Fractions Review: Videos & Practice Problems

To simplify fractions, identify common factors in the numerator and denominator, dividing until no further simplification is possible. For comparison, convert fractions to decimals by dividing the numerator by the denominator, making it easier to determine which is larger. For example, 3/4 equals 0.75 and 4/5 equals 0.8, indicating 4/5 is larger. Understanding these concepts aids in grasping the relationships between fractions, enhancing skills in mathematical reasoning and problem-solving.

Simplifying Fractions

When simplifying fractions, the first step is to identify a common factor that both the numerator (the top part of the fraction) and the denominator (the bottom part) can be divided by. This process involves repeatedly dividing both parts of the fraction by their common factors until no further simplification is possible. For instance, if you have the fraction \(\frac{8}{6}\), you can see that both numbers are even, allowing you to divide by 2, resulting in \(\frac{4}{3}\), which is in its simplest form since 4 and 3 share no common factors.

Another useful technique is to look for zeros at the end of numbers. For example, in the fraction \(\frac{120}{140}\), you can remove the zeros to simplify it to \(\frac{12}{14}\). Again, both numbers are even, so dividing by 2 gives you \(\frac{6}{7}\), which cannot be simplified further.

In cases where the numbers are not even, such as \(\frac{25}{50}\), you can find a common factor by testing divisibility. Here, dividing both by 5 results in \(\frac{5}{10}\), which can be simplified further to \(\frac{1}{2}\).

For fractions like \(\frac{9}{36}\), you can divide both by 3 to get \(\frac{3}{12}\), and then again by 3 to arrive at \(\frac{1}{4}\). This method emphasizes the importance of recognizing common factors, which can sometimes be larger than 2 or 3, as seen when simplifying \(\frac{21000}{140}\) by first removing zeros and then dividing by 7.

Lastly, for fractions like \(\frac{128}{1024}\), repeatedly dividing by 2 leads to \(\frac{1}{8}\), demonstrating that consistent application of common factors can simplify even larger numbers effectively. The key takeaway is to always look for common factors, especially 2, 3, and 5, to simplify fractions efficiently.

Comparing Fractions

When comparing fractions, a straightforward method is to convert each fraction into a decimal by dividing the numerator (the top number) by the denominator (the bottom number). This approach simplifies the comparison, as it is often easier to evaluate decimal values than to compare fractions directly.

For example, to compare the fractions \( \frac{3}{4} \) and \( \frac{4}{5} \), you would perform the following calculations:

1. \( \frac{3}{4} = 3 \div 4 = 0.75 \)

2. \( \frac{4}{5} = 4 \div 5 = 0.80 \)

Since \( 0.80 \) is greater than \( 0.75 \), it follows that \( \frac{4}{5} \) is the larger fraction.

Continuing with another example, consider \( \frac{46}{15} \) and \( \frac{13}{4} \):

1. \( \frac{46}{15} = 46 \div 15 \approx 3.07 \)

2. \( \frac{13}{4} = 13 \div 4 = 3.25 \)

Here, \( 3.25 \) is greater than \( 3.07 \), indicating that \( \frac{13}{4} \) is the larger fraction.

For the fractions \( \frac{2}{17} \) and \( \frac{3}{29} \):

1. \( \frac{2}{17} = 2 \div 17 \approx 0.118 \)

2. \( \frac{3}{29} = 3 \div 29 \approx 0.103 \)

In this case, \( 0.118 \) is greater than \( 0.103 \), so \( \frac{2}{17} \) is the larger fraction.

Lastly, when comparing \( \frac{25}{40} \) and \( \frac{60}{96} \):

1. \( \frac{25}{40} = 25 \div 40 = 0.625 \)

2. \( \frac{60}{96} = 60 \div 96 = 0.625 \)

Both fractions yield the same decimal value, indicating that they are equal.

This method of converting fractions to decimals is a reliable technique for determining which fraction is larger or if they are equal, enhancing your ability to compare fractions effectively.

Here’s what students ask on this topic:

How do you simplify fractions step by step?

To simplify fractions, follow these steps: First, identify a common factor for both the numerator and the denominator. A common factor is a number that both can be divided by. For example, in the fraction $\frac{8}{6}$, both 8 and 6 can be divided by 2. Divide both the numerator and the denominator by this common factor. Repeat this process until no further simplification is possible. For instance, $\frac{8}{6}$ simplifies to $\frac{4}{3}$ after dividing by 2. If both numbers end in zero, you can remove the zeros and simplify further. Continue until you reach the simplest form.

What is the easiest way to compare fractions?

The easiest way to compare fractions is to convert them to decimals. To do this, divide the numerator by the denominator. For example, to compare $\frac{3}{4}$ and $\frac{4}{5}$, divide 3 by 4 to get 0.75 and 4 by 5 to get 0.8. Since 0.8 is greater than 0.75, $\frac{4}{5}$ is larger than $\frac{3}{4}$. This method makes it easier to see which fraction is larger without dealing with the fractions directly.

What are common factors and how do they help in simplifying fractions?

Common factors are numbers that can evenly divide both the numerator and the denominator of a fraction. They help in simplifying fractions by reducing both parts of the fraction by the same factor, making the fraction simpler. For example, in the fraction $\frac{12}{14}$, the common factor is 2. Dividing both 12 and 14 by 2 simplifies the fraction to $\frac{6}{7}$. Identifying and using common factors repeatedly until no further simplification is possible results in the simplest form of the fraction.

How do you know when a fraction is fully simplified?

A fraction is fully simplified when the numerator and the denominator have no common factors other than 1. For example, in the fraction $\frac{4}{3}$, 4 and 3 have no common factors other than 1, so it is fully simplified. Another indicator is when the numerator is 1, as in $\frac{1}{2}$. If you cannot divide both the numerator and the denominator by the same number anymore, the fraction is in its simplest form.

Can you simplify a fraction by dividing by a non-prime number?

Yes, you can simplify a fraction by dividing by any common factor, whether it is a prime number or not. For example, in the fraction $\frac{25}{50}$, you can divide both the numerator and the denominator by 5, which is a prime number, to get $\frac{5}{10}$. You can further simplify by dividing by 5 again to get $\frac{1}{2}$. The key is to find any common factor that can divide both parts of the fraction evenly.