35 Addition and Subtraction of Fractions
Standards about the four operations of fractions:
4.NF.B. Build fractions from unit fractions by applying and extending previous understandings of operations on whole numbers.
3c. Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction.
5.NF.A. Use equivalent fractions as a strategy to add and subtract fractions.
1. Add and subtract fractions with unlike denominators (including mixed numbers) by replacing given fractions with equivalent fractions in such a way as to produce an equivalent sum or difference of fractions with like denominators. For example, [latex]\frac{2}{3}+\frac{5}{4}=...[/latex] [latex]\frac{2}{3}+\frac{5}{4}=\frac{8}{12}+\frac{15}{12}=\frac{23}{12}[/latex]. (In general, [latex]\frac{a}{b}+\frac{c}{d}=\frac{(ad+bc)}{bd}[/latex].)
4.NF.B.4. Multiply a fraction by a whole number. (Denominators are limited to[latex]2, 3, 4, 5, 6, 8, 10, 12,[/latex] and [latex]100[/latex].)
5.NF.B.4. Apply and extend previous understandings of multiplication to multiply a fraction or whole number by a fraction.
a. Interpret the product ([latex]\frac{m}{n}[/latex]) [latex]\times q[/latex] as [latex]m[/latex] parts of a partition of [latex]q[/latex] into [latex]n[/latex] equal parts; equivalently, as the result of a sequence of operations, [latex]m \times q \div n[/latex]. For example, use a visual fraction model to show understanding, and create a story context for [latex](\frac{m}{n})\times q[/latex].
b. Construct a model to develop understanding of the concept of multiplying two fractions and create a story context for the equation. [In general, [latex](\frac{m}{n}) \times (\frac{c}{d}) = \frac{(mc)}{(nd)}[/latex].]
5.NF.B.7. Apply and extend previous understandings of division to divide unit fractions by whole numbers and whole numbers by unit fractions.1
6.NS.A. Apply and extend previous understandings of multiplication and division to divide fractions by fractions.
As we do with other type of numbers, we can add or subtract two fractions. When two fractions are with a common denominator, the addition and subtraction of them are very much like the addition and subtraction of whole numbers or integer, which is demonstrated by examples below.
Example 1 Add and Subtraction Fractions with Common Denominator
Consider two fractions, [latex]\frac{7}{11}[/latex] and [latex]\frac{3}{11}[/latex].
a. If the [latex]C[/latex]-strip [latex]S[/latex] (with a length of [latex]11[/latex]) represents one unit, which strips represent [latex]\frac{7}{11}[/latex] and [latex]\frac{3}{11}[/latex] respectively?
b. Which strips represent the sum and the difference of them?
c. Write the sum and difference as a fraction when the [latex]C[/latex]-strip [latex]S[/latex] represents one unit.
Solution:
a. If the [latex]C[/latex]-strip [latex]S[/latex] (with a length of [latex]11[/latex]) represents one unit, then [latex]K[/latex] (with a length of [latex]7[/latex]) represents [latex]\frac{7}{11}[/latex]; and [latex]L[/latex] (with a length of [latex]3[/latex]) represents [latex]\frac{3}{11}[/latex].
[Figures of [latex]K[/latex], [latex]V[/latex], compared to [latex]S[/latex]]
b. The sum of [latex]K[/latex] and [latex]L[/latex] can be represented by [latex]O[/latex] (with a length of [latex]10[/latex]), and the different of them can be represented by [latex]P[/latex] (with a length of [latex]4[/latex]).
c. When the [latex]C[/latex]-strip [latex]S[/latex] represents one unit, the the sum [latex]O[/latex] represents [latex]\frac{10}{11}[/latex], and the difference [latex]P[/latex] represents [latex]\frac{4}{11}[/latex].
The example above shows us that
[latex]\frac{7}{11}[/latex] [latex]+[/latex] [latex]\frac{3}{11}[/latex] = [latex]\frac{10}{11}[/latex]
[latex]\frac{7}{11} - \frac{3}{11} = \frac{4}{11}[/latex].
The same denominator indicates the same unit, so when we add or subtract two fractions with a common denominator, we just keep the denominator and add or subtract the numerators to get the new numerator. In general, we can write the rule as below:
[latex]\frac{a}{b} + \frac{c}{d} = \frac{(a+c)}{b}[/latex]
[latex]\frac{a}{b} - \frac{c}{d} = \frac{(a-c)}{b}[/latex]
When two fractions are not with a common denominator, we cannot add or subtract them directly. Let's consider [latex]\frac{1}{3}[/latex] and [latex]\frac{1}{2}[/latex]. For [latex]\frac{1}{3}[/latex], we may use [latex]L[/latex] as one unit and the one [latex]W[/latex] then represents [latex]\frac{1}{3}[/latex]. For [latex]\frac{1}{2}[/latex], using [latex]R[/latex] as one unit, then one [latex]W[/latex] also represents [latex]\frac{1}{2}[/latex]. In this way, we can still find the sum of the two [latex]W[/latex]s is the length of [latex]2[/latex], or the [latex]R[/latex]-strip. However, which [latex]C[/latex]-strip should we use to represent one unit? Notice that [latex]\frac{1}{3}[/latex] and [latex]\frac{1}{2}[/latex] have two different units! When we check back the example above, we can find out the key there is the same unit - we always use [latex]S[/latex] to represent one unit.
Then, can we represent [latex]\frac{1}{3}[/latex] and [latex]\frac{1}{2}[/latex] using the same unit? This requires a unit that is divisible by both [latex]3[/latex] and [latex]2[/latex]. In other words, the unit should be one of the common multiples of [latex]3[/latex] and [latex]2[/latex]. For convenience, we usually choose the least common multiple ([latex]LCMO[/latex] of the two denominators, [latex]6[/latex]. Using the [latex]C[/latex]-strip [latex]D[/latex] to represent one unit, then [latex]\frac{1}{3}[/latex] can be represented by [latex]R[/latex] and [latex]\frac{1}{2}[/latex] can be represented by [latex]L[/latex]. Therefore, the sum of [latex]\frac{1}{3}[/latex] and [latex]\frac{1}{2}[/latex]
would be represented by a [latex]C[/latex]-strip with length [latex]5[/latex], that is, [latex]Y[/latex], which represents [latex]\frac{5}{6}[/latex] when [latex]D[/latex] is a unit.
In summary, what we've done with the [latex]C[/latex]-strips can be written as
[latex]\frac{1}{3} + \frac{1}{2} = \frac{2}{6} + \frac{3}{6} = \frac{5}{6}[/latex].
The dot model is another way to visualize this process, especially when the common denominator is beyond [latex]12[/latex].
Example 2
Use the dot model to add [latex]\frac{2}{5}[/latex] and [latex]\frac{3}{7}[/latex].
Solution
Step 1: Let 1 unit = 5 rows of dots by 7 columns of dots, for a total of 35 dots, as shown below. Since there are 35 dots to a unit, each dot = 1/35 of a unit.
[figure of dots in 5 rows by 7 columns]
Step 2: Since there are 5 rows, the unit can be broken up into 5 equal parts by circling the rows, as shown below:
[figure of dots in five groups]
Therefore, each row is 1/5 of a unit. Notice there are 7 dots in 1/5 of a unit:
1/5 = [figure of 7 dots in a row]
Step 3: Similarly, since there are 7 equal columns, each column is 1/7 of a unit:
1/7 = [figure of 5 dots in a column]
Step 4: Now that we have properly defined the unit fraction, 1/5, we are ready to show what 2/5 looks like. Since 1/5 is 1 row of dots, then 2/5 must be 2 rows of dots. Therefore, 2/5 is shown below:
[figure of 14 dots in two rows]
Notice that 25/ contains 14 dots, which is equivalent to 14/35.
Step 5: Similarly, we can show what 3/7 looks like. Since 1/7 is 1 column of dots, then 3/7 must be 3 columns of dots. Therefore, 3/7 is shown below:
[figure of 15 dots in 3 columns]
Notice that 3/7 contains 15 dots, which is equivalent to 15/35.
Step 6: Now, we add the two models together, as shown below.
[figure of 14 dots in two rows adding 15 dots in three columns]
That is, 2/5+3/7=293/5 .
In Example 1, we use the dot model to visualize the following math procedure:
2/5 + 3/7 =
Exercise
1.Add 1/4 and 2/5 using models. Show all of the steps, and explain the procedure as shown in the previous example.
Example 2 Use models to do the following subtraction: 3/7 – 2/5.
Solution
Step 1: Let 1 unit = 5 rows of dots by 7 columns of dots, for a total of 35 dots, as shown below. Since there are 35 dots to a unit, each dot = 1/35 of a unit.
Step 2: Since there are 5 rows, the unit can be broken up into 5 equal parts by circling the rows, as shown below:
Therefore, each row is 1/5 of a unit. Notice there are 7 dots in 1/5 of a unit:
1/5 =
Step 3: Similarly, since there are 7 equal columns, each column is 17 of a unit:
17 =
Step 4: Now that we have properly defined the unit, we are ready to show what 25 looks like. Since 15 is 1 row of dots, then 25 must be 2 rows of dots. Therefore, 25 is shown below:
Notice that 25 contains 14 dots, which is equivalent to 1435.
Step 5: Similarly, we can show what 37 looks like. Since 17 is 1 column of dots, then 37 must be 3 columns of dots. Therefore, 37 is shown below:
Notice that 37 contains 15 dots, which is equivalent to 1435.
Now, we subtract, as shown below. The answer is 1/35.
Exercise
- Do the following subtraction using models: 78 – 23. Show all of the steps, and explain the procedure as shown in the previous example.
summary
Mathematically, what was done above is to write the two fractions into their equivalent forms with a common denominator, then add the numerators. This is because, in order to add or subtract fractions that do not have a common denominator, we must first rewrite each fraction as an equivalent fraction so that both fractions have a common denominator. Sometimes it is convenient to use the product of the two denominators as the common denominator, as summarized below
a/b + c/d = ad/bc + bc/bc = (ad + bc)/bc
Note If you are going to add fractions by rewriting each fraction with a common denominator, it is usually preferable to find the least common denominator, which is the least common multiple of the two denominators. Whether or not you do this, you should always write the answer in simplest form after adding or subtracting.