange soda

      [(900(0.05) + x)/(900 + x)]100 = 10

       (45 + x)/(900 + x) = 10/100

        45 + x = 0.1(900 + x)

        45 + x = 90 - 0.1x

               x + 0.1x = 90 - 45

               0.90x = 45

                     x = 45/0.90    

                     x = 50 gallons    ANSWER

Hope this helps.

teddy boy

Answer 5

You took 5% of 900 gallons and multiplied it by 2 to get 10%. However as you add more orange juice, the total volume goes up too. 90 gallons of the total volume of 945 gallons now does not make 10%. There's where your error lies. Hope u get it :)

However, if u add 50 gallons, your orange juice is now 95 gallons while the total volume is 950 gallons, making 10%.

Workings:

Let vol of orange juice added by X.

Your liquid remains constant at 855 gallons.

(900 + X) x 0.9 = 855

810 + 0.9X = 855

0.9X = 45

X = 50

Answer 6

.05(900) + 1(x) = (900 + x)(.1)

45 + x = 90 + .1x

.9x = 45

x = 50

900(.5) + 50 = 950 (.1)

45 + 50 = 95

Answer 7

I'm confused as well I thought it would be 45 as well!