Mixture replacement math help.

Question

A container contains 64 liters of pure milk. A replacement process takes place three times. A person takes out 16 liters of milk from the container and replaces it with 16 liters of water. Then 16 liters of the mixture is taken out and replaced with 12 liters of wine. Then 8 liters of the mixture is taken out and replaced with 14 liters of water.

A. What is the quantity of milk left after the final replacement?

B. What is the ratio of milk:water:wine in the container after the final replacement?

Answer 1

just track the amount of each

We start with 64 liters of milk.

We remove 16 liters of milk and replace with 16 liters of water, so we have

64 - 16 = 48 liters of milk and 16 liters of water, for 64 liters total

We remove 16 liters of this mixture.

We now have

48 - 16 * 48/64 = 48 - 12 = 36 liters of milk and

16 - 16 * 16/64 = 12 liters of water

12 liters of wine are added, so we have 36 + 12 + 12 liters = 60 liters total

We remove 8 liters, so we have

36 - 8 * 36/60 = 31 1/5 liters of milk

12 - 8 * 12/60 = 10 2/5 liters of water

12 - 8 * 12/60 = 10 2/5 liters of wine

We now add 14 liters of water, so we have

31 1/5 liters of milk

10 2/5 + 14 = 24 2/5 liters of water

10 2/5 liters of wine

a) 31 1/5 liters of milk are left after replacement

milk:water:wine =

31 1/5:24 2/5:10 2/5 =

156/5:122/5:52/5 =

156:122:52 =

78:61:26

Answer 2

Given:

The container starts with 64 liters of pure milk.

1st replacement: take out 16 liters, put back 16 liters of water

2nd replacement: take out 16 liters, put back 12 liters of wine

3rd replacement: take out 8 liters, put back 14 liters of water

The idea behind a mixture replacement problem is that when you scoop out part of a mixture, every liquid in the container loses the same fraction of itself, so we can just multiply each amount by the fraction that stays behind instead of tracking each liquid on its own.

One thing to watch here: the total does not stay at 64, because the 2nd and 3rd replacements put back a different amount than was taken out. So lets recheck the total before every scoop.

Firstly, we can do the 1st replacement:

Milk = 64 - 16 = 48 The container is pure milk, so all 16 liters taken out is milk

Water = 0 + 16 = 16 Put back 16 liters of water

Total = 48 + 16 = 64

_______________________________

Milk 48, Water 16, Wine 0, Total 64

Now the 2nd replacement. This time we are scooping out a mixture, so lets find the fraction that stays behind:

Fraction taken out = 16/64 = 1/4 16 liters out of a total of 64

Fraction that stays = 1 - 1/4 = 3/4

Milk = 48 * 3/4 = 36 Multiply by the fraction that stays

Water = 16 * 3/4 = 12 Multiply by the fraction that stays

Wine = 0 + 12 = 12 Put back 12 liters of wine

Total = 36 + 12 + 12 = 60

_______________________________

Milk 36, Water 12, Wine 12, Total 60

Now the 3rd replacement. The total is 60 now and not 64, so the fraction changes:

Fraction taken out = 8/60 = 2/15 8 liters out of a total of 60

Fraction that stays = 1 - 2/15 = 13/15

Milk = 36 * 13/15 = 31.2 Multiply by the fraction that stays

Water = 12 * 13/15 = 10.4 Multiply by the fraction that stays

Wine = 12 * 13/15 = 10.4 Multiply by the fraction that stays

Water = 10.4 + 14 = 24.4 Put back 14 liters of water

Total = 31.2 + 24.4 + 10.4 = 66

_______________________________

Milk 31.2, Water 24.4, Wine 10.4, Total 66

A. The milk is the 31.2 from the last line.

B. Now lets build the ratio:

Milk : Water : Wine = 31.2 : 24.4 : 10.4

= 156 : 122 : 52 Multiply every term by 5 to clear the decimals

= 78 : 61 : 26 Divide every term by 2

A. 31.2 liters of milk

B. 78 : 61 : 26