Question

Apply the principles of borrowing, and subtract the following: \(5 \frac{1}{2}-3 \frac{3}{4}=\) ______

Short answer

The result is \(1 \frac{3}{4}\).

Step-by-step solution

Convert Mixed Numbers to Improper Fractions

First, convert the mixed numbers into improper fractions to make subtraction easier.For \(5 \frac{1}{2}\), multiply 5 (the whole number) by 2 (the denominator) and add the numerator 1:\[5 \times 2 + 1 = 10 + 1 = 11\]Thus, \(5 \frac{1}{2}\) is equal to \(\frac{11}{2}\).For \(3 \frac{3}{4}\), multiply 3 by 4 and add 3:\[3 \times 4 + 3 = 12 + 3 = 15\]Hence, \(3 \frac{3}{4}\) is equal to \(\frac{15}{4}\).

Find a Common Denominator

To subtract the fractions, they must have a common denominator. The denominators are 2 and 4.The least common multiple (LCM) of 2 and 4 is 4.Hence, convert \(\frac{11}{2}\) to a fraction with a denominator of 4:\[\frac{11}{2} = \frac{11 \times 2}{2 \times 2} = \frac{22}{4}\]

Subtract the Improper Fractions

Now both fractions have the same denominator:\[\frac{22}{4} - \frac{15}{4}\]Subtract the numerators:\[22 - 15 = 7\]The result is \(\frac{7}{4}\).

Convert Improper Fraction to Mixed Number

Convert \(\frac{7}{4}\) back to a mixed number.Divide 7 by 4. The quotient is 1 and the remainder is 3.So, \(\frac{7}{4}\) is equal to \(1 \frac{3}{4}\).

Key concepts

Mixed Numbers

A mixed number is a way of expressing numbers that have both a whole number and a fractional part. Consider 5 \( \frac{1}{2} \) and 3 \( \frac{3}{4} \) from the exercise. The whole numbers are 5 and 3, respectively, while \( \frac{1}{2} \) and \( \frac{3}{4} \) are the fractional parts.
Mixed numbers are often used in daily life, such as in cooking or baking, to make measurements feel more intuitive.

When working with mixed numbers in mathematical operations, such as subtraction, it is usually necessary to convert them into improper fractions first. This simplifies the arithmetic process by allowing you to work with a single fraction rather than juggling two parts at once.

Improper Fractions

An improper fraction occurs when the numerator is greater than or equal to the denominator. For example, \( \frac{11}{2} \) and \( \frac{15}{4} \) are improper fractions resulting from converting the mixed numbers in the exercise.
To transform mixed numbers to improper fractions:

• Multiply the whole number by the denominator.
• Add the numerator to this result.
• Write this sum over the original denominator.

Improper fractions are particularly helpful when performing operations like addition or subtraction because they allow you to handle one combined fraction easily. This avoids the complexity of dealing with both whole and fractional parts, making the process clearer and more direct.

Common Denominator

To subtract fractions, they must share the same denominator, known as the common denominator. In this exercise, the denominators were 2 and 4.

Finding a common denominator ensures that the fractions represent "parts out of the same whole," allowing you to subtract the numerators directly.

• First, find the least common multiple (LCM) of the two denominators. For 2 and 4, the LCM is 4.
• Convert the fractions so they both have this denominator.

For instance, convert \( \frac{11}{2} \) to \( \frac{22}{4} \) by multiplying both the numerator and denominator by 2. Once a common denominator is established, you simply subtract or add the numerators to find the result.

Borrowing in Subtraction

Sometimes in subtraction, especially with fractions, borrowing is needed when the top fraction's numerator is smaller than the bottom's. In normal arithmetic, borrowing involves taking "1" from the next place value to make the subtraction possible. In fractions, it follows a similar concept by adjusting the numbers.

Though the given exercise did not explicitly require borrowing, it's a crucial concept in many subtraction problems that involve mixed numbers. For instance, if you had to subtract \( 2 \frac{3}{4} \) from \( 1 \frac{1}{2} \), you would need to "borrow" 1 from the whole number portion of \( 1 \frac{1}{2} \), turning it into \( \frac{5}{4} \) to make the subtraction feasible.

• Convert the whole number borrowing into an equivalent fraction of the same denominator as the fractional part.
• Subtract the fractions normally.

Understanding borrowing in fraction subtraction helps manage cases where direct subtraction isn't immediately possible.