Question

Divide Rs. 6940 in such a way that \(A\) gets \(\frac{2}{3} \mathrm{rd}\) of what \(B\) gets and \(B\) gets \(\frac{3}{5}\) th of what \(C\) gets? What is the share of \(A\) and \(B\) together? (a) Rs. 1982 (b) Rs. 1388 (c) Rs. 3470 (d) none of these

Short answer

Answer: The combined share of A and B is Rs. 3470.

Step-by-step solution

Writing the relationship between shares of A, B, and C.

According to the exercise, A gets 2/3rd of what B gets, and B gets 3/5th of what C gets. We can write these relationships as: \(A = \frac{2}{3}B\) \(B = \frac{3}{5}C\)

Finding the ratio of shares.

In order to find the ratio of shares, first, we can find the relationship between A and C by substituting the value of B from the second equation into the first equation: \(A = \frac{2}{3}(\frac{3}{5}C)\) \(A = \frac{2}{5}C\) Now, we can write the ratio of A:B:C as: A:B = 2:3 (from the relationship between A and B) B:C = 3:5 (from the relationship between B and C) Thus, the combined ratio becomes (multiply respective parts): A:B:C = 2:3:5

Dividing the total amount according to the ratio.

The total amount given is Rs. 6940, and the ratio of shares is 2:3:5. To divide the total amount according to the ratio, we need to find the value of one unit. To do this, add the parts of the ratio and then divide the total amount by the sum: Total parts = 2 + 3 + 5 = 10 Value of one unit = \(\frac{6940}{10} = 694\) Now, to find the individual shares, multiply the value of one unit with the respective parts of the ratio: A's share = Rs. \((2 * 694) = 1388\) B's share = Rs. \((3 * 694) = 2082\) C's share = Rs. \((5 * 694) = 3470\)

Finding the combined share of A and B.

To find the combined share of A and B, simply add their shares obtained in step 3: A's share + B's share = Rs. \((1388 + 2082) = 3470\) So, the combined share of A and B is Rs. 3470, which corresponds to option (c).

Key concepts

Ratio and Proportion

In mathematics, the concept of ratio is a relationship between two numbers of the same kind. It is expressed as "A:B," where A and B are quantities to be compared. If the relationship involves more than two quantities, it is known as a proportion. Ratios are often utilized when quantities need to be divided or compared.

In the provided exercise, the amounts that A, B, and C get need to be divided according to a specific set of ratios. First, we were given that A receives \( \frac{2}{3} \) of B's share, and B receives \( \frac{3}{5} \) of C's share. Establishing these kinds of relationships helps solve problems involving distributed amounts.

• Ratios are used to divide or compare quantities.
• Ratios can be combined into proportions for multiple quantities.
• They are often employed when evenly distributing resources.

When we expressed the relationship between A, B, and C as \( A:B:C = 2:3:5 \), this meant if we take C's amount as 5 units, then A's amount becomes 2 units, and so on. This ensures that aspects are balanced correctly according to given conditions.

Fractional Relationships

Fractional relationships are instances where quantities relate to each other through fractions, as seen in this exercise. A fraction is composed of a numerator and a denominator, detailing a part of a whole. For instance, when A gets \( \frac{2}{3} \) of what B gets, A's portion is a fraction of B's total.

• Fractions signify parts of a whole.
• They're useful for dividing items into portions.
• Can help establish complex relationships between quantities.

Fractional relationships become evident when substituting B's amount into A's equation, resulting in A receiving \( \frac{2}{5} \) of C's amount. Mastering fractional relationships can simplify complex calculations and help in understanding the division of quantities among different parties. Recognizing how these apply to situations ensures clear, accurate mathematical problem-solving.

Algebraic Expressions

An algebraic expression is a mathematical phrase that can involve numbers, variables, and operators. These expressions can be employed to denote various relationships or calculations. In the step-by-step solution provided, we used algebraic expressions to identify the relation between A, B, and C.

• Expressions can capture a variety of mathematical operations.
• They help in formulating equations based on given conditions.
• Essential for solving real-world math problems involving unknowns.

When writing formulas like \( A = \frac{2}{3}B \), it becomes clear how one quantity is derived from another. Algebraic expression formulation makes it straightforward to substitute values and calculate unknowns as demonstrated when B's expression was replaced to simplify A's calculation in terms of C.

Word Problems in Mathematics

Word problems translate everyday situations into mathematical challenges to solve. They require reading comprehension, identification of math concepts, and problem-solving skills. This exercise is also categorized as a word problem.

• Word problems present real-life scenarios to apply math concepts.
• Require careful reading and interpretation.
• Involve converting narratives into mathematical expressions.

The initial scenario told us to allocate a set amount according to specific shared fractions, which necessitated understanding and extracting mathematical relationships from text. Often, translating a narrative into expressions helps solve these scenarios effectively, allowing the calculation of amounts as demonstrated. Tackling word problems can develop the ability to apply mathematics to practical situations.