Problem 77 - Elementary Algebra

Question

An inheritance of \(\$ 180,000\) is to be divided between a child and the local cancer fund in the ratio of 5 to 1 . How much money will the child receive?

Step-by-Step Solution

Answer

The child will receive $150,000.

Step 1: Determine Total Parts of the Ratio

The first step is to determine the total number of parts into which the inheritance will be divided. The given ratio between the child and the cancer fund is 5:1, which means there are 5 parts for the child and 1 part for the cancer fund. Therefore, the total parts is calculated as follows: 5 parts (for the child) + 1 part (for the cancer fund) = 6 parts.

Step 2: Calculate the Value of One Part

Now that we know the total number of parts is 6, we can calculate the monetary value of each part. The total inheritance is $180,000, so each part is worth:\[ \frac{180,000}{6} = 30,000 \text{ dollars per part} \]

Step 3: Calculate the Child's Share

To find out how much money the child receives, multiply the value of one part by the number of parts allocated to the child. Based on the ratio, the child receives 5 parts.Thus, the child's share is:\[ 5 \times 30,000 = 150,000 \text{ dollars} \]

Key Concepts

Understanding Inheritance Division

When dealing with inheritance division, the objective is to allocate a sum of money, such as an inheritance, between different parties according to a specified ratio. In this case, this involves dividing an inheritance amount of $180,000 between a child and a cancer fund with a preset distribution ratio of 5:1.

• The ratio indicates how the total amount should be split between the two parties—in this case, 5 parts for the child and 1 part for the cancer fund.
• Understanding this proportion is crucial as it ensures the fair and intended division of the inheritance as per wishes or agreements.

By breaking down the inheritance into parts as dictated by the ratio, it allows for an easy calculation of each party’s share by determining the value of each part and multiplying by the number of parts each party receives.

Proportion Calculations Simplified

Proportion calculations are at the heart of solving ratio problems like the inheritance division. They provide a mathematical way to determine how much each element within a ratio should receive. You figure out the size of each 'part' by dividing the total amount by the sum of the parts in the ratio.

• First, add the numbers in the ratio to find the total number of parts. For example, a 5:1 ratio means there are 5 + 1 = 6 total parts.
• Next, divide the total amount (\(180,000\) dollars) by this total to find the value of one part. This would mean dividing 180,000 by 6 to get 30,000 dollars per part in this example.

These steps are straightforward and form the crux of proportion calculations in ratio problems. With each part having a clear value, it's simple to allocate funds according to the specified proportions.

Mastering Algebraic Problem Solving

Algebraic problem solving is an important skill when dealing with ratio and proportion calculations. This involves using basic algebra to derive and solve equations that represent real-world problems.

• Firstly, establish an equation based on the problem statement, like\(5x + x = 180,000\) for the ratio and inheritance problem.
• Next, solve for\(x\) to find out how much each part of the ratio is worth. We already calculated\(x\) to be 30,000.
• Finally, multiply\(x\) by the number of parts each party receives to find the total allocation, such as\(5 imes 30,000 = 150,000\) for the child.

This method provides a structured approach to logically solve complex division problems by breaking them down into simpler, manageable steps using algebra.