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Understand division

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5373583
A rectangular array contains \(42\) dots with \(7\) dots in each row. Find the number of rows. Write the matching division equation and multiplication equation.
Figure for problem 537358

Hints

- How many groups of \(7\) are in \(42\)? - Use a known multiplication fact.

Solution

1. Find how many groups of \(7\) make \(42\): \(42 \div 7 = 6\). 2. The array has \(6\) rows. 3. The related multiplication equation is \(6 \times 7 = 42\).

Answer

\(6\) rows; \(42 \div 7 = 6\) and \(6 \times 7 = 42\).
5373613
Forty-eight beads are shared equally among \(8\) bowls. How many beads go in each bowl? Explain how you can read the answer from the rectangular array.
Figure for problem 537361

Hints

- Read the array as \(8\) equal groups. - How many dots are in one group?

Solution

1. Read the array as \(8\) equal rows. 2. Each row contains \(6\) dots, so each bowl receives \(6\) beads. 3. The division equation is \(48 \div 8 = 6\).

Answer

\(6\) beads per bowl; \(48 \div 8 = 6\). The array has \(8\) equal rows with \(6\) dots in each row.
5373643
The array shows \(32\) dots in \(8\) equal rows. What number belongs in the blank? \(32 \div 8 = \square\) Write a related multiplication equation to check your answer.
Figure for problem 537364

Hints

- Read the number of dots in each row. - To check, multiply the quotient by the divisor.

Solution

1. Each of the \(8\) rows contains \(4\) dots, so \(32 \div 8 = 4\). 2. The related multiplication equation is \(8 \times 4 = 32\).

Answer

\(32 \div 8 = 4\); check: \(8 \times 4 = 32\).
5373623
The same array can be described by \(54 \div 6\) and \(54 \div 9\). Explain what question each division equation answers.
Figure for problem 537362

Hints

- Distinguish the number of groups from the size of each group. - Use the multiplication equation \(6 \times 9 = 54\).

Solution

1. The array has \(6\) rows with \(9\) dots in each row. 2. \(54 \div 6 = 9\) answers: How many dots are in each of the \(6\) rows? 3. \(54 \div 9 = 6\) answers: How many rows of \(9\) dots are there?

Answer

\(54 \div 6 = 9\): the number of dots in each row. \(54 \div 9 = 6\): the number of rows.
5373663
Each rectangular array contains \(36\) dots. For each array, write a division equation whose quotient is the number of rows.
Figure for problem 537366

Hints

- The number of columns tells you how many dots are in each row. - Divide \(36\) by the number of dots in each row.

Solution

1. In array a), each row has \(9\) dots, so \(36 \div 9 = 4\) rows. 2. In array b), each row has \(6\) dots, so \(36 \div 6 = 6\) rows. 3. In array c), each row has \(4\) dots, so \(36 \div 4 = 9\) rows.

Answer

a) \(36 \div 9 = 4\) b) \(36 \div 6 = 6\) c) \(36 \div 4 = 9\)
5373793
A wildlife camera records \(54\) animal tracks. The image groups the tracks into three equal sections. How many tracks are in each section? Write a multiplication equation and a division equation that describe the image.
Figure for problem 537379

Hints

- Compare the sizes of the three sections. - The total is separated into three equal groups.

Solution

1. Each section contains \(18\) tracks. 2. The \(54\) tracks are divided into \(3\) equal groups. 3. The matching equations are \(3 \times 18 = 54\) and \(54 \div 3 = 18\).

Answer

Each section has \(18\) tracks. \(3 \times 18 = 54\) and \(54 \div 3 = 18\).
5194083
Leon has \(18\) marbles. He considers two ways to use them: A) He shares all the marbles equally among \(6\) friends. B) He puts \(6\) marbles in each bag until none are left. Explain how the two situations are different. What is the result in each situation?

Hints

- In A, decide whether the unknown is the number of groups or the number in each group. - In B, decide what the quotient counts. - Compare the meaning of the quotient in the two situations.

Solution

1. In situation A, the number of groups is known, and the number in each group is unknown: \(18 \div 6 = 3\). Each friend gets \(3\) marbles. 2. In situation B, the size of each group is known, and the number of groups is unknown: \(18 \div 6 = 3\). Leon fills \(3\) bags. 3. The division equation is the same, but the quotient represents marbles per friend in A and number of bags in B.

Answer

A) Each of the \(6\) friends gets \(3\) marbles. B) Leon fills \(3\) bags with \(6\) marbles in each bag.
5194093
A garden center harvested \(320\) tulips. 1) Write a word-problem question for \(320 \div 4\) in which the answer represents the number of tulips in each vase. 2) Write a different word-problem question for \(320 \div 4\) in which the answer represents the number of bouquets. 3) Calculate and interpret the answer in both situations.

Hints

- In the first situation, divide a total equally among a known number of groups. - In the second situation, make groups of a known size. - The calculation is the same, but the meaning of the quotient changes.

Solution

1. One possible sharing question is, “How many tulips go in each vase if \(320\) tulips are shared equally among \(4\) vases?” 2. One possible grouping question is, “How many bouquets can be made if each bouquet uses \(4\) tulips?” 3. In both cases, \(320 \div 4 = 80\). The first answer is \(80\) tulips per vase, and the second is \(80\) bouquets.

Answer

1) Example: “How many tulips go in each vase if \(320\) tulips are shared equally among \(4\) vases?” 2) Example: “How many bouquets can be made from \(320\) tulips if each bouquet has \(4\) tulips?” 3) \(80\) tulips per vase and \(80\) bouquets
5200583
Lara has \(12\) marbles and considers two situations: Situation A: She shares the marbles equally among \(4\) bags. Situation B: She puts \(4\) marbles in each bag. 1. Calculate \(12 \div 4\). 2. In which situation does the quotient tell how many bags Lara needs? 3. In which situation does the quotient tell how many marbles are in each bag?

Hints

- In situation A, identify what is fixed and what is unknown. - Do the same for situation B. - The quotient is the same, but its unit is different.

Solution

1. Calculate the quotient: \(12 \div 4 = 3\). 2. In situation B, the group size is \(4\) marbles, so the quotient \(3\) is the number of bags. 3. In situation A, the number of groups is \(4\), so the quotient \(3\) is the number of marbles in each bag.

Answer

1. \(12 \div 4 = 3\). 2. Situation B; the quotient is the number of bags. 3. Situation A; the quotient is the number of marbles in each bag.
5200593
A baker uses the equation \(12 \div 4 = 3\) to plan two ways to pack rolls. Fill in each blank so the statement matches the equation. 1) “I put \(\_\_\_\) rolls in each bag. Then I fill \(3\) bags.” 2) “I share the rolls equally among \(\_\_\_\) bags. Then each bag has \(3\) rolls.”

Hints

- Use all three numbers in \(12 \div 4 = 3\). - In statement 1, decide what the divisor represents. - In statement 2, decide what the divisor represents.

Solution

1. In statement 1, the quotient \(3\) is the number of bags, so the divisor is the number of rolls in each bag. The missing number is \(4\). 2. In statement 2, the quotient \(3\) is the number of rolls in each bag, so the divisor is the number of bags. The missing number is \(4\).

Answer

1) Put \(4\) rolls in each bag. 2) Share the rolls equally among \(4\) bags.
5373653
Both diagrams show \(36\) dots. In situation A, the dots are shared equally among \(6\) children. In situation B, groups of \(9\) dots are made. Find both quotients and explain how the two division questions are different.
Figure for problem 537365

Hints

- For each situation, identify whether the number of groups or the group size is given. - Write a division equation for each diagram.

Solution

1. Situation A asks for the number in each group: \(36 \div 6 = 6\), so each child gets \(6\) dots. 2. Situation B asks for the number of groups: \(36 \div 9 = 4\), so there are \(4\) groups. 3. Both situations use division, but A gives the number of groups and asks for the size of each group, while B gives the group size and asks for the number of groups.

Answer

A: Each child gets \(6\) dots because the number of groups is known. B: There are \(4\) groups because the size of each group is known.

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