5.12: Fundamentals 11B - Fractions–Adding and Subtracting
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By the end of this lesson, you should understand
- fractions with the same denominator can be added (or subtracted) by adding (or subtracting) the numerators and keeping the common denominator
- in order to add (or subtract) fractions with different denominators, we must find a common denominator and convert the fractions to equivalent fractions with the common denominator
By the end of this lesson, you should be able to
- add and subtract any two fractions.
FUNDAMENTALS OF THE LESSON
Two fractions are said to be equivalent if they represent the same quantity. Equivalent fractions can be obtained by multiplying the numerator and denominator of a fraction by the same number. (Note: By doing so, we are essentially multiplying by “1.”)
Example: \(\dfrac{7}{4}\) and \(\dfrac{35}{20}\) are equivalent fractions. They both equal the decimal number 1.75. Also, when we multiply both the numerator and denominator of \(\dfrac{7}{4}\) by 5, we get the fraction . This is shown below.
\[\dfrac{7}{4} = \dfrac{7\times 5}{4\times 5} = \dfrac{35}{20}\nonumber \]
1. Every fraction has an infinite number of equivalent fractions. We can find additional fractions that are equivalent to \(\dfrac{7}{4}\) by multiplying the numerator and denominator by other numbers.
A. What is the equivalent fraction if you multiply the numerator and denominator of \(\dfrac{7}{4}\) by 2?
B. What is the equivalent fraction if you multiply the numerator and denominator of \(\dfrac{7}{4}\) by 4?
C. What is the equivalent fraction if you multiply the numerator and denominator of \(\dfrac{7}{4}\) by 10?
We will use the concept of equivalent fractions to add and subtract fractions when they have different denominators. But, before we do this, let’s explore some situations involving adding and subtracting fractions with common denominators.
2. Fill in the blanks in order to obtain the correct answer on the right:
A. 1 apple + 2 apples = _______ apples
B. 1 fourth + 2 fourths = _______ fourths
C. \(\dfrac{1}{4} + \dfrac{2}{4} = \dfrac{}{4}\)
D. \(\dfrac{2}{5} + \dfrac{3}{5} = \dfrac{}{5} =\)
E. \(\dfrac{6}{7} - \dfrac{4}{7} = \dfrac{}{7}\)
Leftover Pizza
Sarah has some leftover pizza from a party. She has 1/3 of one pizza and 1/2 of another pizza. The pizzas, before being eaten, were the same size. She wants to feed 5 children who came over to eat the leftovers.
3. How can Sarah divide the leftover slices if she wants each child to get the same amount of pizza while at the same time giving away all of the pizza?
Hint: Try to figure your problem out with the help of pictures. What leftover amount is larger? Which one is smaller? How should you divide each of them into slices for the children?
4. Given how the pizza was divided above, how do you know that each of the five children will receive the same amount? How can you show mathematically that the amount of pizza that each of the five children receive are the same?
5. Write fractions that represent the following amounts of pizza
- The smallest leftover amount was _______ of the pizza.
- The largest leftover amount was _______ of the pizza.
- The total leftover amount before the children ate it all was ______ of the two pizzas.
- The amount each child received was ______ of the pizzas.
6. What do you notice about the denominators of your answer to the previous question? How are they related?
The least common multiple (LCM) of the denominators of two fractions is called the least common denominator (LCD) of the two fractions. 6 is the LCD of 1/3 and 1/2. To add ⅓ and ½, we must convert these fractions to equivalent fractions with 6 in the denominator.
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To find the fraction equivalent to \(\dfrac{1}{2}\) with the 6 in the denominator, we must multiply the denominator (2) by 3. So, we multiply the numerator and denominator by 3. |
To find the fraction equivalent to \(\dfrac{1}{3}\) with the 6 in the denominator, we must multiply the denominator (3) by 3. So, we multiply the numerator and denominator by 3. |
7. The sum of \(\dfrac{1}{2} + \dfrac{1}{3}\) is equivalent to the sum \(\dfrac{3}{6} + \dfrac{2}{6}\). What is the sum of these fractions? This sum represents the total amount of pizza left over.
8. Let’s look at a different example. Find the LCD of the fractions 5/4 and 7/6.
9. Add 5/4 and 7/6.
A. What numbers should we multiply the top and bottom of each fraction by so the products have the least common denominator of 12?
\[\dfrac{5}{4} + \dfrac{7}{6} = \dfrac{5\times\;}{4\times\;} + \dfrac{7\times\;}{6\times\;} = \dfrac{}{12} + \dfrac{}{12}\nonumber \]
B. After multiplying, what are the equivalent fractions?
C. What is the sum?
NEXT STEPS
More Leftover Pizza
Paul ordered one pepperoni pizza and one cheese pizza, both of the same size. He had \(\dfrac{2}{3}\) of the pepperoni pizza left over, and had \(\dfrac{3}{4}\) of the cheese pizza left over. Then Luis and Akisha came to visit. He gave the pepperoni pizza leftovers to Luis and gave the cheese pizza leftovers to Akisha. He wants to determine how much more pizza Akisha received than Luis.
10. What mathematical operation does Paul need to perform to solve this problem?
11. To subtract \(\dfrac{2}{3}\) from \(\dfrac{3}{4}\), we must convert the fractions to equivalent fractions with a common denominator.
A. What is the least common denominator (LCD) of the fractions \(\dfrac{2}{3}\) and \(\dfrac{3}{4}\)?
B. \(\dfrac{3}{4} - \dfrac{2}{3} = \dfrac{3\times\;}{4\times\;} - \dfrac{2\times\;}{3\times\;}\)
C. After multiplying, what are the equivalent fractions?
D. What is the difference?
TRY THESE
Similar to negative integer numbers, there are also negative fractions. We can also use the LCD of negative fractions to carry out an addition or a subtraction problem, and the negative sign is assigned to the numerator of the fraction. Below is an example:
\(-\dfrac{8}{9} + \dfrac{5}{6} = -\dfrac{8\times 2}{9\times 2} + \dfrac{5\times 3}{6\times 3} = \dfrac{-16 + 15}{18} = \dfrac{-1}{18} = -\dfrac{1}{18}\)
We can write negative fractions in several ways, because there are several expressions involving a division that yield the same result. For example, the fraction \(-\dfrac{1}{18}\) can be written in the following three ways.
12. Hannah bought 3 pizzas for a family gathering. Her relative ate \(\dfrac{11}{6}\) of the 3 pizzas. How much pizza does Hannah have left?
13. Perform the following calculations:
A. \(-\dfrac{14}{3} + \dfrac{5}{3} =\)
B. \(-\dfrac{5}{7} + (-\dfrac{4}{3}) =\)
C. \(2 - \dfrac{13}{5} =\)
D. \(-\dfrac{3}{2} - ( -\dfrac{11}{4}) =\)
E. \(-\dfrac{7}{6} - \dfrac{3}{8} =\)
Questions: Fractions–Adding and Subtracting
1. For each pair of fractions, find the least common denominator (LCD). Hint: What are the multiples of the denominators? What is the least common multiple?
A. Find the LCD of \(\dfrac{2}{7}\) and \(\dfrac{1}{6}\).
B. Find the LCD of \(\dfrac{1}{9}\) and \(\dfrac{2}{6}\).
C. Find the LCD of \(\dfrac{1}{6}\) and \(\dfrac{1}{3}\).
D. Find the LCD of \(\dfrac{6}{25}\) and \(\dfrac{5}{10}\).
2. Liz needs to use \(\dfrac{3}{4}\) of a cup of flour to make a sponge cake. She wants to cover it with a cream recipe that calls for \(\dfrac{1}{6}\) of a cup of flour.
A. How much flour will she need in total?
B. Liz has 4 cups of flour. If she uses \(\dfrac{11}{12}\) cups of flour for the cake recipe, how much flour will she have left over? Write your answer as a fraction.
3. Lou bought \(\dfrac{3}{4}\) of a meter of ribbon to decorate a vase, then he found out he only needed \(\dfrac{1}{6}\) of a meter. How much ribbon did he have left after he finished his decoration?
4. Jamal decided to make a long trip on foot. On the first day he walked \(\dfrac{2}{9}\) of the way, but he got really tired. On the second day he decided to hitchhike and traveled \(\dfrac{1}{4}\) of the way.
A. What total fraction of the trip did Jamal cover on the first two days?
B. What fraction of the trip does he still need to travel in order to finish his journey?
5. Carla decided to share 14 yards of lace she inherited from her mother with her sisters. She decided to give them the amount they needed for their projects: \(\dfrac{11}{4}\) of a yard to Maria, \(\dfrac{8}{3}\) of a yard to Cecilia and \(\dfrac{6}{5}\) of a yard to Ana.
A. How much lace did Carla give away? Write as a fraction.
B. How much lace does she have left? Write as a fraction.