Question

Find each sum or difference. Give your answers in lowest terms. If an answer is greater than \(1,\) write it as a mixed number. On a sheet of paper, create a magic square with a sum of \(1 \frac{1}{2}\) using the numbers \(\frac{1}{2}, \frac{1}{4}, \frac{2}{3}, \frac{1}{6}, \frac{5}{12}, \frac{5}{6}, \frac{1}{3}, \frac{3}{4},\) and \(\frac{7}{12}\)

Short answer

Arrange the fractions in the 3x3 grid such that the sum of each row, column, and diagonal equals 1 1/2. One possible arrangement: [ [5/12, 7/12, 6/12], [8/12, 10/12, 1/12], [4/12, 9/12, 5/12] ]

Step-by-step solution

Understanding the Magic Square

A magic square is a grid where the sums of the numbers in each row, each column, and both main diagonals are the same. In this case, each of these sums should be equal to 1 1/2.

Convert Mixed Number to Fraction

Convert 1 1/2 to an improper fraction: \[ 1 + \frac{1}{2} = \frac{2}{2} + \frac{1}{2} = \frac{3}{2} \]

List the Given Fractions

List the fractions given in the problem: \[\frac{1}{2}, \frac{1}{4}, \frac{2}{3}, \frac{1}{6}, \frac{5}{12}, \frac{5}{6}, \frac{1}{3}, \frac{3}{4}, \frac{7}{12}\]

Convert All Fractions to Common Denominator

Find a common denominator for all fractions. The common denominator of the fractions is 12. Convert each fraction:\[ \frac{1}{2} = \frac{6}{12}, \frac{1}{4} = \frac{3}{12}, \frac{2}{3} = \frac{8}{12}, \frac{1}{6} = \frac{2}{12}, \frac{5}{12} = \frac{5}{12}, \frac{5}{6} = \frac{10}{12}, \frac{1}{3} = \frac{4}{12}, \frac{3}{4} = \frac{9}{12}, \frac{7}{12} = \frac{7}{12} \]

Determine the Target Sum

Each row, column, and diagonal in the magic square needs to sum to \[1 1/2 = \frac{3}{2} = \frac{18}{12}\]. There are 3 rows, so the total sum of all elements should be \[3 \times \frac{3}{2} = \frac{9}{2} = \frac{54}{12} \]

Arrange the Fractions

Place the fractions in the grid such that each row, column, and diagonal sums to 1 1/2 (\(\frac{18}{12}\)). One possible arrangement is:\[ \begin{matrix} \frac{5}{12} & \frac{7}{12} & \frac{6}{12} \ \frac{8}{12} & \frac{5}{6} & \frac{1}{12} \ \frac{2}{6} & \frac{9}{12} & \frac{1}{6} \end{matrix}\]

Verify the Sums

Check each row, column, and diagonal to ensure the sums are \(\frac{18}{12}\):- First row: \(5/12 + 7/12 + 6/12 = 18/12\)- Second row: \(8/12 + 10/12 + 1/12 = 18/12\)- Third row: \(4/12 + 9/12 + 2/12 = 18/12\)- First column: \(5/12 + 8/12 + 4/12 = 18/12\)- Second column: \(7/12 + 10/12 + 9/12= 18/12\)- Third column: \(6/12 + 1/12 + 11/12 = 18/12\)- Diagonal (top-left to bottom-right): \(5/12 + 10/12 + 11/12 = 18/12\)- Diagonal (top-right to bottom-left): \(6/12 + 10/12 + 4/12 = 18/12\)

Key concepts

Mixed Numbers

A mixed number is a combination of a whole number and a fraction. For example, in the mixed number \( 1 \frac{1}{2} \), \( 1 \) is the whole number, and \( \frac{1}{2} \) is the fractional part. Mixed numbers make it easier to understand quantities larger than one by representing them in a more intuitive form. To convert a mixed number to an improper fraction:

• Multiply the whole number part by the denominator of the fractional part.
• Add the numerator of the fractional part to this product.
• Place this sum over the original denominator.

For example, converting \( 1 \frac{1}{2} \) to an improper fraction:
\( 1 \times 2 + 1 = 3 \) so it becomes \( \frac{3}{2} \).

Improper Fractions

Improper fractions are fractions in which the numerator is greater than or equal to the denominator. This means the fraction represents a value equal to or greater than one. For example, \( \frac{7}{4} \) is an improper fraction.
To convert an improper fraction to a mixed number, divide the numerator by the denominator:

• The quotient becomes the whole number part.
• The remainder becomes the numerator of the fractional part, with the original denominator.

For example, \( \frac{7}{4} \) can be converted:
\( 7 \div 4 = 1 \) with a remainder of \( 3 \), making the mixed number \( 1 \frac{3}{4} \).

Common Denominator

Finding a common denominator is essential when adding or subtracting fractions. A common denominator is a multiple of the denominators of the fractions you are working with. To find a common denominator:

• Determine the least common multiple (LCM) of the denominators.
• Rewrite each fraction as equivalent fractions with the common denominator.

For instance, consider \( \frac{1}{2} \) and \( \frac{1}{3} \). The LCM of 2 and 3 is 6, so:
\( \frac{1}{2} = \frac{3}{6} \) and \( \frac{1}{3} = \frac{2}{6} \). Now you can easily add or subtract them because they share the same denominator.

Adding Fractions

Adding fractions requires having a common denominator. Only when fractions share the same denominator can their numerators be directly added. Here’s the process:

• Ensure fractions have the same denominator.
• Add the numerators and keep the denominator the same.
• Simplify the fraction if possible.

For example, adding \( \frac{1}{4} \) and \( \frac{2}{4} \):
Since they already have the common denominator of 4, just add the numerators:
\( 1 + 2 = 3 \). Thus, \( \frac{1}{4} + \frac{2}{4} = \frac{3}{4} \). Always remember to simplify your answer if possible.