DSQ: Understanding mixture problems - GMAT Quantitative

Suppose we only know that 200 ml of the 40% solution is used. Then we can call the amount of 25% solution used, and the total amount made is . The solution equation becomes,

So 100 ml of the 25% solution is used.

Similarly, if we only know that 100 ml of the 25% solution is used, then we can call the amount of 40% solution used, and the total amount made is . The solution equation becomes,

So 200 ml of the 10% solution is used.

Either way, we get that

and 300 ml of solution is created.

The answer is that either statement is sufficient to answer the question.

Using statement 1, we can find that he has 10 green boxes and 10 brown. We solve mixture problems using 2 equations. If we let be the number of green boxes, and be the number of brown boxes. We can set the equations

1. and

2. 

Solving equation 1) for , we get .

Subsituting for into equation 2) results in

Simplifying we get or

Therefore, this answer would be 10 green boxes and 10 brown.

The second statement is irrelevant in finding our answer.

Let be the number of pounds of Vanilla Heaven coffee beans in the mixture.

If we assume Statement 1 alone, then the total price of the Vanilla Dream coffee beans is $10 per pound times pounds, or . Similarly, the total price of the Mountain Goodness beans is , and the overall price of the beans in the mixture is . Add these together to get the equation

.

If we assume Statement 2 alone, the price of the Vanilla Dream beans is again . However, the price of the Mountain Goodness beans is and the overall price of the beans is , so we can set up this equation:

From either equation we can solve for and get the answer to the question.

Assume both statements are true.

Let be the amount of Chocolate Explosion and be the amount of Cherry Cherry Delight. Then the price of the Chocolate and Cherry coffees will be dollars and dollars, respectively. The total price of the beans is , so we can set up the equation:

However, there is no further information we can use to set up a second equation, so there is insufficient information to answer the question.

Using Statement (1), we can set up the following equation:

The amount of acid in the original solution (40% multiplied by 15 liters) should equal the amount of acid in the final diluted solution (remember there is no acid in water!!!)

Therefore knowing the value of x, we are able to calculate y. We can now find the amount of water used to dilute the solution by subtracting x from y.

Using Statement (2), we set up the following equation:

We can also calculate the amount of water used since we can calculate x from knowing y.

Each Statement ALONE is sufficient to answer the question.

(1) The chemist would like the final solution to be 15% chlorine.

The final solution is 150 ml and contains 15% of chlorine. Let x be the amount of water, then 150-x is the amount of the original chlorine solution. The amount of chlorine in the original solution is the same in the final solution, only the concentration changes.

(2) There is no chlorine in water.

Statement (2) is not helpful as it does not help in finding the amount of chlorine in the final solution.

(1) A pound of almonds is twice as expensive as a pound of walnuts.

Let x be the price per pound of walnuts and y the price per pound of almonds.

We don't know the respective amounts of almonds and walnuts and the prices of any of the nuts aren't clearly stated.

Statement(1) Alone is not sufficient.

(2) The mixture contains 0.7 pounds of walnuts and 0.3 pounds of almonds.

Statement (2) alone is not sufficient, it indicates the quantities of almonds and walnuts in a pound of mixture. However the price per pound of each type of nuts is unknown.

Combining both statements, we can write the price of one pound of the mixture as:

We do not know x, the price per pound of walnuts, therefore both statements together are not sufficient.

(1) A mixture of half almonds and half walnuts sells for $14 per pound.

Let x be the price per pound of walnuts and y the price per pound of almonds.

Since we do not know the price per pound of walnuts or the price per pound of almonds, statement(1) is not sufficient.

(2) A pound of almonds is twice as expensive as the pound of walnuts.

Let x be the price per pound of walnuts and y the price per pound of almonds.

Statement (2) alone is not sufficient.

Combining both statements:

The price per pound of almonds is:

(1) The final solution contains 20% of acid.

Using Statement (1), we know the concentration of acid in the final solution. However, we cannot find the quantity of the final solution as we do not know what quantity of acid was diluted.

So Statement (1) Alone is not sufficient.

(2) x=20 ml

Using statement (2) we know the quantity of the original acid solution, but we do not know what quantity of water was added or the concentration of the final solution.

So Statement (2) Alone is not sufficient.

Combining both Statements,

We have 20 ml of a 100% acid solution. Note that there is 0% acid in water. After diluting the solution, we obtain y ml of a solution containing 20% of acid. The amount (in ml) of acid in the final solution equals the amount of acid of the initial solution:

Therefore 100 ml of solution is obtained by diluting the original acid solution with water.

Both Statements Together are sufficient.

Suppose we only know that 200 ml of the 40% solution is used. Then we can call the amount of 25% solution used, and the total amount made is . The solution equation becomes,

So 100 ml of the 25% solution is used.

Similarly, if we only know that 100 ml of the 25% solution is used, then we can call the amount of 40% solution used, and the total amount made is . The solution equation becomes,

So 200 ml of the 10% solution is used.

Either way, we get that

and 300 ml of solution is created.

The answer is that either statement is sufficient to answer the question.

Using statement 1, we can find that he has 10 green boxes and 10 brown. We solve mixture problems using 2 equations. If we let be the number of green boxes, and be the number of brown boxes. We can set the equations

1. and

2. 

Solving equation 1) for , we get .

Subsituting for into equation 2) results in

Simplifying we get or

Therefore, this answer would be 10 green boxes and 10 brown.

The second statement is irrelevant in finding our answer.

Let be the number of pounds of Vanilla Heaven coffee beans in the mixture.

If we assume Statement 1 alone, then the total price of the Vanilla Dream coffee beans is $10 per pound times pounds, or . Similarly, the total price of the Mountain Goodness beans is , and the overall price of the beans in the mixture is . Add these together to get the equation

.

If we assume Statement 2 alone, the price of the Vanilla Dream beans is again . However, the price of the Mountain Goodness beans is and the overall price of the beans is , so we can set up this equation:

From either equation we can solve for and get the answer to the question.

Assume both statements are true.

Let be the amount of Chocolate Explosion and be the amount of Cherry Cherry Delight. Then the price of the Chocolate and Cherry coffees will be dollars and dollars, respectively. The total price of the beans is , so we can set up the equation:

However, there is no further information we can use to set up a second equation, so there is insufficient information to answer the question.

Using Statement (1), we can set up the following equation:

The amount of acid in the original solution (40% multiplied by 15 liters) should equal the amount of acid in the final diluted solution (remember there is no acid in water!!!)

Therefore knowing the value of x, we are able to calculate y. We can now find the amount of water used to dilute the solution by subtracting x from y.

Using Statement (2), we set up the following equation:

We can also calculate the amount of water used since we can calculate x from knowing y.

Each Statement ALONE is sufficient to answer the question.

(1) The chemist would like the final solution to be 15% chlorine.

The final solution is 150 ml and contains 15% of chlorine. Let x be the amount of water, then 150-x is the amount of the original chlorine solution. The amount of chlorine in the original solution is the same in the final solution, only the concentration changes.

(2) There is no chlorine in water.

Statement (2) is not helpful as it does not help in finding the amount of chlorine in the final solution.

(1) A pound of almonds is twice as expensive as a pound of walnuts.

Let x be the price per pound of walnuts and y the price per pound of almonds.

We don't know the respective amounts of almonds and walnuts and the prices of any of the nuts aren't clearly stated.

Statement(1) Alone is not sufficient.

(2) The mixture contains 0.7 pounds of walnuts and 0.3 pounds of almonds.

Statement (2) alone is not sufficient, it indicates the quantities of almonds and walnuts in a pound of mixture. However the price per pound of each type of nuts is unknown.

Combining both statements, we can write the price of one pound of the mixture as:

We do not know x, the price per pound of walnuts, therefore both statements together are not sufficient.

(1) A mixture of half almonds and half walnuts sells for $14 per pound.

Let x be the price per pound of walnuts and y the price per pound of almonds.

Since we do not know the price per pound of walnuts or the price per pound of almonds, statement(1) is not sufficient.

(2) A pound of almonds is twice as expensive as the pound of walnuts.

Let x be the price per pound of walnuts and y the price per pound of almonds.

Statement (2) alone is not sufficient.

Combining both statements:

The price per pound of almonds is:

(1) The final solution contains 20% of acid.

Using Statement (1), we know the concentration of acid in the final solution. However, we cannot find the quantity of the final solution as we do not know what quantity of acid was diluted.

So Statement (1) Alone is not sufficient.

(2) x=20 ml

Using statement (2) we know the quantity of the original acid solution, but we do not know what quantity of water was added or the concentration of the final solution.

So Statement (2) Alone is not sufficient.

Combining both Statements,

We have 20 ml of a 100% acid solution. Note that there is 0% acid in water. After diluting the solution, we obtain y ml of a solution containing 20% of acid. The amount (in ml) of acid in the final solution equals the amount of acid of the initial solution:

Therefore 100 ml of solution is obtained by diluting the original acid solution with water.

Both Statements Together are sufficient.

Suppose we only know that 200 ml of the 40% solution is used. Then we can call the amount of 25% solution used, and the total amount made is . The solution equation becomes,

So 100 ml of the 25% solution is used.

Similarly, if we only know that 100 ml of the 25% solution is used, then we can call the amount of 40% solution used, and the total amount made is . The solution equation becomes,

So 200 ml of the 10% solution is used.

Either way, we get that

and 300 ml of solution is created.

The answer is that either statement is sufficient to answer the question.

Using statement 1, we can find that he has 10 green boxes and 10 brown. We solve mixture problems using 2 equations. If we let be the number of green boxes, and be the number of brown boxes. We can set the equations

1. and

2. 

Solving equation 1) for , we get .

Subsituting for into equation 2) results in

Simplifying we get or

Therefore, this answer would be 10 green boxes and 10 brown.

The second statement is irrelevant in finding our answer.