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1 Free Pre-Algebra Lesson page Lesson Multiplying Fractions Here s a band with people. Notice that the word of in natural language translates to a multiplication in the corresponding math sentence. Natural Language Picture Math of the bands have green hats. 8 people have green hats. 8 / of the band has green hats. people have green hats. How to Multiply a Fraction and a Whole Number To find a fraction of a group, multiply the fraction by the number in the group. / of a group of people is written. Here are the steps for the multiplication: Step Step Step Step 4 Step Write the problem. Rewrite the whole number as a fraction with denominator. Convert the improper fraction to a mixed or whole number. Combine the fractions by multiplying the numerators, then the denominators. Interpret the result in the context of the problem r0 / of people is people. Multiplication is commutative, so note that has the same numeric result as. However, the picture interpreting it is different. pieces, each / of a whole, makes two wholes:. Example: What is / of a group of 0? r / of 0 is 8. / of 0 is Cheryl Wilcox

2 Free Pre-Algebra Lesson! page If the number in the group is not divisible by the denominator of the fraction, the result will be a mixed number. This doesn t make much sense in groups of people, but might make sense in other contexts where items in the group are themselves divisible. Example: What is / of a group of apples? = = = = r = / of is and / apples. / of apples is / apples. Example: What is / of 0? 0 = 0 = 0 = 0 0 = r 0 = / of 0 is and /. / of the 0 squares are green. / squares are green. A Fraction Times a Fraction The group that you are finding a fraction of may itself be a fraction or mixed number. Just multiply as before. Example: You have inherited / of a / acre parcel of land. How many acres have you inherited? (What is / of /?) = = / of / is /. The whole rectangle is an acre. / acre is shaded. When the / acre is divided into rds, the whole acre is divided in ths. / of / acre is / acre. I ve inherited / acre. 00 Cheryl Wilcox

3 Free Pre-Algebra Lesson page When multiplying fractions, you can cancel any common factors BEFORE multiplying. A factor in either numerator may be cancelled with a factor in either denominator. This is called cross-cancelling. It s convenient because you won t have to simplify your answer the result will already be in lowest terms. Example: What is / of /? Cancelling to Simplify After Multiplication: Cross-Cancelling Before Multiplication / of / is /. This makes sense! Imagine / of a pizza. Then split it in half. Example: What is / of /? / of / is /9. 9 Example: What is 8/ of 0/? Note that divides both and 0, and divides both 8 and. Cross-cancelling before multiplication saves a lot of work in multiplying large numbers. Without cross-cancelling, the previous problem would have been: Cheryl Wilcox

4 Free Pre-Algebra Lesson page 4 A Fraction Times a Mixed Number Example: You have inherited / of a 8 / acre parcel of land. How many acres have you inherited? (What is / of 8 /?) We know we need to multiply 8. First let s just think about it with pictures. You have 8 acres and then some, and you re going to take / of that. / of the 8 acres is 9 acres, so that part seems clear. Then all that s left to do is to divide the / acre in half. This is just like a previous problem, / of /, which we found was /. The answer should be 9 / acre. I ve inherited 9 and / acres. Method : Just as we reasoned above. Multiply the fraction by each part of the mixed number separately, then add the results Method : Convert the mixed number to an improper fraction, then multiply. This method is more work, but it is easier in cases where the whole number is not divisible by the denominator of the fraction. Step Convert the mixed number to an improper fraction 8 8 Step Multiply. I ve inherited 9 and acres. Step Convert the result to a mixed number. 9r 9 9 Example: You have inherited / of a 8 / acre parcel of land. How many acres have you inherited? Since the whole number of acres, 8, is not divisible by the denominator of the fraction,, it is easier to use method. 8 8 r I ve inherited and / acres. 00 Cheryl Wilcox

5 Free Pre-Algebra Lesson page Multiplying a Mixed Number by a Mixed Number Example: You re making / times a recipe that calls for / 4 cup of rice. How much rice should you add? times 4 cup rice is how much rice? Re-write both mixed numbers as improper fractions before you begin multiplying: (Although we have a and a 4, they are both in the denominator. Nothing can be cancelled.) Finally, convert /8 back to a mixed number: I should add 4 /8 cups rice. 00 Cheryl Wilcox

6 Free Pre-Algebra Lesson page Lesson : Multiplying Fractions Worksheet Name. Use the picture to show that / 4 of a group of is.. What is / of? What does this mean, in terms of a group of? Write out the steps for the multiplication 4. What is /? What does this mean, in terms of halves?. Draw a sketch to illustrate each multiplication. a. 4. Write out the steps for each multiplication. a. b. b.. Sketch a picture to show that / of / is /.. Sketch a picture to show that / of 4 / is /. 00 Cheryl Wilcox

7 Free Pre-Algebra Lesson page. Mulltiply the fractions. Use cross-cancelling if possible. Do not convert answers to mixed numbers. 8. Convert the mixed numbers to improper fractions before multiplying. Write the answer as a mixed number. a. a. b. 4 b. 8 c. 4 c. 4 d. 0 d You are making / times a recipe with the following ingredients: / cup butter; 4 eggs; / cups flour How much of each will you use? 0. When you multiply a whole number with a fraction, (as in the problem 4 0 ), is the answer greater than or less than the whole number? Why? 00 Cheryl Wilcox

8 Free Pre-Algebra Lesson! page 8 Lesson : Multiplying Fractions Homework A. Simplify! 4( + ). Name. Draw a line / 8 inches long. Convert / 8 to an improper fraction. Show the eighths on the ruler.. List all the factor pairs of the numbers, then find the GCF Find the prime factorization of the numbers. Use the prime factorizations to simplify the fractions. Use the GCF to simplify the fractions: Write fractions equivalent to.. Simplify 8 0 to lowest terms. Could the mixed number 8 0 terms? be simplified to lowest 00 Cheryl Wilcox

9 Free Pre-Algebra Lesson! page 9. Change the mixed numbers to improper fractions. a Change the improper fractions to mixed numbers. a. b. 4 b. c. 00 c Use the picture to answer the questions. 0. Multiply the fractions. Use cross-cancelling if possible. a. 0 of 4 a. What is / of? b. What is / of? c. What is /4 of? d. What is / of? e. What is / of? b. c There are inches in a foot. f. What fraction of a foot is inches? g. What fraction of a foot is inches? h. What fraction of a foot is inches? d. 0 of 0 i. What fraction of a foot is inch?. Write the answer. Then write the four related equations. a. The product of 4 and is. c. The sum of 8 and is. b. The difference of 9 and is. d. The quotient of 0 and is. 00 Cheryl Wilcox

10 Free Pre-Algebra Lesson page 0 Lesson : Multiplying Fractions Homework A Answers. Simplify Draw a line / 8 inches long. Convert / 8 to an improper fraction. Show the eighths on the ruler List all the factor pairs of the numbers, then find the GCF. 4. Find the prime factorization of the numbers. Use the prime factorizations to simplify the fractions. Use the GCF to simplify the fractions: Write fractions equivalent to Simplify 8 0 to lowest terms. 8 0 Could the mixed number 8 be simplified to lowest 0 terms? Cheryl Wilcox

11 Free Pre-Algebra Lesson page. Change the mixed numbers to improper fractions. a Change the improper fractions to mixed numbers. a. r0 b b. r c c. 4 4 r Use the picture to answer the questions. 0. Multiply the fractions. Use cross-cancelling if possible. a. What is / of? b. What is / of? 4 c. What is /4 of? d. What is / of? e. What is / of? a. b. 0 of There are inches in a foot. f. What fraction of a foot is inches? / g. What fraction of a foot is inches? /4 h. What fraction of a foot is inches? / i. What fraction of a foot is inch? /. Write the answer. Then write the four related equations. a. The product of 4 and is b. The difference of 9 and is c. d of c. The sum of 8 and is d. The quotient of 0 and is Cheryl Wilcox

12 Free Pre-Algebra Lesson! page Lesson : Multiplying Fractions Homework B. Simplify 4! ( + ). Name. Draw a line / 4 inches long. Convert / 4 to an improper fraction. Show the fourths on the ruler.. List all the factor pairs of the numbers, then find the GCF Find the prime factorization of the numbers. 0 Use the prime factorizations to simplify the fractions. Use the GCF to simplify the fractions: Write fractions equivalent to 4..Change to a mixed number. The mixed number is not in the correct format, because / is an improper fraction. What would be the correct way to write it? 00 Cheryl Wilcox

13 Free Pre-Algebra Lesson! page. Change the mixed numbers to improper fractions. a. 4 9 b. c. 8. Change the improper fractions to mixed numbers. Simplify the improper fraction before you begin or simplify the fraction part of the result if not in lowest terms. a. 9 b. 9 c Use the picture to answer the questions. 0. Multiply the fractions. Use cross-cancelling if possible. a. 8 of 8 b There are 0 minutes in one hour. a. How many minutes in / hour? What is / of 0? b. What is / of 0? c. What is /4 of 0? d. What is / of 0? e. What is / of 0? f. What fraction of an hour is minutes? g. What fraction of a hour is 0 minutes? h. What fraction of an hour is minutes? i. What fraction of an hour is 4 minutes?. Write the answer. Then write the four related equations. a. The difference of 8 and is. c d. of 8 c. The product of 8 and 4 is. b. The quotient of and is. d. The sum of 0 and is. 00 Cheryl Wilcox