Question

Apply the principles of borrowing, and subtract the following: \(9 \frac{1}{4}-\frac{3}{4}=\) ______

Short answer

The result is \(8 \frac{1}{2}\).

Step-by-step solution

Understand the Problem

We need to subtract the fraction \(\frac{3}{4}\) from the mixed number \(9 \frac{1}{4}\). The subtraction involves fractions, so we might need to borrow if the fractional part is too small.

Check the Fractional Parts

The fractional part of \(9 \frac{1}{4}\) is \(\frac{1}{4}\), and we are subtracting \(\frac{3}{4}\). Since \(\frac{1}{4}\) is smaller than \(\frac{3}{4}\), we need to borrow to perform the subtraction.

Borrow from the Whole Number

Borrow 1 from the whole number 9, so we have 8 whole parts now. Convert the borrowed 1 into \(\frac{4}{4}\) to add to the existing \(\frac{1}{4}\). Now the fractional part is \(\frac{4}{4} + \frac{1}{4} = \frac{5}{4}\). The new mixed number is \(8 \frac{5}{4}\).

Subtract the Fractions

Subtract \(\frac{3}{4}\) from \(\frac{5}{4}\). The result is \(\frac{5}{4} - \frac{3}{4} = \frac{2}{4} = \frac{1}{2}\) after simplification.

Combine the Results

Now, combine the whole number part 8 with the new fractional part \(\frac{1}{2}\). The final result of \(9 \frac{1}{4} - \frac{3}{4}\) is \(8 \frac{1}{2}\).

Key concepts

Borrowing in Fractions

Borrowing in fractions is essential when the fractional part of a mixed number is smaller than the fraction being subtracted. In our case, we had \(9 \frac{1}{4}\), and we needed to subtract \(rac{3}{4}\). Since \( rac{1}{4} \) is not enough to subtract \( rac{3}{4} \), we need to "borrow" from the whole number part.

The process works by decreasing the whole number (in this case, 9) by one to make it 8. Then, the 1 we borrowed is turned into a fraction equivalent to 1, which is \( rac{4}{4} \). This borrowed fraction is added to the original fractional part of the mixed number. In our example, we transform \( rac{1}{4} \) into \( rac{5}{4} \) by adding \( rac{4}{4} \).

This method is similar to borrowing in whole number subtraction, where you take from a higher place value to allow the subtraction to happen smoothly.

Converting Mixed Numbers

Converting mixed numbers involves changing the composition of the number for easier calculations. In the exercise, we converted the original mixed number \(9 \frac{1}{4}\) into \(8 \frac{5}{4}\) by borrowing. This step ensures that the subtraction of the fractions becomes manageable.

When you convert a mixed number, you essentially reimagine the fraction part by borrowing from the whole number. The entirety of 1 taken from the whole 9 is expressed as an equivalent fraction, \( rac{4}{4} \), which can then be combined with the existing fraction part of \( rac{1}{4} \).

The result is a larger fraction that is easier to work with when performing the subtraction, while the whole number portion is adjusted accordingly. Converting mixed numbers this way often simplifies operations, saving you from more complex fractional arithmetic.

Subtraction of Fractions

Subtraction of fractions requires that both fractions have the same denominator. In our example, the fractions \( rac{5}{4} \) and \( rac{3}{4} \) already share the common denominator of 4, making it straightforward to perform the subtraction.

To subtract fractions, simply subtract the numerators while keeping the denominator the same: \( rac{5}{4} - rac{3}{4} = rac{2}{4} \). After performing the subtraction, the result may need further processing, such as simplification.

This approach not only keeps calculations simple but also clears the path for any further simplifications or adjustments required to reach a proper or simplest form of the fraction result.

Simplifying Fractions

Simplifying fractions involves reducing the fraction to its simplest form. This is done by dividing both the numerator and the denominator by their greatest common divisor (GCD).

After subtracting \( rac{3}{4} \) from \( rac{5}{4} \), we were left with the fraction \( rac{2}{4}\). To simplify, note that the GCD of 2 and 4 is 2. Dividing the numerator and the denominator by 2 gives \( rac{2}{4} = rac{1}{2} \).

Simplifying helps express the fraction in its most reduced form, making it easier to understand and utilize in further calculations. This final step ensures the result is neat and fully optimized for future use or interpretation.