Lesson Objectives

- Demonstrate an understanding of how to find the LCD
- Demonstrate an understanding of inequality relationships using "<, >, or ="
- Learn how to compare the size of fractions with the LCD method
- Learn how to compare the size of fractions with the same numerator
- Learn how to compare the size of fractions with cross products

## How to Compare Fractions

When working with two whole numbers or integers, it is easy to identify the larger and smaller number. The relationship between two fractions is not always so easy to ascertain. Let’s suppose we were asked to determine which is larger: 5/7 or 7/9? $$\frac{5}{7}\hspace{.25em}? \hspace{.25em}\frac{7}{9}$$ There are a few ways to approach this problem. First and foremost, we could convert each fraction into an equivalent fraction with the LCD as its denominator:

LCD = LCM(7, 9) = 63 $$\frac{5}{7}\cdot \frac{9}{9}=\frac{45}{63}$$ $$\frac{7}{9}\cdot \frac{7}{7}=\frac{49}{63}$$ Now that we have a common denominator, we can easily determine which fraction is larger. If the denominators are the same, the larger numerator belongs to the larger fraction. The reason for this is quite simple. Fractions represent a division of the numerator by the denominator. For a given denominator, as the numerator increases, the result will be larger. As the numerator decreases, the result will be smaller. Therefore if the denominators are the same, we can compare numerators. $$\frac{45}{63}\hspace{.25em}?\hspace{.25em}\frac{49}{63}$$ 45 is smaller than 49 $$45 < 49$$ $$\frac{45}{63}\hspace{.25em}<\hspace{.25em}\frac{49}{63}$$ $$\frac{5}{7}\hspace{.25em}< \hspace{.25em}\frac{7}{9}$$ In some cases, we will have a situation where the numerators are the same and the denominators are different. In this case, the smaller denominator belongs to the larger number. This is because we are dividing the same numerator by a smaller amount and the result is a larger number. As an example, suppose we are asked to determine which is larger: 9/20 or 9/13 $$\frac{9}{20}\hspace{.25em}?\hspace{.25em}\frac{9}{13}$$ We can see that the numerators are the same (9), so the smaller denominator (13) belongs to the larger number. Dividing 9 by 13 gives us a larger result than dividing 9 by 20. $$\frac{9}{20}< \frac{9}{13}$$ Let's take a look at a few examples.

Example 1: Replace the ? with the correct inequality symbol: $$\frac{4}{15}\hspace{.25em}?\hspace{.25em}\frac{6}{23}$$ LCD = LCM(15, 23) = 345

Convert each into an equivalent fraction where 345 is the denominator $$\frac{4}{15}\cdot \frac{23}{23}=\frac{92}{345}$$ $$\frac{6}{23}\cdot \frac{15}{15}=\frac{90}{345}$$ 92 is larger than 90 $$92 > 90$$ $$\frac{92}{345}\hspace{.25em}>\hspace{.25em}\frac{90}{345}$$ $$\frac{4}{15}\hspace{.25em}>\hspace{.25em}\frac{6}{23}$$ Example 2: Replace the ? with the correct inequality symbol: $$\frac{25}{13}\hspace{.25em}?\hspace{.25em}\frac{25}{19}$$ In this case, we have the same numerator (25). We only need to compare denominators. The smaller denominator (13) belongs to the larger fraction.

13 < 19

This tells us 25/13 will be larger than 25/19. Dividing 25 by 13 will result in a larger value than dividing 25 by 19. $$\frac{25}{13}\hspace{.25em}>\hspace{.25em}\frac{25}{19}$$

5 x 3 = 15

4 x 4 = 16

The larger fraction will have the larger number beside it. In this case, 16 is beside 4/5, so it is the larger fraction: $$\frac{4}{5}> \frac{3}{4}$$ Why does this work? If we were to transform each fraction into an equivalent fraction where 20 is the denominator: $$\frac{4}{5}\cdot \frac{4}{4}=\frac{16}{20}$$ $$\frac{3}{4}\cdot \frac{5}{5}=\frac{15}{20}$$ Notice how we end up comparing the same two values: 16 and 15. This shortcut allows us to simply compare numerators without the extra effort of transforming each fraction. Let's try a few examples:

Example 3: Replace the ? with the correct inequality symbol: $$\frac{9}{22}\hspace{.25em}?\hspace{.25em}\frac{8}{17}$$ Cross Multiply:

17 x 9 = 153

22 x 8 = 176

This tells us that 8/17 is larger since the larger number (176) is next to 8/17. $$\frac{9}{22}\hspace{.25em}<\hspace{.25em}\frac{8}{17}$$ Example 4: Place the fractions in order from least to greatest $$\frac{2}{5}, \frac{3}{4}, \frac{12}{13}$$ Start by comparing any two fractions. Let's compare 2/5 and 3/4 $$\frac{2}{5}\hspace{.25em}?\hspace{.25em}\frac{3}{4}$$ Cross Multiply: 5 x 3 = 15

4 x 2 = 8

This tells us that 3/4 is larger since the larger number (15) is next to 3/4. $$\frac{2}{5}<\frac{3}{4}$$ Now let's compare 3/4 to 12/13. $$\frac{3}{4}\hspace{.25em}?\hspace{.25em}\frac{12}{13}$$ Cross Multiply: 13 x 3 = 39

4 x 12 = 48

This tells us that 12/13 is larger since the larger number (48) is next to 12/13. $$\frac{2}{5}<\frac{3}{4}< \frac{12}{13}$$ The correct order from least to greatest: $$\frac{2}{5}, \frac{3}{4}, \frac{12}{13}$$

LCD = LCM(7, 9) = 63 $$\frac{5}{7}\cdot \frac{9}{9}=\frac{45}{63}$$ $$\frac{7}{9}\cdot \frac{7}{7}=\frac{49}{63}$$ Now that we have a common denominator, we can easily determine which fraction is larger. If the denominators are the same, the larger numerator belongs to the larger fraction. The reason for this is quite simple. Fractions represent a division of the numerator by the denominator. For a given denominator, as the numerator increases, the result will be larger. As the numerator decreases, the result will be smaller. Therefore if the denominators are the same, we can compare numerators. $$\frac{45}{63}\hspace{.25em}?\hspace{.25em}\frac{49}{63}$$ 45 is smaller than 49 $$45 < 49$$ $$\frac{45}{63}\hspace{.25em}<\hspace{.25em}\frac{49}{63}$$ $$\frac{5}{7}\hspace{.25em}< \hspace{.25em}\frac{7}{9}$$ In some cases, we will have a situation where the numerators are the same and the denominators are different. In this case, the smaller denominator belongs to the larger number. This is because we are dividing the same numerator by a smaller amount and the result is a larger number. As an example, suppose we are asked to determine which is larger: 9/20 or 9/13 $$\frac{9}{20}\hspace{.25em}?\hspace{.25em}\frac{9}{13}$$ We can see that the numerators are the same (9), so the smaller denominator (13) belongs to the larger number. Dividing 9 by 13 gives us a larger result than dividing 9 by 20. $$\frac{9}{20}< \frac{9}{13}$$ Let's take a look at a few examples.

Example 1: Replace the ? with the correct inequality symbol: $$\frac{4}{15}\hspace{.25em}?\hspace{.25em}\frac{6}{23}$$ LCD = LCM(15, 23) = 345

Convert each into an equivalent fraction where 345 is the denominator $$\frac{4}{15}\cdot \frac{23}{23}=\frac{92}{345}$$ $$\frac{6}{23}\cdot \frac{15}{15}=\frac{90}{345}$$ 92 is larger than 90 $$92 > 90$$ $$\frac{92}{345}\hspace{.25em}>\hspace{.25em}\frac{90}{345}$$ $$\frac{4}{15}\hspace{.25em}>\hspace{.25em}\frac{6}{23}$$ Example 2: Replace the ? with the correct inequality symbol: $$\frac{25}{13}\hspace{.25em}?\hspace{.25em}\frac{25}{19}$$ In this case, we have the same numerator (25). We only need to compare denominators. The smaller denominator (13) belongs to the larger fraction.

13 < 19

This tells us 25/13 will be larger than 25/19. Dividing 25 by 13 will result in a larger value than dividing 25 by 19. $$\frac{25}{13}\hspace{.25em}>\hspace{.25em}\frac{25}{19}$$

### Comparing Fractions by Cross Multiplying

We don't need to find the LCD and transform fractions to compare the size. Finding the cross products allows us to quickly determine which fraction is larger. When we discuss cross products, we are referring to the denominator of one fraction times the numerator of the other. Let's suppose we wanted to find the cross products of: 4/5 and 3/4: We can see that we simply multiply:5 x 3 = 15

4 x 4 = 16

The larger fraction will have the larger number beside it. In this case, 16 is beside 4/5, so it is the larger fraction: $$\frac{4}{5}> \frac{3}{4}$$ Why does this work? If we were to transform each fraction into an equivalent fraction where 20 is the denominator: $$\frac{4}{5}\cdot \frac{4}{4}=\frac{16}{20}$$ $$\frac{3}{4}\cdot \frac{5}{5}=\frac{15}{20}$$ Notice how we end up comparing the same two values: 16 and 15. This shortcut allows us to simply compare numerators without the extra effort of transforming each fraction. Let's try a few examples:

Example 3: Replace the ? with the correct inequality symbol: $$\frac{9}{22}\hspace{.25em}?\hspace{.25em}\frac{8}{17}$$ Cross Multiply:

17 x 9 = 153

22 x 8 = 176

This tells us that 8/17 is larger since the larger number (176) is next to 8/17. $$\frac{9}{22}\hspace{.25em}<\hspace{.25em}\frac{8}{17}$$ Example 4: Place the fractions in order from least to greatest $$\frac{2}{5}, \frac{3}{4}, \frac{12}{13}$$ Start by comparing any two fractions. Let's compare 2/5 and 3/4 $$\frac{2}{5}\hspace{.25em}?\hspace{.25em}\frac{3}{4}$$ Cross Multiply: 5 x 3 = 15

4 x 2 = 8

This tells us that 3/4 is larger since the larger number (15) is next to 3/4. $$\frac{2}{5}<\frac{3}{4}$$ Now let's compare 3/4 to 12/13. $$\frac{3}{4}\hspace{.25em}?\hspace{.25em}\frac{12}{13}$$ Cross Multiply: 13 x 3 = 39

4 x 12 = 48

This tells us that 12/13 is larger since the larger number (48) is next to 12/13. $$\frac{2}{5}<\frac{3}{4}< \frac{12}{13}$$ The correct order from least to greatest: $$\frac{2}{5}, \frac{3}{4}, \frac{12}{13}$$

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