Show a complete solution to each problem. Dividing the...

Show a complete solution to each problem. Dividing the estate. Uncle Albert's estate is to be divided among his three nephews. The will specifies that Daniel receive one-half of the amount that Brian receives and that Raymond receive $\$ 1000$ less than one-third of the amount that Brian receives. If the estate amounts to $\$ 25,400,$ then how much does each inherit?

Show a complete solution to each problem.
Dividing the estate. Uncle Albert's estate is to be divided among his three nephews. The will specifies that Daniel receive one-half of the amount that Brian receives and that Raymond receive $\$ 1000$ less than one-third of the amount that Brian receives. If the estate amounts to $\$ 25,400,$ then how much does each inherit?

Key Concepts

Solution Verification

After arriving at potential solutions, it's important to cross-check them by substituting back into the original equations or conditions. This ensures that all relationships and constraints are met, confirming the accuracy of the solution.

Substitution Method

This is a strategy for solving systems of equations where one variable is expressed in terms of another from one equation and then substituted into the other equations. This technique reduces the system to fewer equations, making it easier to solve for the remaining unknowns.

Algebraic Representation

This concept involves converting verbal descriptions and relationships into algebraic expressions using variables. By clearly defining unknown quantities with symbols, one can create equations that encapsulate the problem’s conditions, allowing for systematic analysis and solution.

System of Linear Equations

When a problem involves multiple relationships between unknown quantities, it often leads to a system of linear equations. This set of equations can be solved simultaneously, ensuring that the values found satisfy all the relationships and constraints described in the problem.

Transcript

00:01 Okay, so in this following, we're asked to show a complete solution given the following.

00:05 So there are assets being divided between three people.

00:08 Person a is to receive one half of the amount that person b receives, and person c is to get $1 ,000 less than one third the amount that person b got.

00:18 And if the total value of the assets adds to $25 ,400, how much does each person receive? so we're trying to figure out how much each of these people receive.

00:29 So let's go ahead and start formulating an equation.

00:34 And just off the top of my head, i know that person a plus person b plus person c, all of these people added together are going to equal $25 ,400.

00:47 I just need to figure out a way to represent this equation using one variable.

00:53 And through reading this problem, it looks like everybody kind of is revolving around person b.

00:58 So let's go ahead and use person b as our main variable.

01:03 So i'm going to go ahead and use b as our variable this time.

01:09 And on either side of that person will be a and c.

01:14 So let's start with person a.

01:16 We know that person a is to receive one half of the amount that person b receives.

01:20 So that one's pretty simple.

01:22 We're going to take one half multiplied by whatever person b gets.

01:29 Okay.

01:30 And then person c is to get $1 ,000 less.

01:33 Okay, so i know that we're going to have to subtract $1 ,000 from whatever else is going to come next.

01:39 $1 ,000 less than one -third the amount that person b got.

01:46 So one -third of person b's inheritance minus $1 ,000.

01:52 That's going to give us person c.

01:54 So if the total value of the assets is $25 ,400, i now have an equation that i can work with.

02:04 And i have three like terms on the left side that i can first combine.

02:08 So off to the side here in a little green, going to add one half plus one b, which is going to be represented by 2 over 2, plus 1 3rd b, and let's go ahead and find common denominators for these guys.

02:38 So we're going to have to have a common denominator of six.

02:41 So for this one, we're going to have to multiply by three.

02:46 Same with this guy.

02:49 Three over three, over three...

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