Grade 6, Unit 4 - Practice Problems (2024)

Order from smallest to largest:

• Number of pennies in a stack that is 1 ft high
• Number of books in a stack that is 1 ft high
• Number of dollar bills in a stack that is 1 ft high
• Number of slices of bread in a stack that is 1 ft high

Use each of the numbers 4, 40, and 4000 once to make true statements.

1. The value of $\underline{\hspace{1in}} \div 40.01$ is close to 1

2. The value of $\underline{\hspace{1in}}\div 40.01$ is much less than 1.

3. The value of $ \underline{\hspace{1in}}\div 40.01$ is much greater than 1.

Without computing, decide whether the value of each expression is much smaller than 1, close to 1, or much greater than 1.

1. $100 \div \frac{1}{1000}$
2. $50\frac13 \div 50\frac14$
3. $4.7 \div 5.2$
4. $2 \div 7335$
5. $2,\!000,\!001 \div 9$
6. $0.002 \div 2,\!000$

A rocking horse has a weight limit of 60 pounds.

1. What percentage of the weight limit is 33pounds?
2. What percentage of the weight limit is 114pounds?

3. What weight is 95% of the limit?

Compare using $>$, $=$, or $<$.

1. 0.7 ______ 0.70

2. $0.03 + \frac{6}{10}$ ______ $0.30 + \frac{6}{100}$

3. 0.9 ______ 0.12

Diego has 90 songs on his playlist. How many songs are there for each genre?

1. 40% rock
2. 10% country
3. 30% hip-hop
4. The rest is electronica

A garden hose emits 9 quarts of water in 6 seconds. At this rate:

1. How long will it take the hose to emit 12 quarts?
2. How much water does the hose emit in 10 seconds?

Twenty pounds of strawberries are being shared equally by a group of friends. The equation $20 \div 5=4$ represents the division of strawberries.

A sixth-grade science club needs \$180 to pay for the tickets to a science museum.All ticketscost the same amount.

What could $180 \div 15$ mean in this context?Describe two interpretations of the expression.Then, find the quotient and explain what it means in each interpretation.

Write a multiplication equation that corresponds to each division equation.

Write a division or multiplication equation that represents each situation. Use a “?” for the unknown quantity.

1. 2.5 gallons of water are poured into 5 equally sized bottles. How much water is in each bottle?
2. A large bucket of 200 golf balls is divided into 4 smaller buckets. How many golf balls are in each small bucket?

3. Sixteen socks areput into pairs. How many pairs are there?

Find a value for $a$ thatmakes each statementtrue.

Complete the table. Write each percentage as a percent of 1.

Write a multiplication equation and a division equation that this diagram could represent.

Mai has \$36 to spend on movie tickets. Each movie ticket costs \$4.50. How many tickets can she buy?

1. Write a multiplication equation and a division equation to represent this situation.

Kiran said that this diagram can show the solution to $16\div 8 = {?}$ or $16 \div 2={?}$, depending on how we think about the equations and the “?”.

Write a sentence describing a situation that could berepresented by the equation $4 \div 1\frac13 = {?}$.

Noah said, “When you divide a number by a second number, the result will always be smaller than the first number.”

Jada said, “I think the result could be larger or smaller, depending on the numbers.”

Do you agree with Noah or Jada? Show or explain your reasoning.

Mini muffins cost \$3.00 per dozen.

• Andre says, “I have \$2.00, so I can afford 8 muffins.”
• Elena says, “I want to get 16 muffins, so I’ll need to pay \$4.00."

Do you agree with either, both, or neither of them? Explain your reasoning.

A family has a monthly budget of \$2,400. How much money is spent on each category?

A shopper buys cat food in bags of 3lbs. Her cat eats $\frac34$ lb each week. How many weeks does one bag last?

1. Draw a diagram to represent the situation and label your diagram so it can be followed by others. Answer the question.

2. Write a multiplication or division equation to represent the situation.

3. Multiply your answer in the first question (the number of weeks)by $\frac34$. Did you get 3 as a result? If not, revise your previous work.

Use thediagram to answer the question: How many $\frac13$s are in $1\frac23$? Thehexagon represents 1 whole. Explain or show your reasoning.

Which question can be represented by the equation ${?}\boldcdot \frac18=3$?

Write two division equations for each multiplication equation.

1. $15\boldcdot \frac25 = 6$
2. $6 \boldcdot \frac43 = 8$
3. $16\boldcdot \frac78 = 14$

Write a division expression with a quotient that is:

1. greater than $8 \div 0.001$
2. less than $8 \div 0.001$
3. between $8 \div 0.001$ and $8 \div \frac{1}{10}$

Find each unknown number.

Use the tape diagram to represent and find the value of$\frac12\div\frac13$.

What is the value of$\frac12\div\frac13$? Use pattern blocks to represent and find thisvalue. The yellow hexagon represents 1 whole. Explain or show your reasoning.

Use a standard inch ruler to answer each question. Then, write a multiplication equation and a division equation that answerthequestion.

1. How many $\frac12$s are in 7?

2. How many $\frac38$s are in 6?

3. How many $\frac{5}{16}$s are in $1\frac78$?

Use the tape diagram to represent and answer the question: How many $\frac25$s are in $1\frac12$?

Write a multiplication equation and a division equation to represent each question, statement, or diagram.

A recipe uses 5 cups of flour for every 2 cups of sugar.

1. How much sugar is used for 1 cup of flour?
2. How much flour is used for 1 cup of sugar?
3. How much flour is used with 7 cups of sugar?
4. How much sugar is used with 6 cups of flour?

How many groups of $\frac12$ days are in 1 week?

1. Write a multiplication equation or a division equation to represent the question.
2. Draw a tape diagram to show the relationship between the quantities and to answer the question. Use graph paper, if needed.

Diego said that the answer to the question “How many groups of $\frac56$ are in 1?” is $\frac 65$ or $1\frac15$. Do you agree with his statement? Explain or show your reasoning.

Select all equations that can represent the question: “How many groups of $\frac45$ are in 1?”

1. ${?} \boldcdot 1=\frac45$
2. $1 \boldcdot \frac45 = {?}$
3. $\frac45 \div 1 = {?}$
4. ${?} \boldcdot \frac45 =1$
5. $1\div \frac45 = {?}$

Calculate each percentage mentally.

A recipe calls for $\frac12$ lbof flour for 1 batch. How many batches can be madewith each of the following amounts?

1. 1 lb
2. $\frac34$ lb
3. $\frac14$ lb

Whiskers the cat weighs $2\frac23$ kg. Piglio weighs $4$ kg. For each question,write a multiplication and a division equation, decide whether the answer is greater or less than 1, and then answer the question.

1. How many times as heavy as Piglio is Whiskers?
2. How many times as heavy as Whiskers is Piglio?

Andre is walking from home to a festival that is $1\frac58$ kilometers away. He takes a quick rest after walking $\frac13$ kilometers. In this situation, which question can be represented by the equation: ${?} \boldcdot 1\frac58= \frac13$?

1. What fraction of the trip has Andre completed?
2. How many more kilometers does he have to walk to get to the festival?
3. What fraction of the trip is left?
4. How many kilometers is it from home to the festival and back home?

How many groups of $\frac34$ are in each of the following quantities?

1. $\frac{11}{4}$
2. $6\frac12$

Which question can be represented by the equation $4\div \frac27 = {?}$

For each scenario, use the given tape diagram to help you answer the question. Mark up and label the diagrams as needed.

1. Mai has picked 1cup of strawberries for a cake, whichis enough for $\frac34$ of the cake. How many cups does she need for the whole cake?

   

2. Priya has picked $1\frac12$ cups of raspberries, whichis enough for $\frac34$ of a cake. How many cups does she need for the whole cake?

   

Tyler painted $ \frac92$ square yards of wall area with 3 gallons of paint. How many gallons of paint does it take to paint each square yard of wall?

1. Write multiplication and division equations to represent the situation.

2. Draw a diagram to represent the situation and to answer the question.

After walking $\frac 14$ mile from home, Han is $\frac 13$ of his way to school. What is the distance between his home and school?

1. Write multiplication and division equations to represent this situation.

2. Use the given diagram to help you answer the question. Mark up and label it as needed.

   

Here is a division equation: $\frac45 \div \frac23 = {?}$

1. Write a multiplication equation that corresponds to the division equation.

2. Draw a diagram to represent and answer the question.

A set of books thatare each 1.5 inches wide are being organized on a bookshelf that is 36 inches wide. How many books can fit on the shelf?

1. Write a multiplication equation and a division equation to represent this question.

1. Without calculating, order theexpressionsbased ontheir values, from smallest to largest.

   $56\div8$

   $56\div8,\!000,\!000$

   $56\div 0.000008$

2. Explain how you decidedthe order of the three expressions.
3. Find a number$n$ so that $56\div n$is greater than 1but less than 7.

A group of friends is sharing $2\frac12$ pounds of berries.

1. If each friend received $\frac54$ of a pound of berries, how many friends are sharing the berries?
2. If 5 friends are sharing the berries, how many pounds of berries does each friend receive?

After raining for $\frac34$ of an hour, a rain gauge is $\frac25$ filled. If it continues to rain at that rate for 15 more minutes, what fraction of the rain gauge will be filled?

3 tickets to the museum cost \$12.75. At this rate, what is the cost of:

Elena went 60 meters in 15 seconds. Noah went 50 meters in 10 seconds. Elena and Noah both moved at a constant speed.

1. How far did Elena go in 1 second?
2. How far did Noah go in 1 second?

3. Who went faster? Explain or show your reasoning.

The first row in the table shows a recipe for 1batch of trail mix. Complete the remaining rows with recipes for 2, 3, and 4 batches of the same type of trail mix.

Priya is sharing 24 apples equally with some friends. She uses division to determine how many people can have a share if each person gets a particular number of apples. For example, $24 \div 4 = 6$ means that if each person gets 4 apples, 6 people can have apples. Here are some other calculations:

1. Priya thinks the “?” represents a number less than 24. Do you agree? Explain or show your reasoning.

2. In the case of $24 \div \frac12 = {?}$, how many people can have apples?

Here is a centimeter ruler.

3. Use your workfrom the first partto find each quotient.

   1. $18 \div \frac{1}{10}$

   2. $4\div \frac{2}{10}$

   3. $4\div \frac{8}{10}$

Find each quotient.

It takes one week for a crew of workers to pave$\frac35$ kilometer of a road. At that rate, how long will it take to pave 1 kilometer?

Write a multiplication equation and a division equation that represent the question and then answer the question.Show your reasoning.

Calculate each percentage mentally.

Select all statements that show correct reasoning for finding $\frac{14}{15}\div \frac{7}{5}$.

1. Multiplying $\frac{14}{15}$ by 5 and then by $\frac{1}{7}$.
2. Dividing $\frac{14}{15}$ by 5, and then multiplying by $\frac{1}{7}$.
3. Multiplying $\frac{14}{15}$ by 7, and then multiplying by $\frac{1}{5}$.
4. Multiplying $\frac{14}{15}$ by 5 and then dividing by 7.

Clare said that $\frac{4}{3}\div\frac52$ is $\frac{10}{3}$. Shereasoned: $\frac{4}{3} \boldcdot 5=\frac{20}{3}$ and $\frac{20}{3}\div 2=\frac{10}{3}$.

Explain why Clare’s answer and reasoning are incorrect. Find the correct quotient.

Kiran has $2\frac34$ pounds of flour. When he divides the flour into equal-sized bags, he fills $4\frac18$ bags. How many pounds fit in each bag?

Write a multiplication equation and a division equation to represent the question and then answer the question. Show your reasoning.

Divide $4\frac12$ by the following unit fractions.

After charging for $\frac13$ of an hour, a phone is at $\frac25$ of its full power. How long will it take the phone to charge completely?

Decide whether each equation can represent the situation.

Elena and Noah are each filling a bucket with water. Noah’s bucket is $\frac25$ full and the water weighs $2\frac12$ pounds. How much does Elena’s bucket weigh if her bucket is full and her bucket is identical to Noah’s?

1. Write multiplication and division equations to represent the question.
2. Draw a diagram to show the relationship between the quantities and to answer the question.

One inch is around $2\frac{11}{20}$ centimeters.

3. What question can be answered by finding $10 \div 2\frac{11}{20}$?

A zookeeper is $6\frac14$ feet tall. A young giraffe in his care is $9\frac38$ feet tall.

A rectangular bathroom floor is covered with square tiles that are $1\frac12$ feet by $1\frac12$ feet. The length of the bathroom floor is $10\frac12$ feet and the width is $6\frac12$ feet.

The Food and Drug Administration (FDA) recommends a certain amount of nutrient intake per day called the “daily value.” Food labels usually show percentages of the daily values for several different nutrients—calcium, iron, vitamins, etc.

In $\frac34$ cup of oatmeal, there is $\frac{1}{10}$ of the recommended daily value of iron. What fraction of the daily recommended value of iron is in 1 cup of oatmeal?

Write a multiplication equation and a division equation to represent the question, and then answer the question. Show your reasoning.

1. Find the unknown side length ofthe rectangle if its area is 11 m². Show your reasoning.

   

2. Check your answer by multiplying it bythe given side length ($3\frac 23$). Is the resulting product 11? If not, revisit your work forthe first question.

A television screen haslength $16\frac12$ inches, width $w$ inches, and area 462 square inches. Select all equations that represent the relationship of the side lengths and area of the television.

A bookshelf is 42 inches long.

1. How many books of length $1\frac12$ inches will fit on the bookshelf? Explain your reasoning.
2. A bookcase has 5 of these bookshelves. How many feet of shelf space is there? Explain your reasoning.

How many groups of $1\frac23$ are in each of the following quantities?

Thetriangle has an area of $7\frac{7}{8}$ cm² and a base of $5\frac14$ cm.

1. Which of the followingexpressionscan be usedto find how many cubes with edge length of $\frac13$ unit fit in a prism that is5 units by 5 units by 8 units? Explain or show your reasoning.

   1. $(5 \boldcdot \frac 13) \boldcdot (5 \boldcdot \frac 13) \boldcdot (8 \boldcdot \frac 13)$

   2. $5 \boldcdot 5 \boldcdot 8$

   3. $(5 \boldcdot 3) \boldcdot (5 \boldcdot 3) \boldcdot (8 \boldcdot 3)$

   4. $(5 \boldcdot 5 \boldcdot 8) \boldcdot (\frac 13)$

2. Mai says that we can also find the answer by multiplying the edge lengths of the prism and then multiplying the resultby 27. Do you agree with her statement? Explain your reasoning.

Find the value of each expression. Show your reasoning and check your answer.

A bucketcontains$11\frac23$ gallons of water and is $\frac56$ full. How many gallons of water would be in a full bucket?

Write a multiplication and a division equation to represent the situation, and then find the answer. Show your reasoning.

1. Lin wants to save \$75 for a trip to the city. If she has saved \$37.50 so far, what percentage of her goal has she saved? What percentage remains?
2. Noah wants to save \$60 so that he can purchase a concert ticket. If he has saved \$45 so far, what percentage of his goal has he saved? What percentage remains?

A rectangular prism measures $2\frac25$ inches by $3\frac15$ inches by 2 inch.

1. Priya said, “It takes more cubes with edge length$\frac25$ inch thancubes with edge length $\frac15$ inch to pack the prism.” Do you agree with Priya’s statement? Explain or show your reasoning.

1. Here is a right triangle. What is its area?

To give their animals essential minerals and nutrients, farmers and ranchers often have a block of salt—called “salt lick”—available for their animals to lick.

2. The box that contains the salt lick is $1\frac14$ feet by $1\frac23$ feet by $\frac56$ feet. How many cubes of salt lick fit in the box? Explain or show your reasoning.

1. How many groups of $\frac13$ inch are in $\frac34$ inch?
2. How many inches are in $1\frac25$ groups of $1\frac23$ inches?

Here is a table that shows the ratio of flour to water in an art paste. Complete the table with values in equivalent ratios.

An orange has about $\frac14$ cup of juice. How many oranges are needed to make $2\frac12$ cups of juice? Select all equationsthat represent this question.

1. $ {?} \boldcdot \frac 14= 2\frac12$
2. $\frac14 \div 2\frac12 = {?}$
3. ${?} \boldcdot 2\frac12 = \frac14$
4. $2\frac12 \div \frac14 = {?}$

Mai, Clare, and Tyler arehikingfrom aparking lot to the summit of a mountain. They pass a sign that gives distances.

Mai says: “We are one third of the way there.” Clare says: “We have to go twice as far as we have already gone.” Tyler says: “The total hike is three times as long as what we have already gone.”

Can they all be correct? Explain how you know.

Priya’scat weighs $5\frac12$ pounds and her dog weighs $8\frac14$ pounds. Estimate the missing number in each statementbefore calculatingthe answer. Then, compareyour answer to the estimateand explain any discrepancy.

1. The catis_______ asheavy as the dog.

2. Their combined weight is _______ pounds.

3. Thedog is _______ pounds heavier thanthe cat.

Before refrigerators existed, some people had blocks of ice delivered to their homes. A delivery wagon had a storage box in the shape of a rectangular prism that was $7\frac12$ feet by 6 feet by 6 feet. The cubic ice blocks stored in the box had side lengths $1\frac12$ feet. How many ice blocks fit in the storage box?

Fill in the blanks with 0.001, 0.1, 10, or 1000 so that the value of each quotientis in the correctcolumn.

Jada has some pennies and dimes. The ratio of Jada’s pennies to dimes is 2 to 3.