Question

\(4 \mathrm{~kg}\) of a metal contains \(\frac{1}{5}\) copper and rest is zinc. Another 5 \(\mathrm{kg}\) of metal contains \(\frac{1}{6}\) copper and rest is zinc. The ratio of copper and zinc into the mixture of these two metals : (a) \(49: 221\) (b) \(39: 231\) (c) \(94-181\) (d) none of these

Short answer

Answer: The ratio of copper to zinc in the mixture is 49:221.

Step-by-step solution

Find the amount of copper in the first metal

To find the amount of copper in the first metal, multiply the total weight of the metal (4 kg) by the percentage of copper (\(\frac{1}{5}\)) i.e., Copper in the first metal = \(\frac{1}{5} \times 4\) kg

Find the amount of zinc in the first metal

Subtract the amount of copper from the total weight of the metal to find the amount of zinc in the first metal i.e., Zinc in the first metal = \(4 - \frac{1}{5} \times 4 = 4 - \frac{4}{5}\) kg

Find the amount of copper in the second metal

Multiply the total weight of the second metal (\(5\) kg) by the percentage of copper (\(\frac{1}{6}\)) i.e., Copper in the second metal = \(\frac{1}{6} \times 5\) kg

Find the amount of zinc in the second metal

Subtract the amount of copper from the total weight of the second metal to find the amount of zinc in the second metal i.e., Zinc in the second metal = \(5 - \frac{1}{6} \times 5 = 5 - \frac{5}{6}\) kg

Combine the amounts of copper and zinc

Add the amounts of copper and zinc in both metals to find the total amount of copper and zinc in the mixture i.e., Total copper = \(\left(\frac{1}{5} \times 4\right) + \left(\frac{1}{6} \times 5\right)\) Total zinc = \(\left(4 - \frac{4}{5}\right) + \left(5 - \frac{5}{6}\right)\)

Calculate the ratio of copper to zinc

Divide the total amount of copper and zinc by their respective amounts to find the ratio of copper to zinc i.e., Ratio of copper to zinc = \(\frac{\left(\frac{1}{5} \times 4 + \frac{1}{6} \times 5\right)}{\left(4 - \frac{4}{5} + 5 - \frac{5}{6}\right)}\)

Find the answer in the given options

To check which of the given options matches the ratio calculated above, simply substitute the respective values and verify which option gives the correct answer. After checking, we find that the ratio corresponds to option: (a) \(49 : 221\) So the correct answer is (a) \(49 : 221\).

Key concepts

Compound Ratios

A compound ratio is formed when two or more ratios are combined together to form a single ratio. This is accomplished by multiplying the corresponding terms of the given ratios. For example, if you have two separate ratios, say \( A:B \) and \( C:D \) , their compound ratio can be expressed as \( (A \times C):(B \times D) \). In practice, compound ratios help to simplify complex relationships between different quantities and are commonly used in problems involving mixtures or combinations of different materials.

In the context of our original textbook problem, the concept of compound ratios is implicitly used when combining the proportions of copper and zinc separately from two different metals. By understanding how to combine these ratios, students can solve complex mixture problems with greater ease, applying the compound ratio to find the final composition of a metal alloy or any mixture.

Mixture Problems

Mixture problems often involve combining two or more substances with known quantities or concentrations to determine the resulting mixture's composition. These types of problems appear frequently in fields like chemistry, cooking, and even finance.

In mixture problems, the essential step is to determine the amount of each component in the final mix. This usually requires skills in ratio and proportion and is where the understanding of compound ratios comes into play. Algebraic expressions and equations are often constructed to reflect the relationships and constraints of the mixture scenario. Once students master translating the text of a problem into algebraic notation, solving mixture problems becomes a straightforward process of manipulating quantities and ratios.

Quantitative Aptitude

Quantitative aptitude is a measure of an individual's ability to perform mathematical calculations quickly and accurately. This is a key skill in numerous competitive exams and jobs that require numerical reasoning. It consists of various mathematical concepts, including algebra, geometry, arithmetic, and data interpretation.

Problems involving ratios, proportions, mixtures, and algebraic expressions, like the one from the textbook, are meant to test and enhance a student’s quantitative aptitude. These problems encourage logical thinking and the application of mathematical principles to real-world scenarios. Strengthening skills in this area involves practicing a wide range of quantitative problems, learning shortcuts, and mastering the concepts behind them.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can include numbers, variables, and arithmetic operations (addition, subtraction, multiplication, division, and exponentiation). They are pivotal in forming equations that can be solved to find unknown values. These expressions become a language through which real-life problems can be translated into mathematical form, and then manipulated to find a solution.

In the textbook problem, algebraic expressions are used to describe the quantities of copper and zinc in the metals. By substituting values and following algebraic operations, students can determine the final composition of the mixture. Grasping these expressions and operations is fundamental to solving a wide array of problems that demand quantitative skills.