Section C Practice Problems

Problem 1

If 5 pencils cost 95 cents, how much does each pencil cost? Explain or show your reasoning.
If 68 colored pencils are split evenly between 4 students, how many pencils does each student get? Explain or show your reasoning.

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Solution

19 cents. Sample response: $5 \times 10 = 50$ , $5 \times 9 = 45$ , and $50 + 45 = 95$ .
17 pencils. Sample response: $4 \times 10 = 40$ , $4 \times 7 = 28$ , and $40 + 28 = 68$ .

Problem 2

Clare has 194 square tiles. Can Clare put all of her tiles in 6 rows with the same number of tiles in each row? Explain or show your reasoning.

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Solution

No. Sample response: Clare can make 6 rows of 30 and that's 180 tiles. Then she can add 2 more to each row, so that's 192 tiles in 6 rows of 32, but she would have 2 tiles left over.

Problem 3

A long, rectangular hallway is 8 feet wide and has an area of 368 square feet. How long is the hallway?

Write a multiplication equation and a division equation that represent the situation.
Find the length of the hallway. Explain or show your reasoning.

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Solution

Sample response: $8 \times {?} = 368$ , $368 \div 8 = {?}$
46 feet. Sample responses:
- $8 \times 40 =320$ and $8 \times 6 = 48$ , so $8 \times 46 = 368$ . The hallway is 46 feet long.

Problem 4

Here is 378 represented with base-ten blocks.

base ten diagram. 3 hundreds, 7 tens, 8 ones.

Use words, diagrams, or equations to show how to use the base-ten blocks to find the value of $378 \div 6$ .

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Solution

Sample response: I can divide the 3 hundreds into 6 equal groups of 50. Then I can put 1 ten in each of the 6 groups. Finally, I can put 3 ones in each group. I have $50 + 10 + 3$ , which is 63, in each group.

Problem 5

Here are 2 incomplete calculations of $864 \div 4$ . Complete each calculation to find the value of the quotient.

$\begin{align} 800\div 4&= 200\\ 40\div 4 &= \phantom{010}\\ 20 \div 4 &= \phantom{005} \\4 \div 4&= \phantom{001} \\ \overline {\hspace{5mm}864 \div 4} &\overline{\hspace{1mm}= \phantom{000000}}\\ \end{align}$

Divide. 8 hundred sixty 4 divided by 4, 6 rows.

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Solution

Sample response:

$\begin{align} 800\div 4&= 200\\ 40\div 4 &= \phantom{0}10\\ 20 \div 4 &= \phantom{00}5 \\4 \div4&=\phantom{00}1 \\ \overline {\hspace{5mm}864 \div 4} &\overline{\hspace{1mm}= 216 \phantom{000}}\end{align}$

Problem 6

What is the value of $3,420\div5$ ?

Use partial quotients to show your reasoning.

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Solution

684. Sample response: partial quotients 600, 80, 4

Problem 7

Use partial quotients to find the value of $637 \div 4$ .
There are 637 toothpicks and 4 people. What could $637 \div 4$ mean in this situation? What could each step you took in the algorithm mean?
What does the value of the quotient represent in the situation?

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Solution

Sample response:

637 toothpicks are divided equally among 4 people. First, each person gets 100 toothpicks. Then each person gets 50 more and 9 more.
Each person gets 159 toothpicks, but there is 1 toothpick left over.

Problem 8

There are 875 peaches at the orchard. Each box contains 9 peaches. How many boxes are needed for the peaches? Explain your reasoning.

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Solution

Sample response: 90 boxes will hold 810 peaches. That leaves 65 peaches. Seven more boxes will hold 63 of those peaches. This means 97 boxes will hold 873 peaches and there will either be 2 peaches leftover or another box can hold them but it won't be full.

Problem 9

Priya writes the multiples of a number and 63 is on her list. Priya's number is not 1.

What could Priya's number be? Explain your reasoning.
112 is the last number on Priya's list. What is Priya's number? How many numbers are on Priya's list?

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Solution

Priya's number could be 3, 7, 9, or 21. Sample response: These are factors of 63.
Priya's number is 7. Sample response: 3, 9, and 21 are not factors of 112. It is the 16th number on Priya's list because $16 \times 7 = 112$ .

Problem 10

Consider the expression $286 \div 5$ .

Write a division story with a question that can be answered by finding the value of $286 \div 5$ . Then answer the question.
Write a different story with a question that can be answered by finding the value of $286 \div 5$ but with a different answer than your first story. Answer the question.

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Solution

Sample response:

Story and question: There are 286 kids. How many basketball teams can they make if there are 5 kids per team? Answer: 57 teams, but there is 1 kid without a team.
Story and question: There are 286 pounds of sand. Each bag holds 5 pounds of sand. How many bags do you need to hold all of the sand? Answer: 58 bags, but a whole bag is needed for 1 extra pound of sand.

Problem 11

Mai has a special way of knowing that 531 is a multiple of 9. She says, “Each hundred is 11 nines and 1 more and each ten is one nine and 1 more, so 531 is 58 nines and 9 more.”

Make sense of and explain Mai’s reasoning. Is 531 a multiple of 9?
Use Mai's reasoning to decide if 648 is a multiple of 9.

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Solution

Sample responses:

$100 = (11 \times 9) + 1$ , so $500 = (55 \times 9) + 5$ .

$10 = 9 + 1$ , so $30 = (3 \times 9) + 3$ .

$530 = 500 + 30$ , so $530 = (58 \times 9) + 8$ .

Adding 1 to 530 means adding 1 to $(58 \times 9) + 8$ , which gives $(58 \times 9) + 9$ . The additional 9 make a new group of 9, so $513 = 59 \times 9$ . Yes, 531 is a multiple of 9.
Yes, because:

$648 = 600 + 40 + 8$

$600 = (66 \times 9) + 6$

$40 = (4 \times 9) + 4$

There are $66+4$ or 70 groups of 9 in 640, but there are also some remainders: $6 + 4 + 8$ or 18, which is 2 groups of 9. Altogether, there are 72 groups of 9 in 648 and no remainders, or $648 = 72 \times 9$ .