Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators

We included HMH Into Math Grade 6 Answer Key PDF Module 3 Lesson 2 Explore Division of Fractions with Unlike Denominators to make students experts in learning maths.

HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators

I Can divide two fractions with unlike denominators using several methods.

Spark Your Learning
Roselyn is making a stir fry. The recipe calls for 4 cups of broccoli, but Roselyn has only a \(\frac{2}{3}\) cup measure. How many measuring cups of broccoli should she add?
HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators 1
HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators 2
Answer:
Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators-1
6 cups

Explanation:
Given Roselyn is making a stir fry. The recipe calls for 4 cups of
broccoli, but Roselyn has only a \(\frac{2}{3}\) cup measure.
therefore number of measuring cups of broccoli should she add is
6 cups.

Turn and Talk What if the recipe says to add 3 cups of summer squash? How many measuring cups of squash should she add? Explain.
Answer:
4\(\frac{1}{2}\) cups,

Explanation:
If Roselyn says to add 3 cups of summer squash,Number of measuring cups of squash should she add is 3 ÷ \(\frac{2}{3}\) = 3 X \(\frac{3}{2}\)  =
\(\frac{3 X 3}{2}\) = \(\frac{9}{2}\) as numerator is greater than denominator we write as mixed fraction as 4\(\frac{1}{2}\).

Build Understanding

1. Roselyn also needs to add \(\frac{5}{8}\) cup of orange juice to make a sauce for her stir fry. If she uses a \(\frac{1}{4}\)-cup measuring cup, how many measuring cups will she need to add?
A. How can you divide \(\frac{5}{8}\) into groups of \(\frac{1}{4}\)? Complete the following statement.
Since \(\frac{1}{4}\) = HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators 3, you can divide \(\frac{5}{8}\) into groups of _____.
HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators 4
B. Use the number line to show the number of groups of \(\frac{1}{4}\) in \(\frac{5}{8}\).
HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators 5
C. How many measuring cups of orange juice will Roselyn need to add?
_________________________
D. How many measuring cups of juice will Roselyn need to add if she uses a \(\frac{1}{8}\)-cup measuring cup or a \(\frac{3}{8}\)-cup measuring cup? Use your results to complete the first column of the table.
HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators 6
E. Reciprocals can help you find quotients. You find the reciprocal of a fraction by switching the numerator and denominator. Complete the following equations with the correct reciprocals.
HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators 7
Answer:
A. Since \(\frac{1}{4}\) , I can divide \(\frac{5}{8}\)
into groups of 2\(\frac{1}{2}\) cups,
B. Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators-2
C. 2\(\frac{1}{2}\) measuring cups of
orange juice will Roselyn need to add,
D. Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators-3
E. Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators-4

Explanation:
Given Roselyn also needs to add \(\frac{5}{8}\) cup of orange juice to make a sauce for her stir fry. If she uses a \(\frac{1}{4}\)-cup measuring cup,
A.\(\frac{5}{8}\) into groups of \(\frac{1}{4}\) is \(\frac{5}{8}\) ÷ \(\frac{1}{4}\), \(\frac{5}{8}\) X \(\frac{4}{1}\) =
\(\frac{5 X 4}{8}\) = \(\frac{5}{2}\) as numerator is greater than denominator we write as mixed fraction as 2\(\frac{1}{2}\).
Since \(\frac{1}{4}\) , I can divide \(\frac{5}{8}\) into groups of 2\(\frac{1}{2}\) cups,
B. Used the number line to show the number of
2\(\frac{1}{2}\) groups of \(\frac{1}{4}\) in \(\frac{5}{8}\) as shown above.
C. If we divide \(\frac{5}{8}\) ÷ \(\frac{1}{4}\)
we get 2\(\frac{1}{2}\) measuring cups of orange juice will Roselyn need to add.
D. Numb er of measuring cups of juice will Roselyn need to add if she uses a \(\frac{1}{8}\)-cup measuring cup is \(\frac{5}{8}\) ÷ \(\frac{1}{8}\) =
\(\frac{5}{8}\) X \(\frac{8}{1}\) = \(\frac{5 X 8}{8 X 1}\) = 5 cups or a \(\frac{3}{8}\)-cup measuring cup means \(\frac{5}{8}\) ÷ \(\frac{3}{8}\) = \(\frac{5}{8}\) X \(\frac{8}{3}\) = \(\frac{5 X 8}{8 X 3}\) = \(\frac{5}{3}\) as numerator is greater than denominator we write as mixed fraction as 1\(\frac{2}{3}\).
Used my results to complete the first column of the table as shown above.
E. Reciprocals can help me to find quotients.
I found the reciprocal of a fraction by switching the numerator and denominator. Completed the following equations with the correct reciprocals as shown above.

Connect to Vocabulary
Two numbers whose product is 1 are reciprocals or multiplicative inverses.

F. Now complete the second column of the table. Look for a pattern in the table to complete the following statements.
To find the quotient of two fractions, multiply the ____ fraction by the reciprocal the ___ fraction. In each row of the table, the quotient ___ the ____.
HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators 8
Answer:
Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators-5
The statement is ” To find the quotient of two fractions, multiply the \(\frac{a}{b}\) fraction by the reciproca the \(\frac{d}{c}\) fraction.
In each row of the table, the quotient result of the first fraction divides the second fraction \(\frac{ad}{bc}\),

Explanation:
The statement is ” To find the quotient of two fractions, multiply the first fraction by the reciprocal the second fraction. In each row of the table, the quotient
result of the first fraction divides the second fraction \(\frac{ad}{bc}\).

Step It Out

2. Eric and Tom are making small bows for presents. They will need pieces of ribbon like the one shown, which will be cut from a ribbon that is \(\frac{3}{4}\)-yard long. They used two different methods to find the number of \(\frac{2}{9}\)-yard pieces they could cut from a \(\frac{3}{4}\)-yard ribbon.
HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators 9
HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators 10
A. Explain Eric’s solution method.
_________________________
_________________________
_________________________

B. Explain Tom’s solution method.
_________________________
_________________________
_________________________
_________________________

C. How many whole pieces can they cut from the long ribbon?
_________________________

D. What does the \(\frac{3}{8}\) in the quotient 3\(\frac{3}{8}\) mean in this situation?
HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators 11
\(\frac{3}{8}\) is the amount of ribbon left over, or ____ yard of ribbon.
A. Eric’s solution method: Multiplying first fraction with the second fraction reciprocal,

Explanation:
In Eric’s solution method we multiply first fraction with the second fraction reciprocal
\(\frac{3}{4}\) X \(\frac{9}{2}\) \(\frac{3 X 9}{4 X 2}\) = \(\frac{27}{8}\) and as numerator is greater than denominator
writing quotient in mixed fraction as 3\(\frac{3}{8}\).

B. Tom’s solution method:
Making common denominators,

Explanation:
In Tom’s solution method he made both denominators 4, 9 as common 36 multiplying
first fraction \(\frac{3}{4}\) numerator and denominator by 9 and second fraction\(\frac{2}{9}\) numerator and denominator by 4 then take out both
denominators 36 then dividing \(\frac{27}{8}\) as numerator is greater than denominator writing quotient in mixed fraction as 3\(\frac{3}{8}\).

C. 3 whole pieces can they cut from the long ribbon

Explanation:
As we got quotient as 3\(\frac{3}{8}\), so number of whole pieces can they cut from the long ribbon are 3.

D. Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators-6

Explanation:
\(\frac{3}{8}\) in the quotient 3\(\frac{3}{8}\) mean in this situation is the amount of ribbon left over, or \(\frac{1}{12}\)  yard of ribbon.

Turn and Talk Whose method do you prefer and why do you prefer it?
Answer:
Eric’s solution method,

Explanation:
I prefer Eric’s solution method, It very easy and simply doing second fraction reciprocal and multiplying, instead of lengthy making of common denominators
then solving.

3. Tina feeds her dog \(\frac{4}{7}\) pound of dog food per day. If she buys a bag containing 9 pounds of dog food, how many days will it last?
A. What expression could you use to solve the problem?
_________________________
HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators 12
Answer:
A. Expression 9 ÷ \(\frac{4}{7}\),

Explanation:
Given Tina feeds her dog \(\frac{4}{7}\) pound of dog food per day. If she buys a bag containin 9 pounds of dog food,
A. The expression could I use to solve the problem is
9 ÷ \(\frac{4}{7}\).

B. What do you need to do first to find the quotient?
_________________________
Answer:
Multiply second fraction reciprocal,

Explanation:
I need to do first multiply the second fraction reciprocal to find the quotient.

C. What do you need to do next?
_________________________
_________________________
Answer:
Remove common factors and write results in mixed fraction,

Explanation:
After multiplication removing common factors from numerator and denominator and writing results in mixed fraction.

D. Complete the number sentence to find the number of days the bag of dog food will last.
HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators 13
The dog food will last _____ days.
Answer:
Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators-7
The dog food will last \(\frac{63}{4}\) days or 15 \(\frac{3}{4}\) days,

Explanation:
Completed the number sentence to find the number of days the bag of dog food will last a \(\frac{63}{4}\) days or 15 \(\frac{3}{4}\) days above.

Turn and Talk About how many 9-pound bags of dog food will Tina need to feed her dog for an entire year?
Answer:
Number of 9-pound bags of dog food will Tina need to her dog for an enitre year is 23\(\frac{11}{63}\) pound bags about 24 bags of 9 – pound bags,

Explanation:
Asking about how many 9-pound bags of dog food will Tina need to feed her dog for an entire year as there are 365 days in an enitre year as one 9 – pound bag will last for
\(\frac{63}{4}\) days so number of 9-pound bags of dog food will Tina need to her dog for an enitre year is 365 ÷ \(\frac{63}{4}\) = 365 X \(\frac{4}{63}\) = \(\frac{365 X 4}{63}\) = \(\frac{1460}{63}\) = 23\(\frac{11}{63}\) bags approximately about 24 bags of 9 – pound bags.

Check Understanding

Question 1.
Marcus needs to measure out \(\frac{2}{3}\) liter of a solution. He is using a container
that holds \(\frac{1}{6}\) liter. How many groups of \(\frac{1}{6}\) are in \(\frac{2}{3}\)? How many times will Marcus need to fill the container?
Answer:
4 groups, 4 times will Marcus need to fill the container,

Explanation:
Given Marcus needs to measure out \(\frac{2}{3}\) liter of
a solution. He is using a container that holds \(\frac{1}{6}\) liter.
Number of groups of \(\frac{1}{6}\) are in \(\frac{2}{3}\),
Number of times will Marcus need to fill the container are
\(\frac{2}{3}\) ÷ \(\frac{1}{6}\) =
\(\frac{2}{3}\) X \(\frac{6}{1}\) = 2 X 2 = 4,
therefore 4 groups, 4 times will Marcus need to fill the container.

Question 2.
Roberta bought \(\frac{9}{10}\) pound of raisins. She put \(\frac{2}{5}\) pound in bags for her lunch. How many \(\frac{2}{5}\)-pound bags can Roberta fill? Does she have any left over?
_________________________
Answer:
Number of \(\frac{2}{5}\)-pound bags can Roberta fill are 2 and She has left over \(\frac{1}{4}\),

Explanation:
Given Roberta bought \(\frac{9}{10}\) pound of raisins.
She put \(\frac{2}{5}\) pound in bags for her lunch.
Number of \(\frac{2}{5}\)-pound bags can Roberta fill are \(\frac{9}{10}\) ÷ \(\frac{2}{5}\) = \(\frac{9}{10}\) X \(\frac{5}{2}\) =
\(\frac{9 X 5}{10 X 2}\) = \(\frac{9}{4}\) as numerator is greater than denominator writing quotient in mixed fraction as 2\(\frac{1}{4}\),
2, therefore Number of \(\frac{2}{5}\)-pound bags can
Roberta fill are 2 and She has left over \(\frac{1}{4}\).

For Problems 3-6. divide the fractions.

Question 3.
\(\frac{6}{10}\) ÷ \(\frac{2}{5}\) _________________________
Answer:
1\(\frac{1}{2}\),

Explanation
Given to find \(\frac{6}{10}\) ÷ \(\frac{2}{5}\)= \(\frac{6}{10}\) X \(\frac{5}{2}\) = \(\frac{6 X 5}{10 X 2}\) = \(\frac{3}{2}\)
as numerator is greater than denominator writing quotient in mixed fraction as 1\(\frac{1}{2}\).

Question 4.
\(\frac{3}{8}\) ÷ \(\frac{1}{3}\) _________________________
Answer:
1\(\frac{1}{8}\),

Explanation
Given to find \(\frac{3}{8}\) ÷ \(\frac{1}{3}\)= \(\frac{3}{8}\) X \(\frac{3}{1}\) = \(\frac{3 X 3}{8 X 1}\) = \(\frac{9}{8}\)
as numerator is greater than denominator
writing quotient in mixed fraction as 1\(\frac{1}{8}\).

Question 5.
\(\frac{5}{9}\) ÷ \(\frac{2}{3}\) _________________________
Answer:
\(\frac{5}{6}\),

Explanation
Given to find \(\frac{5}{9}\) ÷ \(\frac{2}{3}\)= \(\frac{5}{9}\) X \(\frac{3}{2}\) = \(\frac{5 X 3}{9 X 2}\) = \(\frac{5}{6}\).

Question 6.
10 ÷ \(\frac{2}{5}\) _________________________
Answer:
25,

Explanation
Given to find 10 ÷ \(\frac{2}{5}\) = 10 X \(\frac{5}{2}\) = \(\frac{10 x 5}{1 X 2}\) = 25.

On Your Own

Model with Mathematics For Problems 7-11, write an expression to model each situation. Then answer the question.

Question 7.
Mr. Duale would like to plant a vegetable garden. He has a part of an acre of land, which he plans to divide into \(\frac{2}{5}\)-acre sections. How many sections will he have?
HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators 14
Answer:
Mr.Duale will have 2 sections,

Explanation:
Given Mr. Duale would like to plant a vegetable garden.
He has a part of an acre of land, which he plans to divide into \(\frac{2}{5}\)-acre sections. Number of sections will he have are \(\frac{8}{10}\) ÷ \(\frac{2}{5}\)= \(\frac{8}{10}\) X \(\frac{5}{2}\) = \(\frac{8 X 5}{10 X 2}\) = 2, Therefore Mr.Duale will have 2 sections.

Question 8.
STEM The width of a single atom of aluminum is \(\frac{7}{25}\) nanometer, which is more than 100,000 times smaller than a millimeter. Scientists sometimes use Angstroms to measure distances on an atomic scale. One Angstrom is \(\frac{1}{10}\) nanometer. How many Angstroms wide is a single atom of aluminum?
Answer:
2\(\frac{4}{5}\) Angstroms wide is a single atom of aluminum,

Explanation:
Given the width of a single atom of aluminum is \(\frac{7}{25}\) nanometer, which is more than 100,000 times smaller than a millimeter. Scientists sometimes
use Angstroms to measure distances on an atomic scale. One Angstrom is \(\frac{1}{10}\) nanometer. So Angstroms wide is a single atom of aluminum is
\(\frac{7}{25}\) ÷ \(\frac{1}{10}\) = \(\frac{7}{25}\) X \(\frac{10}{1}\) = \(\frac{7 X 10}{25 X 1}\) =\(\frac{7 X 2}{5 X 1}\) = \(\frac{14}{5}\) as numerator is greater than denominator
writing quotient in mixed fraction as 2\(\frac{4}{5}\).
Therefore 2\(\frac{4}{5}\) Angstroms wide is a single atom of aluminum.

Question 9.
At a school, each class period is \(\frac{3}{4}\) hour long. If there are 6 hours of class time in a school day, how many class periods are there?
Answer:
8 class periods are there,

Explanation:
Given at a school each class period is \(\frac{3}{4}\) hour long.
If there are 6 hours of class time in a school day,
Number of many class periods are there are 6 ÷ \(\frac{3}{4}\) = 6 X \(\frac{4}{3}\) = \(\frac{6 X 4}{3}\)= 2 X 4 = 8, therefore 8 class periods.

Question 10.
A pitcher contains \(\frac{8}{10}\) liter of juice and is used to fill cups that hold \(\frac{1}{5}\) liter. How many cups can be filled?
Answer:
4 cups can be filled,

Explanation:
Given a pitcher contains \(\frac{8}{10}\) liter of juice and is used to fill cups that hold \(\frac{1}{5}\) liter. Number of cups can be filled are \(\frac{8}{10}\) ÷ \(\frac{1}{5}\) = \(\frac{8}{10}\) X \(\frac{5}{1}\) =
\(\frac{8 X 5}{10 X 1}\) =\(\frac{8}{2}\) = 4 cups.

Question 11.
Reason Patrick has \(\frac{7}{10}\) pound of flour. A batch of biscuits requires \(\frac{1}{8}\) pound of flour. How many whole batches of biscuits can Patrick make? Explain your reasoning.
Answer:
5 batches of biscuits Patrick can make,

Explanation:
Given Patrick has \(\frac{7}{10}\) pound of flour.
A batch of biscuits requires \(\frac{1}{8}\) pound of flour.
Number of whole batches of biscuits can Patrick make are \(\frac{7}{10}\) ÷ \(\frac{1}{8}\) = \(\frac{7}{10}\) X \(\frac{8}{1}\) =
\(\frac{7 X 8}{10 X 1}\) =\(\frac{7 X 4}{5 X 1}\) = \(\frac{28}{5}\) as numerator is greater than denominator writing quotient in mixed fraction as 5\(\frac{3}{5}\). So Patrick can make 5 batches of biscuits.

Question 12.
Critique Reasoning Hannah is asked to divide \(\frac{1}{6}\) by \(\frac{1}{2}\). She says that the answer is 3, because the product of \(\frac{1}{2}\) and 6 is equal to 3. Is she correct? Why or why not?
_________________________
_________________________
_________________________
Answer:
Hannah is incorrect, Answer is \(\frac{1}{3}\) not 3,

Explanation:
Given Hannah is asked to divide \(\frac{1}{6}\) by \(\frac{1}{2}\). She says that the answer is 3, because the product of \(\frac{1}{2}\) and
6 is equal to 3. Checking \(\frac{1}{6}\) ÷ \(\frac{1}{2}\) = \(\frac{1}{6}\) X \(\frac{2}{1}\) = \(\frac{1 X 2}{6 X 1}\) = \(\frac{1}{3}\), So Hannah is incorrect it’s not 3 but answer is \(\frac{1}{3}\).

Question 13.
Model with Mathematics Diane had \(\frac{15}{16}\) cup of butter. A recipe for a cake calls for \(\frac{1}{4}\) cup of butter. Diane was able to make 3 whole cakes. How much butter did she use? How much butter does she have left over? Show how to model and solve this problem.
Answer:
Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators-8
Diane was able to make 3 whole cakes.
Butter does she have left over is \(\frac{3}{4}\),

Explanation:
Given Diane had \(\frac{15}{16}\) cup of butter.
A recipe for a cake calls for \(\frac{1}{4}\) cup of butter.
Diane was able to make 3 whole cakes. Much of butter did she use is \(\frac{15}{16}\) ÷ \(\frac{1}{4}\) = \(\frac{15}{16}\) X \(\frac{4}{1}\) =
\(\frac{15 X 4}{16 X 1}\) = \(\frac{15}{4}\) as numerator is greater than denominator writing quotient in mixed fraction as 3\(\frac{3}{4}\).
Diane was able to make 3 whole cakes. Butter does she have left over is \(\frac{3}{4}\).

Question 14.
Sandy is a jeweler. She has 2 grams of gold. If each earring she makes must contain \(\frac{3}{16}\) gram of gold, how many earrings can Sandy make? How many earrings could she make from a gold bar of 1,000 grams of gold? Show your work.
HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators 15
Answer:
Sandy can make 10\(\frac{2}{3}\) earrings in 2 grams of gold,
5333\(\frac{1}{3}\) earrings could she make from a gold bar of 1,000 grams of gold,

Explanation:
Given Sandy is a jeweler. She has 2 grams of gold.
If each earring she makes must contain \(\frac{3}{16}\) gram of gold,
Number of earrings can Sandy make are 2 ÷ \(\frac{3}{16}\) = 2 X \(\frac{16}{3}\) = \(\frac{2 X 16}{3}\) = \(\frac{32}{3}\) as numerator is greater than denominator writing quotient in mixed fraction as 10\(\frac{2}{3}\).
Number of earrings could she make from a gold bar of 1,000 grams of gold are
1,000 ÷ \(\frac{3}{16}\) = 1,000 X \(\frac{16}{3}\) = \(\frac{1000 X 16}{3}\) = \(\frac{16000}{3}\) as numerator is greater than denominator
writing quotient in mixed fraction as 5333\(\frac{1}{3}\).

For Problems 15-18, find the reciprocal.

Question 15.
\(\frac{5}{16}\) ____________
Answer:
\(\frac{16}{5}\),

Explanation:
The reciprocal of \(\frac{5}{16}\) switching numerator and denominator so it is
\(\frac{16}{5}\).

Question 16.
\(\frac{1}{5}\) ____________
Answer:
5,

Explanation:
The reciprocal of \(\frac{1}{5}\) switching numerator and denominator so it is
\(\frac{5}{1}\) = 5.

Question 17.
4 ____________
Answer:
\(\frac{1}{4}\),

Explanation:
The reciprocal of 4 is \(\frac{1}{4}\).

Question 18.
\(\frac{4}{9}\) ____________
Answer:
\(\frac{9}{4}\),

Explanation:
The reciprocal of \(\frac{4}{9}\) switching numerator and denominator so it is
\(\frac{9}{4}\).

For Problems 19-27, divide the fractions.

Question 19.
\(\frac{5}{12}\) ÷ \(\frac{1}{3}\) ____________
Answer:
1\(\frac{1}{4}\),

Explanation
Given to find \(\frac{5}{12}\) ÷ \(\frac{1}{3}\)= \(\frac{5}{12}\) X \(\frac{3}{1}\) = \(\frac{5 X 3}{12 X 1}\) = \(\frac{5}{4}\)
as numerator is greater than denominator writing quotient in mixed fraction as 1\(\frac{1}{4}\).

Question 20.
\(\frac{2}{3}\) ÷ \(\frac{1}{6}\) ____________
Answer:
4,

Explanation
Given to find \(\frac{2}{3}\) ÷ \(\frac{1}{6}\)= \(\frac{2}{3}\) X \(\frac{6}{1}\) = \(\frac{2 X 6}{3 X 1}\) = 4.

Question 21.
\(\frac{1}{2}\) ÷ \(\frac{7}{8}\) ____________
Answer:
\(\frac{4}{7}\),

Explanation
Given to find \(\frac{1}{2}\) ÷ \(\frac{7}{8}\)= \(\frac{1}{2}\) X \(\frac{8}{7}\) = \(\frac{1 X 8}{2 X 7}\) = \(\frac{4}{7}\).

Question 22.
\(\frac{11}{15}\) ÷ \(\frac{3}{5}\) ____________
Answer:
1\(\frac{2}{9}\),

Explanation
Given to find \(\frac{11}{15}\) ÷ \(\frac{3}{5}\) = \(\frac{11}{15}\) X \(\frac{5}{3}\) = \(\frac{11 X 5}{15 X 3}\) = \(\frac{11}{9}\) as numerator is greater than denominator writing quotient in mixed fraction as 1\(\frac{2}{9}\).

Question 23.
\(\frac{1}{6}\) ÷ \(\frac{2}{3}\) ____________
Answer:
\(\frac{1}{4}\),

Explanation
Given to find \(\frac{1}{6}\) ÷ \(\frac{2}{3}\) = \(\frac{1}{6}\) X \(\frac{3}{2}\) = \(\frac{1 X 3}{6 X 2}\) = \(\frac{1}{4}\).

Question 24.
\(\frac{5}{14}\) ÷ \(\frac{5}{7}\) ____________
Answer:
\(\frac{1}{2}\),

Explanation
Given to find \(\frac{5}{14}\) ÷ \(\frac{5}{7}\) = \(\frac{5}{14}\) X \(\frac{7}{5}\) = \(\frac{5 X 7}{14 X 5}\) = \(\frac{1}{2}\).

Question 25.
\(\frac{4}{5}\) ÷ \(\frac{24}{25}\) ____________
Answer:
\(\frac{5}{6}\),

Explanation
Given to find \(\frac{4}{5}\) ÷ \(\frac{24}{25}\) = \(\frac{4}{5}\) X \(\frac{25}{24}\) = \(\frac{4 X 25}{5 X 26}\) = \(\frac{5}{6}\).

Question 26.
\(\frac{5}{6}\) ÷ \(\frac{5}{9}\) ____________
Answer:
1\(\frac{1}{2}\),

Explanation
Given to find \(\frac{5}{6}\) ÷ \(\frac{5}{9}\) = \(\frac{5}{6}\) X \(\frac{9}{5}\) = \(\frac{5 X 9}{6 X 5}\) = \(\frac{3}{2}\)
as numerator is greater than denominator writing quotient in mixed fraction as 1\(\frac{1}{2}\).

Question 27.
\(\frac{7}{8}\) ÷ \(\frac{3}{16}\) ____________
Answer:
4\(\frac{2}{3}\),

Explanation
Given to find \(\frac{7}{8}\) ÷ \(\frac{3}{16}\) = \(\frac{7}{8}\) X \(\frac{16}{3}\) = \(\frac{7 X 16}{8 X 3}\) = \(\frac{14}{3}\)
as numerator is greater than denominator writing quotient in mixed fraction as 4\(\frac{2}{3}\).

I’m in a Learning Mindset!

What strategies do I use to ensure I can complete my work on dividing fractions with different denominators?
Answer: To divide fractions, regardless of the denominators, flip the second fraction (the divisor) upside down and then multiply the result with the first fraction (the dividend),

Explanation:
Regardless of the denominators, Dividing two fractions is the same as multiplying
the first fraction by the reciprocal of the second fraction.
The first step to dividing fractions is to find the reciprocal (reverse the numerator and denominator) of the second fraction.
Next, multiply the two numerators. Then, multiply the two denominators.

Lesson 3.2 More Practice/Homework

Explore Division of Fractions with Unlike Denominators

Question 1.
A small bookshelf is \(\frac{8}{12}\) yard long. How many books can fit on the shelf if the width of each book is \(\frac{1}{24}\) yard? Explain.
HMH Into Math Grade 6 Module 3 Lesson 2 Answer Key Explore Division of Fractions with Unlike Denominators 16
Answer:
16 books,

Explanation:
Given Aasmall bookshelf is \(\frac{8}{12}\) yard long.
Number of books can fit on the shelf if the width of each book is \(\frac{1}{24}\) yard is \(\frac{8}{12}\) ÷ \(\frac{1}{24}\) = \(\frac{8}{12}\) X 24 = \(\frac{8 X 24}{12}\) = 16 books.

Question 2.
Reason Isabella owns a rectangular lot with an area of \(\frac{9}{32}\) square mile. If the length of the western side of her lot is \(\frac{3}{4}\) mile, what is the length of the northern side? How can you find the length?
_________________________
_________________________
Answer:
The length of the northern side is \(\frac{3}{8}\) mile,

Explanation:
Given Isabella owns a rectangular lot with an area of \(\frac{9}{32}\) square mile. If the length of the western side of her lot is \(\frac{3}{4}\) mile,
by dividing area with length of the western side so the length of the northern side is
\(\frac{9}{32}\) ÷ \(\frac{3}{4}\) = \(\frac{9}{32}\) X \(\frac{4}{3}\) = \(\frac{9 X 4}{32 x 3}\) = \(\frac{3}{8}\) mile.

Question 3.
Math on the Spot Show two methods for finding the quotient \(\frac{3}{8}\) ÷ \(\frac{3}{4}\).
Answer:
First method: Multiplying first fraction with the second fraction reciprocal,
Second method:
Making common denominators, The quotient is \(\frac{1}{2}\),

Explanation:
First method:
\(\frac{3}{8}\) ÷ \(\frac{3}{4}\), We multiply first fraction with
the second fraction reciprocal \(\frac{3}{8}\) X \(\frac{4}{3}\)
\(\frac{3 X 4}{8 X 3}\) = \(\frac{1}{2}\),
Second method:
Making both denominators 8, 4 as common 8 multiplying second fraction \(\frac{3}{4}\) numerator and denominator by 2 taking out both denominators 8 then dividing we get \(\frac{3}{8}\) X \(\frac{8}{6}\) = \(\frac{1}{2}\).

Question 4.
Construct Arguments When \(\frac{9}{10}\) is divided by \(\frac{2}{5}\), will the quotient be greater than 1 or less than 1? How do you know?
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Answer:
The quotient is greater than 1,
As the divisor is smaller than the dividend
the quotient is more than 1,

Explanation:
When the divisor is smaller than the dividend the quotient is more than 1 so when \(\frac{9}{10}\) is divided by \(\frac{2}{5}\) the quotient will be
greater than 1, \(\frac{9}{10}\) ÷ \(\frac{2}{5}\) = \(\frac{9}{10}\) X \(\frac{5}{2}\) = \(\frac{9 X 5}{10 X 2}\) = \(\frac{9}{4}\) as numerator is greater than denominator writing quotient in mixed fraction as 2\(\frac{1}{4}\).

For Problems 5-8, find the reciprocal.

Question 5.
\(\frac{7}{8}\) ________
Answer:
\(\frac{8}{7}\),

Explanation:
The reciprocal of \(\frac{7}{8}\) switching numerator and denominator so it is
\(\frac{8}{7}\).

Question 6.
\(\frac{1}{10}\) ________
Answer:
10,

Explanation:
The reciprocal of \(\frac{1}{10}\) switching numerator and denominator so it is 10.

Question 7.
12 ____________
Answer:
\(\frac{1}{12}\),

Explanation:
The reciprocal of 12 is switching numerator and denominator so it is
\(\frac{1}{12}\).

Question 8.
\(\frac{14}{16}\) ________
Answer:
\(\frac{16}{14}\),

Explanation:
The reciprocal of \(\frac{14}{16}\) switching numerator and denominator so it is
\(\frac{16}{14}\).

For Problems 9-17, divide the fractions.

Question 9.
\(\frac{3}{8}\) ÷ \(\frac{2}{3}\)
____________
Answer:
\(\frac{9}{16}\),

Explanation
Given to find \(\frac{3}{8}\) ÷ \(\frac{2}{3}\) = \(\frac{3}{8}\) X \(\frac{3}{2}\) = \(\frac{3 X 3}{8 X 2}\) = \(\frac{9}{16}\).

Question 10.
\(\frac{9}{2}\) ÷ \(\frac{4}{10}\)
____________
Answer:
11\(\frac{1}{4}\),

Explanation
Given to find \(\frac{9}{2}\) ÷ \(\frac{4}{10}\) = \(\frac{9}{2}\) X \(\frac{10}{4}\) = \(\frac{9 X 10}{2 X 4}\) = \(\frac{45}{4}\)
as numerator is greater than denominator writing quotient in mixed fraction as 11\(\frac{1}{4}\).

Question 11.
\(\frac{3}{14}\) ÷ \(\frac{2}{6}\)
____________
Answer:
\(\frac{9}{14}\),

Explanation
Given to find \(\frac{3}{14}\) ÷ \(\frac{2}{6}\) = \(\frac{3}{14}\) X \(\frac{6}{2}\) = \(\frac{3 X 6}{14 X 2}\) = \(\frac{9}{14}\).

Question 12.
\(\frac{5}{8}\) ÷ \(\frac{1}{24}\)
____________
Answer:
15,

Explanation
Given to find \(\frac{5}{8}\) ÷ \(\frac{1}{24}\) = \(\frac{5}{8}\) X \(\frac{24}{1}\) = \(\frac{5 X 24}{8 X 1}\) = 15.

Question 13.
\(\frac{5}{6}\) ÷ \(\frac{5}{24}\)
____________
Answer:
4,

Explanation
Given to find \(\frac{5}{6}\) ÷ \(\frac{5}{24}\) = \(\frac{5}{6}\) X \(\frac{24}{5}\) = \(\frac{5 X 24}{6 X 5}\) = 4.

Question 14.
\(\frac{3}{4}\) ÷ \(\frac{1}{24}\)
____________
Answer:
18,

Explanation
Given to find \(\frac{3}{4}\) ÷ \(\frac{1}{24}\) = \(\frac{3}{4}\) X 24 = \(\frac{3 X 24}{4}\) = 18.

Question 15.
12 ÷ \(\frac{18}{25}\)
____________
Answer:
16\(\frac{2}{3}\),

Explanation
Given to find 12 ÷ \(\frac{18}{25}\) = 12 X \(\frac{25}{18}\) =
\(\frac{12 X 25}{18}\) = \(\frac{50}{3}\) as numerator is greater than denominator writing quotient in mixed fraction as 16\(\frac{2}{3}\).

Question 16.
20 ÷ \(\frac{15}{16}\)
____________
Answer:
21\(\frac{1}{3}\),

Explanation
Given to find 20 ÷ \(\frac{15}{16}\) = 20 X \(\frac{16}{15}\) =
\(\frac{20 X 16}{15}\) = \(\frac{64}{3}\) as numerator is greater than denominator writing quotient in mixed fraction as 21\(\frac{1}{3}\).

Question 17.
16 ÷ \(\frac{10}{11}\)
____________
Answer:
17\(\frac{3}{5}\),

Explanation
Given to find 16 ÷ \(\frac{10}{11}\) = 16 X \(\frac{11}{10}\) =
\(\frac{16 X 11}{10}\) = \(\frac{88}{5}\) as numerator is greater than denominator writing quotient in mixed fraction as 17\(\frac{3}{5}\).

Test Prep

Question 18.
How many \(\frac{1}{2}\) cups are in \(\frac{7}{8}\) cup?
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Answer:
1\(\frac{3}{4}\),

Explanation
Given to number of \(\frac{1}{2}\) cups are in \(\frac{7}{8}\) cup,
\(\frac{7}{8}\) ÷ \(\frac{1}{2}\) = \(\frac{7}{8}\) X \(\frac{2}{1}\) = \(\frac{7 X 2}{8}\) = \(\frac{7}{4}\) as numerator is greater than denominator writing quotient in mixed fraction as 1\(\frac{3}{4}\).

Question 19.
An expression is shown.
\(\frac{2}{10}\) ÷ \(\frac{5}{4}\)
What is the value of the expression?
A. \(\frac{1}{50}\)
B. \(\frac{4}{10}\)
C. \(\frac{8}{50}\)
D. \(\frac{1}{2}\)
Answer:
C. \(\frac{8}{50}\),

Explanation:
Given an expression is \(\frac{2}{10}\) ÷ \(\frac{5}{4}\) the value of the expression is \(\frac{2}{10}\) X \(\frac{4}{5}\) = \(\frac{2 X 4}{10 X 5}\) = \(\frac{8}{50}\) matches with bit C.

Question 20.
A large toy weighs \(\frac{5}{8}\) pound. How many small toys that each weigh \(\frac{5}{16}\) pound have a combined weight equal to the weight of the large toy?
Answer:
2 small toys that each weigh \(\frac{5}{16}\) pound have a combined weight equal to the weight of the large toy,

Explanation:
Given a large toy weighs \(\frac{5}{8}\) pound.
Number of small toys that each weigh \(\frac{5}{16}\) pound
have a combined weight equal to the weight of the large toy so solving
\(\frac{5}{8}\) ÷ \(\frac{5}{16}\) = \(\frac{5}{8}\) X \(\frac{16}{5}\) = \(\frac{5 X 16}{8 X 5}\) = 2, therefore
2 small toys that each weigh \(\frac{5}{16}\) pound have a combined weight equal to the weight of the large toy.

Question 21.
Select all the expressions that have the same value as \(\frac{3}{5}\) ÷ \(\frac{6}{8}\).
A. \(\frac{3}{5}\) ÷ \(\frac{8}{6}\)
B. \(\frac{3}{5}\) × \(\frac{8}{6}\)
C. \(\frac{5}{3}\) × \(\frac{6}{8}\)
D. \(\frac{24}{40}\) ÷ \(\frac{30}{40}\)
E. \(\frac{24}{40}\) × \(\frac{40}{30}\)
Answer:
B. \(\frac{3}{5}\) × \(\frac{8}{6}\),
D. \(\frac{24}{40}\) ÷ \(\frac{30}{40}\) and
E. \(\frac{24}{40}\) × \(\frac{40}{30}\),

Explanation:
Given an expression is \(\frac{3}{5}\) ÷ \(\frac{6}{8}\) the value of the expression is \(\frac{3}{5}\) X \(\frac{8}{6}\) = \(\frac{3 X 8}{5 X 6}\) = \(\frac{4}{5}\), Now checking with bit A. \(\frac{3}{5}\) ÷ \(\frac{8}{6}\) = \(\frac{3}{5}\) X \(\frac{6}{8}\) =
\(\frac{3 X 6}{5 X 8}\) = \(\frac{9}{20}\) which will not match,
checking with bit B. \(\frac{3}{5}\) × \(\frac{8}{6}\) = \(\frac{3 X 8}{5 X 6}\) = \(\frac{4}{5}\) matches, checking with bit C. \(\frac{5}{3}\) × \(\frac{6}{8}\) = \(\frac{5 X 6}{3 X 8}\) = \(\frac{5}{4}\) will not match, checking with bit D. \(\frac{24}{40}\) ÷ \(\frac{30}{40}\) = \(\frac{24}{40}\) X \(\frac{40}{30}\) = \(\frac{24 X 40}{40 X 30}\) = \(\frac{4}{5}\) matches, checking with bit E. \(\frac{24}{40}\) × \(\frac{40}{30}\) = \(\frac{24 X 40}{40 X 30}\) = \(\frac{4}{5}\) matches, therefore bits B. \(\frac{3}{5}\) × \(\frac{8}{6}\),
D. \(\frac{24}{40}\) ÷ \(\frac{30}{40}\) and
E. \(\frac{24}{40}\) × \(\frac{40}{30}\) matches.

Spiral Review

Question 22.
On Monday, the temperature was -5 °F. On Tuesday, the temperature was -8 °F. Which temperature has a greater absolute value? Which temperature is colder?
Answer:
Greater absolute value is -8 °F,
temperature colder is -8 °F,

Explanation:
-8 is to the left of -5, and both are left of 0. The absolute value of a number may be thought of as its distance from zero along real number line.
The number farthest from 0 has the largest absolute value: -8 °F.
The number farthest to the left is colder: -8 °F.

Question 23.
What is the product of 2\(\frac{1}{4}\) and 1\(\frac{1}{4}\)?
Answer:
2\(\frac{13}{16}\),

Explanation:
2\(\frac{1}{4}\) X 1\(\frac{1}{4}\) = \(\frac{2 X 4 + 1}{4}\) X \(\frac{1 X 4 + 1}{4}\), \(\frac{9}{4}\) X \(\frac{5}{4}\) =
\(\frac{9 X 5}{4 X 4}\) = \(\frac{45}{16}\) as numerator is greater than denominator writing quotient in mixed fraction as 2\(\frac{13}{16}\).

Question 24.
Order the numbers from least to greatest.
–\(\frac{9}{4}\), -2.5, 2\(\frac{1}{2}\), 0, -2\(\frac{1}{3}\)
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Answer:
order the numbers from least to greatest -2.5 < -2\(\frac{1}{3}\) < –\(\frac{9}{4}\) < 0 < 2\(\frac{1}{2}\),

Explanation:
Given numbers are –\(\frac{9}{4}\), -2.5, 2\(\frac{1}{2}\), 0, -2\(\frac{1}{3}\) as  –\(\frac{9}{4}\) = -2.25, 2\(\frac{1}{2}\) = \(\frac{5}{2}\) = 2.5, -2\(\frac{1}{3}\) = –\(\frac{7}{3}\) = -2.33,
so -2.5 < -2.33 < -2.25 < 0 < 2.5 there are order the numbers from least to greatest
-2.5 < -2\(\frac{1}{3}\) < –\(\frac{9}{4}\) < 0 < 2\(\frac{1}{2}\).

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