Question

One ounce of Solution \(X\) contains only ingredients \(a\) and \(b\) in a ratio of \(2: 3 .\) One ounce of Solution \(Y\) contains only ingredients \(a\) and \(b\) in a ratio of \(1: 2 .\) If Solution \(Z\) is created by mixing solutions \(X\) and \(Y\) in a ratio of \(3: 11,\) then 630 ounces of Solution \(Z\) contains how many ounces of \(a ?\) A. $$68$$ B. $$73$$ C. $$89$$ D. $$219$$ E. $$236$$

Short answer

The Solution Z consists of approximately 192.15 ounces of 'a'. As this is not an option, the closest answer would be A. $$68$$

Step-by-step solution

Analyze the ratio in each solution

In Solution X, the ratio of a and b is 2:3, so for every 5 ounces, 2 ounces are of a and 3 ounces are of b. Similarly, in Solution Y, the ratio of a and b is 1:2, so for every 3 ounces, 1 ounce is of a and 2 ounces are of b.

Determine the ratio in Solution Z

Solution Z is a mixture of Solutions X and Y in the ratio 3:11, meaning 3 parts of X are mixed with 11 parts of Y. So, for every 14 parts (3+11) of Solution Z, 3 parts are from X and 11 parts are from Y. Therefore, 'a' in solution Z from X = 2/5 * 3 = 0.6 and from Y = 1/3 * 11 = 3.67.

Find the amount of 'a' in Solution Z

The sum of 'a' from both solutions X and Y in Z = 0.6 + 3.67 = 4.27. This means for every 14 oz of Solution Z, 4.27 oz is of 'a'. Therefore, 'a' in 630 oz (which is 45*14 oz) of Solution Z = 4.27 * 45 =192.15 oz approximately.

Key concepts

Ratios in Solutions

Ratios are a way of expressing the relative sizes of different components in a mix. In solution chemistry, understanding ratios helps us determine the composition of mixtures. For example, in Solution X, the ratio of ingredients \(a\) to \(b\) is \(2:3\). This tells us that in 5 ounces of this solution, 2 ounces consist of ingredient \(a\) and the remaining 3 ounces consist of ingredient \(b\).
Similarly, Solution Y has a ratio of \(1:2\), which indicates that in every 3 ounces, 1 ounce is \(a\) and 2 ounces are \(b\). Knowing these ratios is crucial to understanding how to calculate the amounts of each ingredient when these solutions are mixed.
Ratios help simplify the calculations later, allowing us to scale up or down easily and understand the proportional presence of each ingredient in larger mixtures.

Mixture Problems

Mixture problems involve combining different ingredients or solutions to find the proportions or total amounts of components in the final mix. In our exercise, Solution Z is formed by mixing Solution X and Solution Y in a specific ratio of \(3:11\).
To tackle such mixture problems, you need to keep track of how much of each solution contributes to the new mixture.

• Take the ratio \(3:11\), which means for every 14 parts of Solution Z, 3 parts are from Solution X and 11 parts are from Solution Y.
• This tells us the contribution of each solution in forming the mixture and the eventual presence of specific ingredients.

Properly working through mixture problems can help you in various scenarios, whether you're working in a lab, cooking, or solving these in exams like the GRE.

Algebraic Problem Solving

Algebraic problem solving involves formulating equations based on given information and solving them to find unknown values. In this case, we need to find out how much of ingredient \(a\) is in the mixed Solution Z.
We know from Solutions X and Y's ratios how much ingredient \(a\) contributes to each ounce.

• For Solution X: \( \frac{2}{5} \) of each part is \(a\), so when 3 parts are mixed, it contributes \(3 \times \frac{2}{5} = 0.6\).
• For Solution Y: \( \frac{1}{3} \) of each ounce is \(a\), so from 11 parts, it contributes \(11 \times \frac{1}{3} = 3.67\).

We sum these contributions to find that \(4.27\) ounces of \(a\) are present in every 14 ounces of Solution Z.
This formula essentially gives us a way to calculate amounts flexibly and helps tackle more complex mixture scenarios by balancing equations.

Quantitative Reasoning

Quantitative reasoning is key to understanding and solving GRE math problems. It allows you to analyze information, decipher ratios, and mathematically deduce the required outcomes. In this exercise, we use quantitative reasoning to determine the total amount of \(a\) after solving the ratios and mixture equations.
First, we identify the basic elements and perform calculations. This involves:

• Starting with the ratio of each ingredient and solution mix.
• Breaking down these ratios to understand component contributions to a larger mix.

Finally, employ principles of multiplication and summation to find that 630 ounces of Solution Z contains approximately 192.15 ounces of \(a\).
This approach showcases how mathematical thinking helps us break down real-world problems logically and analytically, paving the way for successful problem-solving on tests like GRE.