Free Math Worksheets

- Think of the numbers on a number line. - Find the difference between \(320{,}000\) and each benchmark. - Check whether you are measuring upward or downward from \(320{,}000\).

1. Find how much greater \(320{,}000\) is than \(300{,}000\): \(320{,}000 - 300{,}000 = 20{,}000\). 2. Find how much less \(320{,}000\) is than \(400{,}000\): \(400{,}000 - 320{,}000 = 80{,}000\).

The population is \(20{,}000\) more than \(300{,}000\). The population is \(80{,}000\) less than \(400{,}000\).

- For a), think about the operation used to find how many more. - For b), determine how many groups of the smaller amount fit in the larger amount. - Notice the difference between “how many more” and “how many times as many.”

1. For the additive comparison, subtract: \(35 - 7 = 28\). 2. For the multiplicative comparison, divide: \(35 \div 7 = 5\).

a) The bakery made \(28\) more pretzels. b) The bakery made \(5\) times as many pretzels as croissants.

- Pay attention to the difference between “times as large” and “greater than.” - Decide whether each part calls for addition or multiplication. - Rewrite each statement as a numerical expression.

1. For part a, multiply: \(12 \times 9 = 108\). 2. For part b, add: \(12 + 9 = 21\). 3. For part c, add: \(40 + 15 = 55\). 4. For part d, multiply: \(40 \times 4 = 160\).

a) \(108\) b) \(21\) c) \(55\) d) \(160\)

- Use subtraction to find how much greater one number is. - Use division to find how many times as large one number is. - Check each multiplicative comparison with multiplication.

1. For \(48\) and \(6\): The additive difference is \(48 - 6 = 42\). The multiplicative comparison is \(48 \div 6 = 8\). Thus, \(48\) is \(42\) greater than and \(8\) times as large as \(6\). 2. For \(9\) and \(45\): The additive difference is \(45 - 9 = 36\). The multiplicative comparison is \(45 \div 9 = 5\). Thus, \(45\) is \(36\) greater than and \(5\) times as large as \(9\). 3. For \(7\) and \(42\): The additive difference is \(42 - 7 = 35\). The multiplicative comparison is \(42 \div 7 = 6\). Thus, \(42\) is \(35\) greater than and \(6\) times as large as \(7\).

- \(48\) and \(6\): \(42\) greater; \(8\) times as large - \(9\) and \(45\): \(36\) greater; \(5\) times as large - \(7\) and \(42\): \(35\) greater; \(6\) times as large

- For part a), determine how many times the shorter length fits into the longer length. - For part b), find the difference between the two lengths. - Which operation finds how much longer one object is than another?

1. Compare the lengths multiplicatively: \(240 \div 40 = 6\). 2. Compare the lengths additively: \(240\,\text{cm} - 40\,\text{cm} = 200\,\text{cm}\).

a) The blue rope is \(6\) times as long as the red rope. b) The blue rope is \(200\,\text{cm}\) longer than the red rope.

- For a), use the operation that finds the difference between two numbers. - For b), think about how many equal groups of \(75\) make \(300\). - Doubling means multiplying by \(2\).

1. For a), find the additive difference: \(300 - 75 = 225\). 2. For b), find the multiplicative comparison: \(300 \div 75 = 4\). 3. For c), double \(75\): \(75 \times 2 = 150\). Then \(300 \div 150 = 2\).

a) \(300\) is \(225\) greater than \(75\). b) \(75\) fits into \(300\) four times. c) The new number is \(150\), and it fits into \(300\) twice.

- “Increase by” indicates addition. - “Four times” indicates multiplication. - Subtract the smaller result from the larger result.

1. Part a: Increase by \(4\): \(8 + 4 = 12\). 2. Part b: Four times \(8\): \(8 \times 4 = 32\). 3. Compare the results: \(32 - 12 = 20\).

a) \(12\) b) \(32\) The larger result is \(20\) greater than the smaller result.

- Write half a million as a number. - Find the difference between \(545{,}000\) and each benchmark. - Pay attention to whether the comparison says more or less.

1. Half a million is \(500{,}000\). 2. The amount above half a million is \(545{,}000 - 500{,}000 = 45{,}000\). 3. The amount below \(600{,}000\) is \(600{,}000 - 545{,}000 = 55{,}000\).

That is \(45{,}000\) more than half a million. That is \(55{,}000\) less than \(600{,}000\).

- Write three-quarters of a million as a number. - Find the difference between \(785{,}500\) and each benchmark. - For part b, subtract the given number from \(800{,}000\).

1. Three-quarters of a million is \(750{,}000\). 2. The amount above \(750{,}000\) is \(785{,}500 - 750{,}000 = 35{,}500\). 3. The amount needed to reach \(800{,}000\) is \(800{,}000 - 785{,}500 = 14{,}500\).

a) \(35{,}500\) b) \(14{,}500\)

- First find how many blue balloons there are. - Read carefully: \(40\) is the difference, not the number of blue balloons. - Once you know both quantities, determine how many groups of the smaller quantity make the larger quantity.

1. Find the number of blue balloons: \(48 - 40 = 8\). 2. Compare the quantities multiplicatively: \(48 \div 8 = 6\).

a) She buys \(8\) blue balloons. b) She buys \(6\) times as many red balloons as blue balloons.

- Find the remaining length first. - Compare the two pieces using subtraction for part b). - Use division to find how many times one length fits into the other.

1. Subtract to find the remaining length: \(35 - 7 = 28\,\text{m}\). 2. Find the difference between the pieces: \(28 - 7 = 21\,\text{m}\). 3. Divide to find the multiplicative comparison: \(28 \div 7 = 4\).

a) \(28\,\text{m}\) b) \(21\,\text{m}\) longer c) \(4\) times as long

- Write one question about the difference between the amounts. - Write another question about how many times the original amount fits into the new amount. - Use subtraction for one question and division for the other.

1. An additive comparison question is, “How many books were added?” Subtract: \(72 - 8 = 64\). 2. A multiplicative comparison question is, “How many times as many books are there now as there were originally?” Divide: \(72 \div 8 = 9\).

One possible pair is: 1. “How many books were added?” \(64\) books were added. 2. “How many times as many books are there now?” There are \(9\) times as many books now.

- Find how much water remains first. - Use subtraction to find how much greater one amount is. - Use division to find how many times as great one amount is.

1. Find the amount left: \(48 - 8 = 40\,\text{L}\). 2. Find the additive difference: \(40 - 8 = 32\,\text{L}\). 3. Find the multiplicative comparison: \(40 \div 8 = 5\).

a) The amount left is \(32\,\text{L}\) greater. b) The amount left is \(5\) times as great.

- Use subtraction for the additive difference. - Use division for the multiplicative comparison. - Relate division with multiples of \(10\) to basic multiplication facts.

1. For \(240\) and \(30\): The difference is \(240 - 30 = 210\), and \(240 \div 30 = 8\). The larger number is \(8\) times as large as the smaller number. 2. For \(60\) and \(420\): The difference is \(420 - 60 = 360\), and \(420 \div 60 = 7\). The larger number is \(7\) times as large as the smaller number. 3. For \(810\) and \(90\): The difference is \(810 - 90 = 720\), and \(810 \div 90 = 9\). The larger number is \(9\) times as large as the smaller number.

- \(240\) and \(30\): difference \(210\); \(8\) times as large - \(60\) and \(420\): difference \(360\); \(7\) times as large - \(810\) and \(90\): difference \(720\); \(9\) times as large

- Use subtraction to find how many more. - Use division to find how many times as many. - Notice the difference between additive and multiplicative comparison language.

1. An additive comparison question is, “How many more cards does Elias have than Mia?” Subtract: \(32 - 8 = 24\). 2. A multiplicative comparison question is, “How many times as many cards does Elias have as Mia?” Divide: \(32 \div 8 = 4\).

a) Question: “How many more cards does Elias have than Mia?” Elias has \(24\) more cards. b) Question: “How many times as many cards does Elias have as Mia?” Elias has \(4\) times as many cards.

- Use two different operations for the two statements. - Which operation finds a difference? - Which operation finds how many times one amount fits into another? - Check each statement with its own calculation.

1. Subtraction finds the additive difference: \(15 - 3 = 12\,\text{L}\). The bucket holds \(12\,\text{L}\) more. 2. Division finds the multiplicative comparison: \(15 \div 3 = 5\). The bucket holds \(5\) times as much.

Statement A: \(15 - 3 = 12\), which finds the difference in liters. Statement B: \(15 \div 3 = 5\), which finds how many times as much the bucket holds.

- “Difference” tells you to subtract. - “How many times as great” tells you to divide. - For c), find the new difference and compare it with the answer to a).

1. The original difference is \(360 - 120 = 240\). 2. The multiplicative comparison is \(360 \div 120 = 3\), so \(360\) is \(3\) times as great as \(120\). 3. Double the smaller number: \(120 \times 2 = 240\). 4. The new difference is \(360 - 240 = 120\). 5. Since \(240 \div 2 = 120\), the new difference is half the original difference. The claim is true.

a) \(240\) b) \(360\) is \(3\) times as great as \(120\). c) The claim is true. After \(120\) is doubled to \(240\), the new difference is \(120\), which is half of \(240\).

- Distinguish between “three times as many” and “\(300\) more.” - Find each girl's number of stickers. - Compare the two results.

1. Find Sarah's number of stickers: \(120 \times 3 = 360\). 2. Find Mia's number of stickers: \(120 + 300 = 420\). 3. Since \(420 > 360\), Mia has more stickers.

Mia has more stickers. She has \(420\), while Sarah has \(360\).

- Convert both heights to the same unit. - Use subtraction for the difference. - Use division to find how many times as tall.

1. Convert the toy chest's height: \(480\,\text{mm} = 48\,\text{cm}\). 2. Find the additive difference: \(48 - 8 = 40\,\text{cm}\). 3. Find the multiplicative comparison: \(48 \div 8 = 6\).

1) The toy chest is \(40\,\text{cm}\) taller. 2) The toy chest is \(6\) times as tall.