Problem 41 - Algebra 1: Concepts and Skills

Question

Evaluate the expression. Write the answer as a fraction or as a mixed number in simplest form. (Skills Review pp. 764-765) $$ \frac{2}{7} \div \frac{1}{14}-\frac{5}{4} $$

Step-by-Step Solution

Answer

The simplest form of the expression is \(2 \frac{3}{4}\).

Step 1: Divide Fractions

Start with the division operation. To divide fractions, multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by switching the numerator and the denominator, i.e. the reciprocal of \( \frac{1}{14} \) is \( \frac{14}{1} \). Thus, the operation \( \frac{2}{7} \div \frac{1}{14} \) becomes \( \frac{2}{7} \times \frac{14}{1}\).

Step 2: Multiply Fractions

After changing the division operation to multiplication, perform the multiplication. Multiply the numerators together to get a new numerator and the denominators together to get a new denominator. Thus, \( \frac{2}{7} \times \frac{14}{1} \) equals \( \frac{2*14}{7*1} \) which simplifies to \( \frac{28}{7}\). Simplifying this fraction results in 4.

Step 3: Subtract Fractions

After evaluating the division operation, proceed with the subtraction operation. Subtract \( \frac{5}{4} \) from 4. As 4 is an integer, to subtract a fraction from it, convert it to a fraction first by writing it over 1. Thus, 4 becomes \( \frac{4}{1}\). Then, subtract: \( \frac{4}{1} - \frac{5}{4} \).

Step 4: Normalize Fractions and Subtract

Bring both fractions to a common denominator. The common denominator is the least common multiple (LCM) of the denominators. In this case, the LCM of 1 and 4 is 4. Change the fractions to equivalents with denominator 4: \( \frac{4}{1} \) becomes \( \frac{16}{4}\). Now subtract \( \frac{16}{4} - \frac{5}{4} \) which equals \( \frac{11}{4}\).

Step 5: Convert to Mixed Number

The result of the subtraction operation is \( \frac{11}{4}\), which is an improper fraction. To write it in simplest form, convert it to a mixed number. A mixed number has a whole part and a fractional part. Divide the numerator by the denominator: 11 ÷ 4 equals 2 remainder 3. This means the result is 2 whole and \( \frac{3}{4}\) of another whole, or \(2 \frac{3}{4}\).

Key Concepts

Dividing Fractions

Dividing fractions might sound tricky at first, but it's more straightforward than it seems. The key is to turn the division operation into multiplication.
To divide one fraction by another, you multiply by the reciprocal of the second fraction.
For example, if you have \( \frac{2}{7} \div \frac{1}{14} \), you need to take the reciprocal of \( \frac{1}{14} \), which means flipping it to get \( \frac{14}{1} \).

• First, write the first fraction down as it is.
• Next, change the division sign to a multiplication sign.
• Then, flip the second fraction, turning it upside down to get its reciprocal.

With these steps, we change our problem to \( \frac{2}{7} \times \frac{14}{1} \).
Multiplying fractions can be easier once you have this setup, moving on to the multiplication phase.

Subtracting Fractions

Subtracting fractions is all about finding a common denominator.
This common denominator allows you to combine them properly.
For instance, after finishing the division, we encounter \( 4 - \frac{5}{4} \). Since 4 is a whole number, we express it as a fraction: \( \frac{4}{1} \).

• Find the least common multiple (LCM) of the denominators.
• Convert each fraction to an equivalent fraction with the LCM as the denominator.

So, the LCM for 1 and 4 is 4, making our fraction \( \frac{4}{1} \equiv \frac{16}{4} \).
Now, with common denominators, subtract: \( \frac{16}{4} - \frac{5}{4} = \frac{11}{4} \).
Remember to always simplify, if possible.

Improper Fractions

An improper fraction has a numerator larger than the denominator, like \( \frac{11}{4} \).
Improper fractions tell us that the fraction is more than a whole because the numerator is bigger than or equal to the denominator.
These kinds of fractions can often be converted to something easier to understand.

• Check if the numerator is greater than the denominator.
• If so, divide the numerator by the denominator.

This division finds how many whole parts there are and what fraction is left over.
Improper fractions appear often in math tasks, so knowing how to handle them is useful.

Mixed Numbers

Mixed numbers combine a whole number with a fraction.
This format can make improper fractions simpler to understand.
For example, take the improper fraction \( \frac{11}{4} \). When we divide 11 by 4, we get 2 with a remainder of 3.

• First, identify the whole number: 11 divided by 4 equals 2 with a remainder of 3.
• The remainder becomes the numerator of the fractional part, over the original denominator.

This gives us the mixed number \( 2 \frac{3}{4} \).
Mixed numbers are often friendlier, especially in everyday contexts, as they show both the complete and fractional parts together.