Problem Solving

Darla decides to mix lemonade with limeade to make a new drink called citrusade. The lemonade is 50% water, 30% lemon juice, and 20% sugar. The limeade is 40% water, 28% lime juice, and 32% sugar. If the citrusade is more than 45% water and more than 24% sugar, which of the following could be the ratio of lemonade to limeade?

(A) 3 : 1
(B) 7 : 3
(C) 3 : 2
(D) 4 : 5
(E) 3 : 4

OA [spoiler](C)[/spoiler]

sachin_yadav wrote:Darla decides to mix lemonade with limeade to make a new drink called citrusade. The lemonade is 50% water, 30% lemon juice, and 20% sugar. The limeade is 40% water, 28% lime juice, and 32% sugar. If the citrusade is more than 45% water and more than 24% sugar, which of the following could be the ratio of lemonade to limeade?

(A) 3 : 1
(B) 7 : 3
(C) 3 : 2
(D) 4 : 5
(E) 3 : 4

Since the percentage of water in citrusade (MORE than 45%) is closer to the percentage of water in lemonade (50%) than to the percentage of water in limeade (40%), cirtusade must contain MORE LEMONADE THAN LIMEADE.
Eliminate D and E.

We can PLUG IN THE ANSWERS, which represent the ratio of lemonade to limeade.
When the correct answer choice is plugged in, more than 24% of the citrusade will be sugar.

B) 7:3
Let citrusade consist of 70 liters of lemonade and 30 liters of limeade, for a total of 100 liters.
Since lemonade is 20% sugar and limeade is 32% sugar, the amount of sugar in the 100 liters of citrusade = (20% of 70) + (32% of 30) = 14 + 9.6 = 23.6%.
Too small.
Eliminate B.

To increase the amount of sugar, the proportion of limeade -- which has a HIGHER SUGAR PERCENTAGE -- must INCREASE.

The correct answer is C.

C) 3:2 = 6:4
Let citrusade consist of 60 liters of lemonade and 40 liters of limeade, for a total of 100 liters.
Since lemonade is 20% sugar and limeade is 32% sugar, the amount of sugar in the 100 liters of citrusade = (20% of 60) + (32% of 40) = 12 + 12.8 = 24.8%.
Success!

sachin_yadav wrote:Darla decides to mix lemonade with limeade to make a new drink called citrusade. The lemonade is 50% water, 30% lemon juice, and 20% sugar. The limeade is 40% water, 28% lime juice, and 32% sugar. If the citrusade is more than 45% water and more than 24% sugar, which of the following could be the ratio of lemonade to limeade?

(A) 3 : 1
(B) 7 : 3
(C) 3 : 2
(D) 4 : 5
(E) 3 : 4

You can use allegation.

To use allegation you relate the weighted averages of percentages of the ingredients to the ratio of the parts of the mixture.

For example if you use a 1:1 ratio of the two parts of the mixture, then the weighted average of the percentages of the ingredients will be the mean of the percentages found in the two parts. in other words, in a 1:1 mixture the weighted average of the percentages will be exactly halfway between the percentages found in each of the ingredients.

The mixture is to contain more than 45% water.

45% is the mean of the percentages of water found in the two parts, lemonade, which contains 50% water, and limeade, which contains 40% water. So a mixture containing a 1:1 ratio of lemonade to limeade would contain 45% water.

Since the mixture is to contain more than 45% water, it must contain more lemonade than limeade.

Eliminate answer choices D and E.

The mixture is also to contain more than 24% sugar. 24% is 1/3 of the way from 20%, the proportion of sugar in lemonade, to 32%, the proportion of sugar in limeade. So to get a sugar percentage greater than 24%, we need the mixture to be more than 1/3 limeade.

A indicates that limeade is 1/4 of the mixture. 1/4 < 1/3. Eliminate A.

B indicates that limeade is 3/10 of the mixture. 3/10 < 1/3. Eliminate B.

C indicates that limeade is 2/5 of the mixture. 2/5 > 1/3.

The correct answer is C.

great explanation

thanks to both of you.