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Chapter 2: Problem 92

Show a complete solution to each problem. Mary's assets. Mary Hall's will specifies that her lawyer is to liquidate her assets and divide the proceeds among her three sisters. Lena's share is to be one-half of Lisa's, and Lisa's share is to be one-half of Lauren's. If the lawyer has agreed to a fee that is equal to \(10 \%\) of the largest share and the proceeds amount to \(\$ 164,428,\) then how much does each person get?

Short Answer

Expert verified
Lauren gets \[88,860\], Lisa gets \[44,430\], and Lena gets \[22,215\].

Step by step solution

01

Define the variables

Let Lauren's share be denoted by \(L\). So, Lisa's share will be \(\frac{1}{2}L\), and Lena's share will be \(\frac{1}{4}L\).
02

Determine the lawyer's fee

Since the lawyer's fee is equal to \(10\%\) of the largest share, which is Lauren's share, the lawyer's fee is \(0.1L\).
03

Set up the total proceeds equation

The total proceeds are divided among the three sisters and also pay the lawyer's fee. The total proceeds can be expressed as: \(L + \frac{1}{2}L + \frac{1}{4}L + 0.1L = 164,428\).
04

Simplify the equation

Combine like terms to simplify the equation: \(1.85L = 164,428\).
05

Solve for Lauren's share

Solve the equation for \(L\): \[L = \frac{164,428}{1.85} = 88,860\]
06

Calculate Lisa and Lena's shares

Using \(L = 88,860\), find Lisa's share: \[ \frac{1}{2} \times 88,860 = 44,430\] and Lena's share: \[ \frac{1}{4} \times 88,860 = 22,215 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

variables
In algebra, a variable is a symbol or letter that represents an unknown value. In this problem, we used the variable \(L\) to represent Lauren's share of the proceeds. This helps us set up and solve equations where values change. For example, by defining Lauren's share as \(L\), we can easily express Lisa's and Lena's shares in relation to \(L\).

Our variable breakdown goes like this:
  • Lauren's share: \(L\)
  • Lisa's share: \(\frac{1}{2} L\)
  • Lena's share: \(\frac{1}{4} L\)
equations
Equations are mathematical statements that assert the equality of two expressions. They are essential in solving algebra word problems. Here, we used the total proceeds to set up our main equation.

Considering the lawyer's fee, which is \(10\%\) of Lauren's share, the equation becomes:
\(L + \frac{1}{2} L + \frac{1}{4} L + 0.1 L = 164,428\)

By combining like terms, we simplify this into:
\(1.85L = 164,428\)

This equation allows us to solve for the unknown variable \(L\), which is pivotal to finding each sister's share.
percentages
Percentages represent a part of a whole and are essential in understanding portions of a total amount. In this exercise, the lawyer's fee is expressed as a percentage: \(10\%\) of Lauren's share.

To convert this percentage into a usable figure for our equation, we multiply Lauren's share by \(0.1\) (the decimal equivalent of \(10\%\)). This gives us the lawyer's fee, which is then added to the total distribution:

\(0.1L\) represents \(10\%\) of \(L\).

Including this in our equation ensures we're accounting for every part of the proceeds correctly.
shares
Shares refer to portions of a total amount that are to be divided. In this problem, the total proceeds from Mary Hall's assets need to be divided among her three sisters. Each sister's share is a fraction of Lauren's share.

The relationships given are:
  • Lisa's share: \(\frac{1}{2} L\)
  • Lena's share: \(\frac{1}{4} L\)


Using the solved value \(L = 88,860\), we find:
  • Lisa's share: \(44,430\)
  • Lena's share: \(22,215\)


This structured division ensures each sister gets her rightful share based on the specified ratios.

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Most popular questions from this chapter

Show a complete solution to each problem. First Super Bowl. In the first Super Bowl game in the Los Angeles Coliscum in 1967 , the Green Bay Packers outscored the Kansas City Chiefs by 25 points. If 45 points were scored in that game, then what was the final score?

Show a complete solution to each problem. Tennis. The distance from the bascline to the service line on a tennis court is 3 feet less than the distance from the service line to the net. If the distance from the bascline to the net is 39 feet, then what is the distance from the service line to the net?

Solve problem by using an inequality. Sewing machines. Charles wants to buy a sewing machine in a city with a \(10 \%\) sales tax. He has at most \(\$ 700\) to spend. In what price range should he look?

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Solve each problem by using an absolute value equation or inequality. See Example 10. Famous battles. In the Hundred Years鈥橶ar, Henry V defeated a French army in the battle of Agincourt and Joan of Arc defeated an English army in the battle of Orleans (The Doubleday Almanac). Suppose you know only that these two famous battles were 14 years apart and that the battle of Agincourt occurred in 1415. Use an absolute value equation to find the possibilities for the year in which the battle of Orleans occurred.
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