Question

Perform the indicated operation. \(11-6 \frac{3}{7}\)

Short answer

The result is \(\frac{32}{7}\) or \(4 \frac{4}{7}\).

Step-by-step solution

Convert the Mixed Number to an Improper Fraction

The mixed number \(6 \frac{3}{7}\) is made up of the whole number 6 and the fraction \(\frac{3}{7}\). To convert this mixed number into an improper fraction, use the formula: \( \text{whole number} \times \text{denominator} + \text{numerator} \). Apply this formula: \(6 \times 7 + 3 = 42 + 3 = 45\). So, \(6 \frac{3}{7}\) becomes \(\frac{45}{7}\).

Subtract the Improper Fraction from the Whole Number

Rewrite the whole number 11 as a fraction with the denominator 7: \(\frac{77}{7}\). Now, perform the subtraction operation with fractions: \(\frac{77}{7} - \frac{45}{7}\). Subtract the numerators: \(77 - 45 = 32\), keeping the denominator the same: \(\frac{32}{7}\).

Simplify the Resulting Fraction

The fraction \(\frac{32}{7}\) is already in its simplest form. However, if needed, it can be expressed as a mixed number. Divide 32 by 7, which goes 4 times with a remainder of 4: \(4 \frac{4}{7}\).

Key concepts

Understanding Improper Fractions

Improper fractions are a type of fraction where the numerator is larger than or equal to the denominator. This means the top number (numerator) represents a quantity that is more than a whole. For example, the fraction \(\frac{45}{7}\) is an improper fraction because 45 is greater than 7. When working with improper fractions, particularly in arithmetic operations like addition or subtraction, it can be easier to understand their value by converting them into mixed numbers. However, in operations where you're asked to subtract them from whole numbers or other fractions, you'll often keep them as improper fractions to simplify calculations. Improper fractions, as complicated as they might first appear, are essential for performing various operations between fractions and whole numbers.

The Mixed Numbers Connection

Mixed numbers combine whole numbers and fractions, which can provide a more intuitive understanding of quantities greater than a whole. For example, \(6 \frac{3}{7}\) is a mixed number, with 6 as the whole part and \(\frac{3}{7}\) as the fractional part. To manage arithmetic operations like subtraction more efficiently, it's often necessary to convert mixed numbers into improper fractions. This conversion is done using the formula: - Multiply the whole number by the denominator.- Add the numerator to the result.- Place this sum over the original denominator to form an improper fraction.Using this method, the operation \(6 \frac{3}{7}\) becomes \(\frac{45}{7}\), which can be more straightforward for calculations.

Efficient Fraction Subtraction

Fraction subtraction might seem daunting at first, but it follows straightforward rules that can make the task manageable. To subtract fractions effectively, ensure that the denominators (the bottom numbers) match. If you are subtracting an improper fraction from a whole number, like in the original exercise, first convert the whole number to a fraction with the same denominator.In our exercise, to subtract \(\frac{45}{7}\) from 11, we rewrite 11 as \(\frac{77}{7}\). Now, with matching denominators, subtract the numerators: \(77 - 45 = 32\).This gives us a new fraction, \(\frac{32}{7}\), which represents the solution. The subtraction process highlights how understanding the equivalence of whole numbers and fractions can ease the complexity of operations involving fractions.

Simplifying Fractions Explained

Simplifying fractions involves reducing them to their simplest form, where the numerator and the denominator have no common factors other than 1. However, sometimes fractions might already be in their simplest form.Let's consider the fraction \(\frac{32}{7}\) from our solution. Since 32 and 7 have no common factors besides 1, it's already simplified. However, you can always convert it into a mixed number if needed for easier interpretation or preference.To do this, divide 32 by 7:- The whole number result is 4, as 32 divided by 7 goes 4 full times.- There is a remainder of 4, which forms the fractional part: \(\frac{4}{7}\).Thus, \(\frac{32}{7}\) can also be written as \(4 \frac{4}{7}\). Simplifying fractions isn't always about reducing them; it can also mean expressing them in different, sometimes more understandable formats.