If you missed any of the items for this unit, read the detailed solution below. You may also wish to review this unit. When you have reviewed all the detailed solutions, try the additional practice for this unit.

1. Complete the multiplication for each of the following:

   Rule for multiplication of fractions: When multiplying fractions, you simply multiply the numerators together, and then the denominators together. You should then simplify the result.

   1. 

      First we apply the multiplication rule.
      Often, it is easier to simplify before we complete the multiplication. We do this by factoring the numbers that are being multiplied together, and canceling out any common factors.

      By doing the factoring and canceling before multiplying, we make the multiplication and simplification easier.
      

       

   2. 

      First we apply the multiplication rule.
      The numbers in the numerator are already prime numbers, so we only need to factor the numbers in the denominator. Cancel out any common factors.

      Since 7 is a factor of 14, 5 is a factor of 10, and 3 is a factor of 12, some cancellation can occur. With practice, noticing possible common factors becomes easier.

    

2. Complete the division for each of the following:

   Rule for division of fractions: When you divide two fractions, you take the reciprocal of the second fraction and multiply. (Taking the reciprocal of a fraction means to flip it over.)

   1. 

      First we apply the division rule.
      The numbers in the denominator are already prime numbers, so we only need to factor the numbers in the numerator. Cancel out any common factors.

       

   2. 

      First we apply the division rule. To simplify, we need to factor the numbers in both the numerator and denominator. When we factor this time, we could factor each number down to its prime factors, but let's try factoring only as much as needed. For example, 99 can be factored into 11 and 9. The number 100 can be factored into 20 and 5.
      From here we can see that we can cancel the 20s and the 11s. This leaves us with a result of 9/5.

      
      
      
3. Complete the addition for each of the following:

   Rule for addition of fractions: When adding fractions, you must make sure that the fractions being added have the same denominator. If they do not, find the LCD for the fractions and put each in its equivalent form. Then simply add the numerators of the fractions.

   1. 

      Let's follow the steps for solving these kinds of equations.

      1. Determine whether the fractions have the same denominator. If the denominators are the same, move to step 4.
         They do not have the same denominator, so we must move to step 2.

      2. Find the LCD for the fractions being added.
         
         1. Write the prime factors for the denominator of each fraction.
            
            • The prime factor for 2 is: 2.
              
            • The prime factors for 8 are: 2, 2, and 2.

         2. Note all prime factors that occur. For each prime factor that occurs, determine in which denominator it occurs the most. Write down the prime factor the number of times it occurs in that one denominator.
            The prime factors that occur are 2, 2, and 2.

         3. Calculate the LCD of your fractions. To do this, multiply the factors selected in step 2b.
            2 x 2 x 2 = 8.
            8 is the LCD for our fractions.

      3. Find the equivalent fractions that have the LCD in the denominator.
         Start with 3/8. This fraction already has the LCD as its denominator, so we do not need to do anything.

         For 5/2, we can see that a 2 and 2 are missing in the denominator. Since 2 x 2 = 4, the multiplier for this fraction is 4. Our resulting equivalent fraction is:

         

      4. Add the numerators of the fractions.
         

      5. Simplify the resulting fraction.
         The fraction 23/8 is already in its simplest form, so that is our result.
         

      
      

   2. 
      

      Let's follow the steps for solving these kinds of equations.
      

      1. Determine whether the fractions have the same denominator. If the denominators are the same move to step 4.
         They do not have the same denominators so we must move to step 2.

      2. Find the LCD for the fractions being added.
         
         1. Write the prime factors for the denominator of each fraction.
            
            • The prime factors for 85 are: 5 and 17.
              
            • The prime factors for 45 are: 3, 3, and 5.

         2. Note all prime factors that occur. For each prime factor that occurs, determine in which denominator it occurs the most. Write down the prime factor the number of times it occurs in that one denominator.
            The prime factors that occur are 3, 3, 5, and 17.

         3. Calculate the LCD of your fractions. To do this, multiply the factors selected in step 2b.
            3 x 3 x 5 x 17 = 765.
            765 is the LCD of the fractions.

      3. Find the equivalent fractions that have the LCD in the denominator.
         Start with 13/85: The prime factors for 85 are 5 and 17. The prime factors missing from this denominator are 3 and 3. Since 3 x 3 = 9, the multiplier for this fraction is 9. Our resulting equivalent fraction is:
         

         For 5/45: The prime factors for 45 are: 3, 3, and 5. The only prime factor missing from this denominator is 17, so 17 is the multiplier for this fraction. Our resulting equivalent fraction is:

         

      4. Add the numerators of the fractions.
         

      5. Simplify the resulting fraction.
         
         

4. Complete the subtraction for each of the following:

   Rule for subtraction of fractions: When subtracting fractions, you must make sure that the fractions being subtracted have the same denominator. If they do not, find the LCD for the fractions and put each in its equivalent form. Then simply subtract the numerators of the fractions.

   1. 
      

      Let's follow the steps for solving these kinds of equations.
      

      1. Determine whether the fractions have the same denominator. If the denominators are the same, move to step 4.
         They do not have the same denominator so we must move to step 2.

      2. Find the LCD for the fractions being subtracted.
         
         1. Write the prime factors for the denominator of each fraction.
            
            • The prime factor for 3 is: 3.
              
            • The prime factors for 16 are: 2, 2, 2, and 2.

         2. Note all prime factors that occur. For each prime factor that occurs, determine in which denominator it occurs the most. Write down the prime factor the number of times it occurs in that one denominator.
            The prime factors that occur are 2, 2, 2, 2, and 3.

         3. Calculate the LCD of your fractions. To do this, multiply the factors selected in step 2b.
            2 x 2 x 2 x 2 x 3 = 48.
            48 is the LCD for our fractions.

      3. Find the equivalent fractions that have the LCD in the denominator.
         Start with 13/16: The prime factors for 16 are 2, 2, 2, and 2. The prime factor missing from this denominator is 3, so 3 is the multiplier for this fraction. Our resulting equivalent fraction is:
         

         For 2/3: The prime factor for 3 is just 3. The prime factors missing from this denominator are 2, 2, 2, and 2. Since 2 x 2 x 2 x 2 = 16, the multiplier for this fraction is 16. Our resulting equivalent fraction is:

         

      4. Subtract the numerators of the fractions.
         

      5. Simplify the resulting fraction.
         This fraction is in its simplest form, so 7/48 is our answer.

         
         

   2. 
      Let's follow the steps for solving these kinds of equations.
      
      1. Determine whether the fractions have the same denominator.
         If the denominators are the same, move to step 4.
         They do not have the same denominator, so we must move to step 2.

      2. Find the LCD for the fractions being subtracted.
         
         1. Write the prime factors for the denominator of each
            fraction.
            
            • The prime factor for 2 is: 2.
              
            • The prime factors for 8 are: 2, 2, and 2.

         2. Note all prime factors that occur. For each prime factor that occurs, determine in which denominator it occurs the most. Write down the prime factor the number of times it occurs in that one denominator.
            The prime factors that occur are 2, 2, and 2.

         3. Calculate the LCD of your fractions. To do this, multiply the factors selected in step 2b.
            2 x 2 x 2 = 8.
            8 is the LCD for our fractions.

      3. Find the equivalent fractions that have the LCD in the denominator.
         Start with 3/8: This fraction already has the LCD as its denominator, so we do not need to do anything.
         

         For 5/2, we can see that a 2 and 2 are missing in the denominator. Since 2 x 2 = 4, the multiplier for this fraction is 4. Our resulting equivalent fraction is:
         

         

      4. Subtract the numerators of the fractions.
         

      5. Simplify the resulting fraction.
         The fraction -17/8 is already in its simplest form, so that is our result.
         

REMEMBER: After reviewing this unit, try the additional practice for this unit.