Problem 102 - Elementary Algebra

Question

Blake Scott leaves \(\frac{1}{4}\) of his estate to the Boy Scouts, \(\frac{2}{5}\) to the local cancer fund, and the rest to his church. What fractional part of the estate does the church receive?

Step-by-Step Solution

Answer

The church receives \(\frac{7}{20}\) of the estate.

Step 1: Determine Total Fraction Given Away

Blake Scott leaves \(\frac{1}{4}\) of his estate to the Boy Scouts and \(\frac{2}{5}\) to the local cancer fund. First, we need to add these fractions together to find the total fraction of the estate given away.To add fractions, convert them to a common denominator. The least common multiple of 4 and 5 is 20. Convert each fraction:\[\frac{1}{4} = \frac{5}{20}\] and \[\frac{2}{5} = \frac{8}{20}\].Add these fractions together:\[\frac{5}{20} + \frac{8}{20} = \frac{13}{20}\].So, a total of \(\frac{13}{20}\) of the estate is given to the Boy Scouts and the local cancer fund.

Step 2: Determine Fraction Left for the Church

After giving away \(\frac{13}{20}\) of the estate, the remainder will go to Blake Scott's church. Since the total estate is \(\frac{20}{20}\), the fraction of the estate left for the church is:\[\frac{20}{20} - \frac{13}{20} = \frac{7}{20}\].Thus, the church receives \(\frac{7}{20}\) of the estate.

Key Concepts

Fraction Addition

Adding fractions is an essential skill in math. It involves finding a way to combine fractions, even if their denominators differ. To add fractions, you'll first need a common denominator. For example, if you want to add \(\frac{1}{4}\) and \(\frac{2}{5}\), start by finding a common foundation for these fractions to sit on.

When adding fractions like \(\frac{1}{4}\) and \(\frac{2}{5}\), the first step is not to attempt a hasty addition of the numerators. Instead, the golden rule is to first make sure the fractions have the same denominator. This can be seen when \(\frac{1}{4}\) becomes \(\frac{5}{20}\) and \(\frac{2}{5}\) becomes \(\frac{8}{20}\). Now that they share a foundation, it's easy to combine them: \(\frac{5}{20} + \frac{8}{20} = \frac{13}{20}\).

• Ensure fractions share the same denominator before addition.
• Add numerators while keeping the common denominator.

Least Common Denominator

The least common denominator (LCD) is a critical concept when adding or subtracting fractions with different denominators. It refers to the smallest number into which all denominators in a set of fractions can divide equally. Utilizing the LCD simplifies operations by standardizing denominators.

To find the LCD for fractions like \(\frac{1}{4}\) and \(\frac{2}{5}\), locate the least common multiple (LCM) of their denominators. Here, 4 and 5 both multiply into 20, making 20 the LCD. Converting \(\frac{1}{4}\) to \(\frac{5}{20}\) and \(\frac{2}{5}\) to \(\frac{8}{20}\) helps in accurately carrying out operations on them. This way, fractions become friendly to addition or subtraction.

• Identify a multiple common to all fraction denominators.
• Convert fractions to share this denominator.
• Perform arithmetic operations with ease.

Subtraction of Fractions

Subtraction of fractions follows a similar method to addition, but instead of summing numerators, you subtract them. Before you can perform subtraction, ensure that the fractions share the same denominator. This concept is demonstrated in the calculation of how much of Blake Scott's estate goes to the church.

Start with the total amount \(\frac{20}{20}\) (the whole estate) and subtract the portion given away, \(\frac{13}{20}\). With the same denominator, subtract the numerators: \(20 - 13 = 7\). So, the remaining fraction for the church is \(\frac{7}{20}\). The challenge arises only when the fractions initially have different denominators, which is why having a common denominator first is essential.

• Ensure a common denominator before subtraction.
• Subtract numerators, maintain the common denominator.
• Check results by verifying calculations holistically.