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Relate multiplication and division

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5157983
Use the related multiplication fact to find or check each quotient. a) \(63 \div 9 = \square\); check with \(\square \times 9 = 63\) b) \(32 \div 4 = \square\); check with \(\square \times 4 = 32\) c) \(81 \div 9 = \square\); check with \(\square \times 9 = 81\)

Hints

- Use the related multiplication equation for each part. - The multiplication product must equal the dividend. - Recall the matching multiplication fact.

Solution

1. Since \(7 \times 9 = 63\), \(63 \div 9 = 7\). 2. Since \(8 \times 4 = 32\), \(32 \div 4 = 8\). 3. Since \(9 \times 9 = 81\), \(81 \div 9 = 9\).

Answer

a) \(63 \div 9 = 7\); check: \(7 \times 9 = 63\) b) \(32 \div 4 = 8\); check: \(8 \times 4 = 32\) c) \(81 \div 9 = 9\); check: \(9 \times 9 = 81\)
5158063
Find each quotient. Then write the related multiplication fact to check your answer. a) \(32 \div 4\) b) \(42 \div 7\) c) \(45 \div 5\) d) \(56 \div 8\)

Hints

- Which number multiplied by the divisor gives the dividend? - Use the related multiplication fact to check each quotient. - Look for the dividend in the divisor’s multiplication facts.

Solution

1. \(32 \div 4 = 8\) because \(8 \times 4 = 32\). 2. \(42 \div 7 = 6\) because \(6 \times 7 = 42\). 3. \(45 \div 5 = 9\) because \(9 \times 5 = 45\). 4. \(56 \div 8 = 7\) because \(7 \times 8 = 56\).

Answer

a) \(32 \div 4 = 8\); check: \(8 \times 4 = 32\) b) \(42 \div 7 = 6\); check: \(6 \times 7 = 42\) c) \(45 \div 5 = 9\); check: \(9 \times 5 = 45\) d) \(56 \div 8 = 7\); check: \(7 \times 8 = 56\)
5183303
Find the missing numbers. Use the 8s multiplication facts. \(32 \div 8 = \square\) \(48 \div \square = 6\) \(\square \div 8 = 3\) \(\square \div 8 = 10\)

Hints

- What multiplication fact is related to each division equation? - To find a missing dividend, multiply the divisor and quotient. - Check each completed equation with multiplication.

Solution

1. Since \(4 \times 8 = 32\), \(32 \div 8 = 4\). 2. Since \(6 \times 8 = 48\), the missing divisor in \(48 \div \square = 6\) is \(8\). 3. Since \(3 \times 8 = 24\), the missing dividend in \(\square \div 8 = 3\) is \(24\). 4. Since \(10 \times 8 = 80\), the missing dividend in \(\square \div 8 = 10\) is \(80\).

Answer

\(32 \div 8 = 4\) \(48 \div 8 = 6\) \(24 \div 8 = 3\) \(80 \div 8 = 10\)
5372543
The array shows \(32\) dots. a) Which multiplication equation uses the number of rows as the first factor to describe the array? b) The \(32\) dots are arranged equally in \(4\) rows. How many dots are in each row? c) Write the two related division equations.
Figure for problem 537254

Hints

- Count the rows and columns. - Division can undo multiplication. - Use the number of rows once and the number of columns once as the divisor.

Solution

1. The array has \(4\) rows and \(8\) columns, so the multiplication equation is \(4 \times 8 = 32\). 2. Dividing \(32\) dots among \(4\) rows gives \(32 \div 4 = 8\) dots per row. 3. The two related division equations are \(32 \div 4 = 8\) and \(32 \div 8 = 4\).

Answer

a) \(4 \times 8 = 32\) b) \(8\) dots c) \(32 \div 4 = 8\) and \(32 \div 8 = 4\)
5373553
Arrays a) and b) contain the same number of dots. Find a multiplication equation for array b) and derive it from the square array a).
Figure for problem 537355

Hints

- Use the known square product \(36\). - Which number multiplied by \(4\) gives \(36\)?

Solution

1. Array a) is a square with \(6 \times 6 = 36\) dots. 2. Array b) has \(4\) rows. Since \(36 \div 4 = 9\), each row has \(9\) dots. 3. Therefore, array b) shows \(4 \times 9 = 36\).

Answer

a) \(6 \times 6 = 36\) b) Since \(36 \div 4 = 9\), \(4 \times 9 = 36\).
5381213
The graph shows the kinds of books students read. Which kind was read half as many times as comics?
Figure for problem 538121

Hints

- Read the comics bar first. - Find half of that number. - Look for the bar with the matching value.

Solution

1. Comics were read \(20\) times. 2. Half of \(20\) is \(10\). 3. The mystery-book bar shows \(10\), so mystery books were read half as many times as comics.

Answer

Mystery books were read half as many times as comics.
5158073
Find each missing number. Use related multiplication facts. a) \(\square \div 6 = 7\) b) \(63 \div \square = 9\) c) \(\square \div 8 = 4\) d) \(81 \div 9 = \square\)

Hints

- When the dividend is missing, multiply the divisor and quotient. - When the divisor is missing, use the related multiplication equation. - Recall the square facts for the last equation.

Solution

1. For \(\square \div 6 = 7\), multiply \(7 \times 6 = 42\), so the dividend is \(42\). 2. For \(63 \div \square = 9\), solve \(9 \times \square = 63\). Since \(9 \times 7 = 63\), the divisor is \(7\). 3. For \(\square \div 8 = 4\), multiply \(4 \times 8 = 32\), so the dividend is \(32\). 4. Since \(9 \times 9 = 81\), \(81 \div 9 = 9\).

Answer

a) \(42 \div 6 = 7\) b) \(63 \div 7 = 9\) c) \(32 \div 8 = 4\) d) \(81 \div 9 = 9\)
5158163
What division rule is shown in the table? Find the divisor in the table heading, complete the missing entries, and add your own example in the last row. <table> <tr><th colspan="2">\(\div \square\)</th></tr> <tr><td>\(18\)</td><td>\(2\)</td></tr> <tr><td>\(54\)</td><td></td></tr> <tr><td></td><td>\(9\)</td></tr> <tr><td></td><td></td></tr> </table>

Hints

- Which divisor makes the first row true? - To work from the quotient back to the dividend, use multiplication. - Use multiplication facts for the divisor you found to create your own row.

Solution

1. Since \(18 \div 9 = 2\), the divisor is \(9\). 2. Complete the second row: \(54 \div 9 = 6\). 3. Work backward in the third row: \(9 \times 9 = 81\), so \(81 \div 9 = 9\). 4. Any correct division fact with divisor \(9\) works in the last row, such as \(90 \div 9 = 10\).

Answer

The rule is divide by \(9\). The missing entries are \(6\) and \(81\). One possible final row is \(90\) and \(10\), because \(90 \div 9 = 10\).
5187123
Complete the multiplication table. First find the missing factors in the top row and first column. <table> <tr><td>\(\times\)</td><td>\(\square\)</td><td>\(7\)</td><td>\(\square\)</td></tr> <tr><td>\(2\)</td><td>\(16\)</td><td>\(\square\)</td><td>\(20\)</td></tr> <tr><td>\(\square\)</td><td>\(\square\)</td><td>\(21\)</td><td>\(\square\)</td></tr> </table>

Hints

- How can you find a missing factor when the product and one factor are known? - Use the inverse operation. - Complete the headers before filling the remaining products.

Solution

1. Since \(16 \div 2 = 8\), the first missing factor in the top row is \(8\). 2. Since \(20 \div 2 = 10\), the last missing factor in the top row is \(10\). 3. Since \(21 \div 7 = 3\), the missing factor in the first column is \(3\). 4. Complete the products: \(2 \times 7 = 14\), \(3 \times 8 = 24\), and \(3 \times 10 = 30\).

Answer

The completed table is: <table> <tr><td>\(\times\)</td><td>\(8\)</td><td>\(7\)</td><td>\(10\)</td></tr> <tr><td>\(2\)</td><td>\(16\)</td><td>\(14\)</td><td>\(20\)</td></tr> <tr><td>\(3\)</td><td>\(24\)</td><td>\(21\)</td><td>\(30\)</td></tr> </table>
5363013
Fill in the missing numbers in the product wall. Each brick contains the product of the two bricks directly below it. Every value is at most \(100\).
Figure for problem 536301

Hints

- Division is the inverse of multiplication, so it can help you find a missing lower brick. - Begin where only one number in a group of three connected bricks is missing.

Solution

1. Let the bottom-left number be \(x\). Since \(x \times 4 = 8\), \(x = 8 \div 4 = 2\). 2. Let the second-row right number be \(y\). Since \(8 \times y = 64\), \(y = 64 \div 8 = 8\). 3. Let the bottom-right number be \(z\). Since \(4 \times z = 8\), \(z = 8 \div 4 = 2\).

Answer

The bottom row is \(2, 4, 2\). The missing number in the second row is \(8\).
5381203
Young trees were counted in a park. Which kind of tree has exactly twice as many trees as another kind? Name both kinds.
Figure for problem 538120

Hints

- Read the number for each kind of tree. - Look for two numbers where one is two times the other. - Check the relationship with multiplication.

Solution

1. The graph shows \(18\) oak trees and \(9\) beech trees. 2. Since \(2 \times 9 = 18\), the number of oak trees is twice the number of beech trees.

Answer

There are exactly twice as many oak trees as beech trees.
5381553
Which bar is three times as tall as another bar? State the comparison using the bar names.
Figure for problem 538155

Hints

- Read all three bar values. - Look for two values where one is three times the other. - Check the relationship with multiplication.

Solution

1. Bar B has a value of \(24\), and Bar A has a value of \(8\). 2. Since \(3 \times 8 = 24\), Bar B is three times as tall as Bar A.

Answer

Bar B is three times as tall as Bar A.
5382003
The pie chart shows votes for fruit. 1) State one correct comparison using “twice as many votes as.” 2) State another correct comparison using “twice as many votes as.”
Figure for problem 538200

Hints

- Read all four slice values. - Look for a value that is two times another value. - Find two different comparisons.

Solution

1. Apple has \(12\) votes, and pear has \(6\) votes. Since \(2 \times 6 = 12\), apple has twice as many votes as pear. 2. Pear has \(6\) votes, and melon has \(3\) votes. Since \(2 \times 3 = 6\), pear has twice as many votes as melon.

Answer

1) Apple has twice as many votes as pear. 2) Pear has twice as many votes as melon.

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