Question

A man makes a will and dies leaving his wife pregnant. His will disposes of 100 écus such that if his wife has a daughter, the mother should take twice as much as the daughter, but if she has a son, he should have twice as much as the mother. [Sexist problem!] The mother gives birth to twins, a son and a daughter. How should the estate be split, respecting the father's intentions?

Short answer

Answer: The mother should receive approximately 28.58 écus, the son should receive approximately 57.16 écus, and the daughter should receive approximately 14.29 écus.

Step-by-step solution

Identifying variables

Let's denote the portions of the écus to be received by the mother as M, the son as S, and the daughter as D. The total amount of écus to be split is 100.

Setting up equations based on the father's intentions

According to the father's will, if the wife has a daughter, the mother should take twice as much as the daughter: M = 2D. And if she has a son, he should have twice as much as the mother: S = 2M. Since there's a son and a daughter, both of these conditions must apply.

Express S in terms of D

We can use the first equation to express M in terms of D: M = 2D. Then we can substitute this expression into the second equation: S = 2(2D) = 4D.

Set up an equation for the total amount to be split

The sum of the portions allocated to the mother, the son, and the daughter must equal 100 écus: M + S + D = 100. Now substitute the expressions we found for M and S in terms of D: (2D) + (4D) + D = 100.

Solve for D

Simplify the equation: 7D = 100. To find the value of D, divide both sides of the equation by 7: D = 100/7 ≈ 14.29.

Calculate M and S in terms of D

Now that we have the value for D, we can calculate the portions for M and S: M = 2D = 2(14.29) ≈ 28.58, and S = 4D = 4(14.29) ≈ 57.16.

Distribute the écus based on calculated values

According to our calculations, the mother should receive approximately 28.58 écus, the son should receive approximately 57.16 écus, and the daughter should receive approximately 14.29 écus. Please note that due to rounding, the total might be slightly different from 100 écus, but this distribution best respects the father's intentions for splitting his estate.

Key concepts

Algebraic Equations

Algebraic equations are at the heart of many mathematical word problems. They enable us to represent real-world situations in a structured mathematical format. In the case of resolving the distribution of an estate, we defined variables that stand for the respective shares of the mother (M), the son (S), and the daughter (D). By translating the conditions of the will into algebraic equations (\( M = 2D \) and \( S = 2M \) ), we created a system of equations that reflect the constraints given in the problem.

Mastering algebraic equations requires familiarity with variables, constants, and the arithmetic operations that connect them. It's also essential to understand how to manipulate these equations through processes like substitution (\( M = 2D \) becoming \( S = 4D \) when substituted into the second equation) and simplification (combining like terms to construct the equation \( 7D = 100 \)). This simplification is a crucial skill, as it directly leads to finding the solution for the unknown variable.

Problem-Solving Strategies

Problem-solving strategies in mathematics allow students to approach complex scenarios systematically. When faced with a word problem, it is essential to follow a step-by-step approach, often starting with carefully reading the problem and identifying all relevant information.

In our example, the initial step was to define the variables (Step 1) to represent the unknown quantities. The next strategic move (Step 2) was to convert the conditions of the problem into mathematical equations, which are the building blocks for any algebra-based solution. Subsequent steps involved mathematical techniques like expressing one variable in terms of another (Step 3) and forming an equation that encompasses the entire scenario (Step 4). Finally, solving for the single variable (Step 5) and back-substituting to find the other unknowns (Step 6) brought us to the solution.

A reflective and iterative process is beneficial – it's vital to review each step for accuracy and revisit earlier steps if the solution doesn't seem to fit. As seen in the provided solution, rounding errors may occur, so re-checking with the original conditions can ensure the solution satisfies the problem's requirements (Step 7).

Proportional Reasoning

Proportional reasoning is a key component when solving problems that involve ratios and comparisons – much like how an estate ought to be divided according to a person's will. This mathematical concept helps us understand and solve problems that deal with the relative size of two quantities.

In the given problem, we see proportional relationships between the shares of the mother, son, and daughter. By establishing equations that relate these shares (\( M = 2D \) and \( S = 2M \)), we're using the concept of proportionality. These relationships suggest that the son's share is twice the mother's and the mother's share is twice the daughter's, leading to a proportional distribution of the estate based on the father's intentions.

Proportional reasoning also highlights the importance of considering the whole picture. When the entirety of the estate (100 écus) is taken into account, the allocation for each relative must be proportionally accurate according to the total sum, as seen in the final equation \( M + S + D = 100 \). Through proportional reasoning, we could decipher and solve this allocation problem to respect the balance mandated by the father's will.