Problem 52 - Prealgebra

Question

The following problems all involve the concept of borrowing. Subtract in case. \(7 \frac{1}{6}-6 \frac{5}{6}\)

Step-by-Step Solution

Answer

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

Step 1: Understand the Problem

We have the expression \(7 \frac{1}{6} - 6 \frac{5}{6}\). This means we need to subtract one mixed number from another. In this context, borrowing might be necessary because the fraction \(\frac{1}{6}\) is smaller than \(\frac{5}{6}\).

Step 2: Borrowing from the Whole Number

Since the fraction \(\frac{1}{6}\) in \(7 \frac{1}{6}\) is less than \(\frac{5}{6}\), we need to borrow 1 from the whole number 7. This makes the whole number 6 and adds \(\frac{6}{6}\) to the fraction part resulting in \(6 \frac{7}{6}\).

Step 3: Perform the Subtraction

Now subtract \(6 \frac{5}{6}\) from \(6 \frac{7}{6}\). Subtract the whole numbers: \(6 - 6 = 0\) and the fractions: \(\frac{7}{6} - \frac{5}{6} = \frac{2}{6}\), which simplifies to \(\frac{1}{3}\).

Step 4: Simplify the Fraction

Simplify \(\frac{2}{6}\) by dividing the numerator and the denominator by their greatest common divisor, which is 2. This simplifies the fraction to \(\frac{1}{3}\). Thus, the answer to the subtraction is \(\frac{1}{3}\).

Key Concepts

Borrowing in Subtraction

When subtracting mixed numbers, borrowing can become necessary, especially when the fractional part of the first number is smaller than the fractional part of the second.
This is similar to borrowing in basic arithmetic when subtracting whole numbers. In our example, with the expression \(7 \frac{1}{6} - 6 \frac{5}{6}\), we notice \(\frac{1}{6}\) is smaller than \(\frac{5}{6}\).
To solve this, we "borrow" 1 from the whole number 7. This borrowing transforms 7 into 6 and adds an extra "whole" to the fraction.

• The new fraction becomes \(\frac{7}{6}\) because \(1 = \frac{6}{6}\), and we already had \(\frac{1}{6}\).
• Now, the expression looks like \(6 \frac{7}{6} - 6 \frac{5}{6}\).
• By borrowing in this way, the fraction is "fixed" so that subtraction can proceed smoothly.

Simplifying Fractions

Simplifying fractions is essential in ensuring your final answer is in its most reduced form. This step comes especially handy after subtracting, as in our previous result from the expression \(6 \frac{7}{6} - 6 \frac{5}{6}\).
After borrowing and subtracting we ended up with \(\frac{2}{6}\). We simplify fractions by dividing both the numerator and the denominator by their greatest common divisor (GCD).

• First, identify the GCD. For 2 and 6, the GCD is 2.
• Thus, divide both the numerator and the denominator by 2.
• The simplified fraction becomes \(\frac{1}{3}\).

Making sure to simplify your fractions not only keeps your answers clean but also ensures correctness in future calculations.

Mixed Numbers

Mixed numbers are numbers involving both a whole number and a fraction. They are frequently used in everyday life to represent quantities that are not whole.
An example of a mixed number is \(7 \frac{1}{6}\). Here, 7 is the whole number, and \(\frac{1}{6}\) is the fractional part. In mathematical operations such as subtraction, it is crucial to manage both parts correctly.

• Adding and subtracting mixed numbers involves handling both the whole and fractional parts.
• Sometimes, like in our problem, the fraction part isn't large enough to simply "subtract" another fraction from it, requiring strategies like borrowing.

Understanding how to manipulate mixed numbers ensures you can perform operations smoothly, while borrowing, simplifying fractions and combining results accurately.