Problem 61 - Elementary Algebra

Question

If two grandchildren each received \(\frac{1}{32}\) of an estate and their mother received \(\frac{1}{3}\) of the estate, find the total fraction of the estate received by the mother and her two children.

Step-by-Step Solution

Answer

The total fraction of the estate received by the mother and her two children is \(\frac{19}{48}\).

Step 1: Determine the fraction received by both grandchildren

Each grandchild received \(\frac{1}{32}\) of the estate. Since there are two grandchildren, we need to add their shares together: \[\frac{1}{32} + \frac{1}{32} = \frac{2}{32} = \frac{1}{16}\]

Step 2: Confirm the mother's fraction

It is given that the mother received \(\frac{1}{3}\) of the estate.

Step 3: Add the fractions received by the mother and her children

We need to add the fractions received by the mother \(\frac{1}{3}\) and the grandchildren \(\frac{1}{16}\): \[\frac{1}{3} + \frac{1}{16}\]

Step 4: Find a common denominator

The least common denominator for 3 and 16 is 48. Convert each fraction to have this denominator: \[\frac{1}{3} = \frac{16}{48}\] and \[\frac{1}{16} = \frac{3}{48}\]

Step 5: Add the converted fractions

Now, add \(\frac{16}{48}\) and \(\frac{3}{48}\): \[\frac{16}{48} + \frac{3}{48} = \frac{19}{48}\]

Key Concepts

common denominator

When working with fractions, it is essential to have a common denominator to add or compare them easily. The common denominator is a shared multiple of the denominators in the given fractions. This allows the fractions to be expressed with the same denominator, simplifying operations like addition.

To find a common denominator, you can:

• Identify the denominators of each fraction
• Determine the least common multiple (LCM) of these denominators
• Convert each fraction to an equivalent fraction with the found common denominator

By applying this method, you can seamlessly add fractions, like in the problem where \(\frac{1}{3}\) and \(\frac{1}{16}\) were converted to fractions with a common denominator of 48.

least common multiple (LCM)

The least common multiple (LCM) is the smallest positive number that is a multiple of two or more numbers. It is particularly useful in operations involving fractions with different denominators, as it helps find a common denominator.

To find the LCM of two or more numbers:

• List the multiples of each number
• Identify the smallest multiple that is common to all lists

In the exercise, the denominators were 3 and 16. The multiples of 3 are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, 48, ...
The multiples of 16 are 16, 32, 48, 64, ...
Here, 48 is the smallest common multiple, making it the LCM. Using this LCM helps convert fractions to have a common denominator, making addition simpler.

fraction conversion

Fraction conversion involves changing a fraction to an equivalent fraction with a different denominator. This process often uses the least common multiple (LCM) to align denominators, enabling easy addition or subtraction.

Steps for fraction conversion:

• Identify the LCM of the denominators
• Adjust each fraction so that its denominator equals the LCM
• Multiply the numerator and denominator of each fraction by the same number to maintain equivalence

For example, to add \(\frac{1}{3}\) and \(\frac{1}{16}\), the LCM is 48. Convert each fraction: \(\frac{1}{3} = \frac{1 \times 16}{3 \times 16} = \frac{16}{48}\)
and \(\frac{1}{16} = \frac{1 \times 3}{16 \times 3} = \frac{3}{48}\). Now both fractions share the same denominator, simplifying the addition.

sum of fractions

Adding fractions, especially with different denominators, requires converting them to equivalent fractions with a common denominator. Once the fractions share the same denominator, you can directly add the numerators and keep the common denominator.

In the given exercise, two fractions were added: \(\frac{1}{3}\) and \(\frac{1}{16}\). After converting them to equivalent fractions with a common denominator of 48, the fractions became \(\frac{16}{48}\) and \(\frac{3}{48}\). Adding these results in:
\( \frac{16}{48} + \frac{3}{48} = \frac{19}{48} \)
With this technique, the total fraction of the estate received by the mother and her two children is found to be \(\frac{19}{48}\). This method makes it easier to add fractions accurately while maintaining their mathematical integrity.