Aimathic – Interpret numerical expressions

Interpret numerical expressions

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Difficulty: 1 – 5

5217725

Describe the structure of the numerical expression in words: $120 \div (4 \times 6)$

Hints

- Identify the outermost operation first. - Describe the complete divisor as one quantity. - Use terms such as sum, difference, product, and quotient.

Solution

1. The outermost operation is division, so the entire expression is a quotient. 2. The dividend is $120$, and the divisor is the product of $4$ and $6$. 3. One complete description is: “The quotient of $120$ and the product of $4$ and $6$.”

Answer

The quotient of $120$ and the product of $4$ and $6$.

5217735

Describe the structure of the numerical expression in words: $(75 + 25) \times (12 - 8)$

Hints

- Identify the outermost operation first. - Treat each grouped quantity as one factor. - Describe the main structure before the parts inside the grouping symbols.

Solution

1. The outermost operation is multiplication, so the entire expression is a product. 2. The first factor is the sum of $75$ and $25$. The second factor is the difference between $12$ and $8$. 3. One complete description is: “The product of the sum of $75$ and $25$ and the difference between $12$ and $8$.”

Answer

The product of the sum of $75$ and $25$ and the difference between $12$ and $8$.

5217745

Describe the structure of the numerical expression in words: $(15 \times 4) - (80 \div 2)$

Hints

- Identify the outermost operation first. - Describe the complete quantity on each side of the subtraction sign. - Use product and quotient for the grouped calculations.

Solution

1. The outermost operation is subtraction, so the entire expression is a difference. 2. The first quantity is the product of $15$ and $4$. The second quantity is the quotient of $80$ and $2$. 3. One complete description is: “The difference between the product of $15$ and $4$ and the quotient of $80$ and $2$.”

Answer

The difference between the product of $15$ and $4$ and the quotient of $80$ and $2$.

5174315

Evaluate each expression. a) $156 \div 4$ b) $19 \times 18$ c) $441 \div 9$ d) $24 \times 15$

Hints

- Break numbers into easier parts. - To divide by $4$, halve twice. - To multiply by $15$, add the products for multiplying by $10$ and by $5$.

Solution

1. Divide by $4$: $156 \div 4 = 39$. 2. Multiply: $19 \times 18 = 342$. 3. Divide: $441 \div 9 = 49$. 4. Multiply: $24 \times 15 = 360$.

Answer

a) $39$ b) $342$ c) $49$ d) $360$

5179005

Describe the structure of the expression using mathematical terms, then evaluate it: $4200 + (150 - 75)$

Hints

- Identify the operation performed last. - Name each part of the expression based on its operation. - Evaluate inside the grouping symbols before finding the final sum.

Solution

1. The expression is a sum. Its first addend is $4200$, and its second addend is the difference $150 - 75$. 2. Evaluate the difference: $150 - 75 = 75$. 3. Evaluate the sum: $4200 + 75 = 4275$.

Answer

Structure: the sum of $4200$ and the difference of $150$ and $75$ Value: $4275$

5179725

A sum has $8$ addends. Each addend is increased by $12$. By how much does the sum increase?

Hints

- How many times is the increase of $12$ added to the total? - Represent the repeated increase with multiplication.

Solution

1. The increase of $12$ occurs once for each of the $8$ addends. 2. The total increase is $8 \times 12 = 96$.

Answer

The sum increases by $96$.

5179825

Insert $+$ or $-$ in each circle so that every equation is true. a) $450 \bigcirc 120 \bigcirc 230 = 560$ b) $1240 \bigcirc 850 \bigcirc 310 = 700$ c) $3000 \bigcirc 1450 \bigcirc 670 \bigcirc 220 = 2000$

Hints

- Compare the target value with the first number. - Test the signs systematically from left to right. - Check the completed equation exactly.

Solution

1. a) $450 - 120 + 230 = 330 + 230 = 560$. 2. b) $1240 - 850 + 310 = 390 + 310 = 700$. 3. c) $3000 - 1450 + 670 - 220 = 1550 + 670 - 220 = 2000$.

Answer

a) $450 - 120 + 230 = 560$ b) $1240 - 850 + 310 = 700$ c) $3000 - 1450 + 670 - 220 = 2000$

5186195

Insert $\times$ or $\div$ to make each equation true. a) $(82 - 74)\ \_\_\ 6 = 48$ b) $(100 - 64)\ \_\_\ 4 = 9$ c) $(53 - 44)\ \_\_\ 9 = 81$ d) $(95 - 67)\ \_\_\ 7 = 4$

Hints

- Evaluate each grouped subtraction first. - Decide whether multiplication or division connects that value to the target. - Check the completed equation.

Solution

1. Evaluate the grouped differences: $8$, $36$, $9$, and $28$. 2. a) $8 \times 6 = 48$, so use $\times$. 3. b) $36 \div 4 = 9$, so use $\div$. 4. c) $9 \times 9 = 81$, so use $\times$. 5. d) $28 \div 7 = 4$, so use $\div$.

Answer

a) $\times$ b) $\div$ c) $\times$ d) $\div$

5193365

Evaluate each set of whole-number expressions. a) Multiply $450$, $1205$, and $30{,}040$ by $8$. b) Divide $960$, $4836$, and $56{,}008$ by $4$. c) Multiply $250$, $1500$, and $12{,}300$ by $60$. d) Divide $3500$, $42{,}000$, and $210{,}000$ by $70$.

Hints

- For multiplication by $60$, multiply by $6$ and then by $10$. - For division by $70$, first divide both the dividend and divisor by $10$. - Use place value to check the size of each result.

Solution

1. Multiply by $8$: $450 \times 8 = 3600$, $1205 \times 8 = 9640$, and $30{,}040 \times 8 = 240{,}320$. 2. Divide by $4$: $960 \div 4 = 240$, $4836 \div 4 = 1209$, and $56{,}008 \div 4 = 14{,}002$. 3. Multiply by $60$: $250 \times 60 = 15{,}000$, $1500 \times 60 = 90{,}000$, and $12{,}300 \times 60 = 738{,}000$. 4. Divide by $70$: $3500 \div 70 = 50$, $42{,}000 \div 70 = 600$, and $210{,}000 \div 70 = 3000$.

Answer

a) $3600$; $9640$; $240{,}320$ b) $240$; $1209$; $14{,}002$ c) $15{,}000$; $90{,}000$; $738{,}000$ d) $50$; $600$; $3000$

5194215

Evaluate each expression. Then classify the entire expression as a sum, difference, product, or quotient according to its outermost operation. a) $25 + 5 \times 8$ b) $(25 + 5) \times 8$

Hints

- Apply the order of operations when there are no grouping symbols. - Notice how grouping symbols change which operation is performed last. - Classify the entire expression by its outermost operation.

Solution

1. a) Multiply first: $5 \times 8 = 40$. Then $25 + 40 = 65$. The outermost operation is addition, so the expression is a sum. 2. b) Evaluate the grouping symbols first: $25 + 5 = 30$. Then $30 \times 8 = 240$. The outermost operation is multiplication, so the expression is a product.

Answer

a) Value: $65$; classification: sum b) Value: $240$; classification: product

5194225

Evaluate each expression. Then classify the entire expression as a sum, difference, product, or quotient according to its outermost operation. a) $120 - 40 \div 4$ b) $(120 - 40) \div 4$

Hints

- Compare the order of operations with and without grouping symbols. - Identify the operation that combines the largest parts of the expression. - Classify the entire expression by its outermost operation.

Solution

1. a) Divide first: $40 \div 4 = 10$. Then $120 - 10 = 110$. The outermost operation is subtraction, so the expression is a difference. 2. b) Evaluate the grouping symbols first: $120 - 40 = 80$. Then $80 \div 4 = 20$. The outermost operation is division, so the expression is a quotient.

Answer

a) Value: $110$; classification: difference b) Value: $20$; classification: quotient

5194355

Evaluate each expression. Then classify the entire expression as a sum, difference, product, or quotient according to its outermost operation. a) $90 \div (6 + 9)$ b) $90 \div 6 + 9$

Hints

- Apply the order of operations in each expression. - Compare the effect of the grouping symbols. - Classify the entire expression by its outermost operation.

Solution

1. a) Evaluate the grouping symbols first: $6 + 9 = 15$. Then $90 \div 15 = 6$. The outermost operation is division, so the expression is a quotient. 2. b) Divide before adding: $90 \div 6 = 15$. Then $15 + 9 = 24$. The outermost operation is addition, so the expression is a sum.

Answer

a) Value: $6$; classification: quotient b) Value: $24$; classification: sum

5194665

Consider the expression $80 \div (12 + 8)$. a) Describe the expression in words. b) Evaluate the expression.

Hints

- Identify which quantity is evaluated first. - Determine the complete divisor. - Make the wording clear about what is divided by what.

Solution

1. a) The grouped sum is the divisor, so one description is: “Divide $80$ by the sum of $12$ and $8$.” 2. b) Evaluate the grouped sum: $12 + 8 = 20$. 3. Divide: $80 \div 20 = 4$.

Answer

a) Divide $80$ by the sum of $12$ and $8$. b) $4$

5198845

Calculate mentally. a) $600 \times 4000$ b) $48{,}000 \div 800$ c) $802 \times 300$

Hints

- Use place value when multiplying numbers that end in zeros. - Divide both numbers by the same power of $10$ to simplify a quotient. - Break $802$ into $800 + 2$.

Solution

1. Multiply $6 \times 4 = 24$, then account for the five zeros: $600 \times 4000 = 2{,}400{,}000$. 2. Divide both numbers by $100$: $48{,}000 \div 800 = 480 \div 8 = 60$. 3. Use the distributive property: $802 \times 300 = 800 \times 300 + 2 \times 300 = 240{,}600$.

Answer

a) $2{,}400{,}000$ b) $60$ c) $240{,}600$

5206795

Evaluate each expression when possible. a) $24\,\text{m}\times 6$ b) $\$15.80\times 4$ c) $12\,\text{kg}+450\,\text{m}$ d) $1230\,\text{m}\div 6$

Hints

- Check whether the measurements describe the same attribute before adding. - Multiplying or dividing a measurement by a number keeps the unit. - Carry the unit through each valid calculation.

Solution

1. $24\times 6=144$, so the result is $144\,\text{m}$. 2. $15.80\times 4=63.20$, so the result is $\$63.20$. 3. A mass and a length cannot be added because they measure different attributes. 4. $1230\div 6=205$, so the result is $205\,\text{m}$.

Answer

a) $144\,\text{m}$ b) $\$63.20$ c) Not possible d) $205\,\text{m}$

5217605

Calculate mentally using multiplication and division facts and place value. 1) $4 \times 70$ 2) $280 \div 4$ 3) $40 \times 70$ 4) $2800 \div 70$

Hints

- Reduce each expression to a basic multiplication or division fact. - Track how factors of $10$ affect the result. - Simplify a quotient by dividing both numbers by the same power of $10$.

Solution

1. Use $4 \times 7 = 28$: $4 \times 70 = 280$. 2. Use $28 \div 4 = 7$: $280 \div 4 = 70$. 3. Use $4 \times 7 = 28$ and place value: $40 \times 70 = 2800$. 4. Divide both numbers by $10$: $2800 \div 70 = 280 \div 7 = 40$.

Answer

1) $280$ 2) $70$ 3) $2800$ 4) $40$

5331935

A rainwater tank in a school garden is observed over five days. The graph shows the amount of water in the tank, in liters, at the indicated times. Lucas wants to know how much water is in the tank at the end of Day 5. He writes two expressions: Expression 1: $120 + 30 - 10 + 40 - 20 + 50$ Expression 2: $120 + (30 + 40 + 50) - (10 + 20)$ a) Evaluate both expressions. b) Explain what the two grouped expressions in Expression 2 represent in this situation.

Hints

- Follow the graph from left to right. - Think about what an increase or decrease in the graph means. - Compare the numbers that are added with the numbers that are subtracted. - Locate the changes from the expressions in the graph.

Solution

1. Evaluate Expression 1 from left to right: $120 + 30 = 150$, $150 - 10 = 140$, $140 + 40 = 180$, $180 - 20 = 160$, and $160 + 50 = 210$. The result is $210\,\text{L}$. 2. In Expression 2, $30 + 40 + 50 = 120$ and $10 + 20 = 30$, so $120 + 120 - 30 = 210$. The result is also $210\,\text{L}$. 3. The group $(30 + 40 + 50)$ is the total amount of water added during the five days. The group $(10 + 20)$ is the total amount of water removed during the same period.

Answer

a) Both expressions equal $210\,\text{L}$. b) The expression $(30 + 40 + 50)$ represents the total water added, and $(10 + 20)$ represents the total water removed.

5352395

Which expression represents the expression tree? Then evaluate it. A) $100 - 30 + 20$ B) $100 - (30 + 20)$ C) $(100 - 30) + 20$

Hints

- Identify which numbers are directly connected by addition. - Use grouping symbols to show which operation occurs first. - Compare each choice with the tree structure.

Solution

1. The tree first adds $30$ and $20$, and then subtracts that sum from $100$. 2. Expression B matches the tree: $100 - (30 + 20)$. 3. Evaluate: $100 - 50 = 50$.

Answer

B) $100 - (30 + 20) = 50$

5353455

The calculation tree represents the cost of $3$ items at $\$4.50$ each and one item that costs $\$6.50$. Use the tree to find the total cost.

Hints

- Evaluate the multiplication branch before the addition. - Line up the decimal points when you add the two money amounts.

Solution

1. Evaluate the multiplication branch: $3 \times \$4.50 = \$13.50$. 2. Add the remaining cost: $\$13.50 + \$6.50 = \$20.00$.

Answer

The total cost is $\$20.00$.

5173555

Calculate each result using an appropriate standard method. a) $254 \times 36$ b) $9144 \div 24$

Hints

- For multiplication, find the partial products for the tens and ones digits. - For division, estimate each quotient digit and check that each remainder is less than $24$. - Verify the quotient by multiplying it by the divisor.

Solution

1. Use partial products: $254 \times 30 = 7620$ and $254 \times 6 = 1524$. Then $7620 + 1524 = 9144$. 2. Divide $9144$ by $24$. The quotient is $381$ because $24 \times 381 = 9144$.

Answer

a) $9144$ b) $381$

5173565

Find each missing value. a) $123 \times 123 = \square$ b) $4000 - \triangle = 1567$

Hints

- Break $123$ into $100 + 20 + 3$ for the multiplication. - Rewrite part b) as $1567 + \triangle = 4000$. - Check each missing value in the original equation.

Solution

1. Use partial products: $123 \times 100 = 12{,}300$, $123 \times 20 = 2460$, and $123 \times 3 = 369$. Their sum is $15{,}129$. 2. The missing subtrahend is $4000 - 1567 = 2433$.

Answer

a) $\square = 15{,}129$ b) $\triangle = 2433$

5174305

Evaluate and compare each pair of expressions. Write $<$, $>$, or $=$. a) $378 \div 6 \quad \square \quad 14 \times 4$ b) $26 \times 11 \quad \square \quad 858 \div 3$ c) $168 \div 7 \quad \square \quad 5 \times 5$

Hints

- Evaluate both expressions before comparing them. - Record the two values so you can compare accurately. - For multiplication by $11$, use $n \times 10 + n$.

Solution

1. $378 \div 6 = 63$ and $14 \times 4 = 56$, so $63 > 56$. 2. $26 \times 11 = 286$ and $858 \div 3 = 286$, so the values are equal. 3. $168 \div 7 = 24$ and $5 \times 5 = 25$, so $24 < 25$.

Answer

a) $>$ b) $=$ c) $<$

5179015

Describe the structure of the expression using mathematical terms, then evaluate it: $(780 + 220) - (150 - 60)$

Hints

- Identify the operation performed last. - Name the expression on each side of the final subtraction. - Evaluate the grouped expressions before the final difference.

Solution

1. The entire expression is a difference. The minuend is the sum $780 + 220$, and the subtrahend is the difference $150 - 60$. 2. Evaluate the grouped expressions: $780 + 220 = 1000$ and $150 - 60 = 90$. 3. Evaluate the final difference: $1000 - 90 = 910$.

Answer

Structure: the difference between the sum of $780$ and $220$ and the difference of $150$ and $60$ Value: $910$

5179025

Describe the structure of the expression using mathematical terms, then evaluate it: $(3400 - 1200) - (500 - 150)$

Hints

- Identify the operation performed last. - Name the minuend and subtrahend as expressions. - Evaluate each grouped difference before the final subtraction.

Solution

1. The entire expression is a difference. The minuend is the difference $3400 - 1200$, and the subtrahend is the difference $500 - 150$. 2. Evaluate the grouped expressions: $3400 - 1200 = 2200$ and $500 - 150 = 350$. 3. Evaluate the final difference: $2200 - 350 = 1850$.

Answer

Structure: the difference between the difference of $3400$ and $1200$ and the difference of $500$ and $150$ Value: $1850$

5179735

A sum has $5$ addends. Two addends are each increased by $15$, and the other three addends are each decreased by $8$. By how much does the sum change?

Hints

- Find the total increase from the first two addends. - Find the total decrease from the other three addends. - Combine the two changes to find the net effect.

Solution

1. The two increases add $2 \times 15 = 30$ to the sum. 2. The three decreases subtract $3 \times 8 = 24$ from the sum. 3. The net change is $30 - 24 = 6$, so the sum increases by $6$.

Answer

The sum increases by $6$.

5179745

A sum has $4$ addends. Three addends are each decreased by $9$. How must the fourth addend change so that the sum stays the same?

Hints

- Find the combined decrease from the three addends. - The change to the fourth addend must exactly cancel that decrease.

Solution

1. The total decrease from the first three addends is $3 \times 9 = 27$. 2. To keep the sum unchanged, the fourth addend must offset that decrease by increasing by $27$.

Answer

The fourth addend must increase by $27$.

5193125

For each problem, estimate first by rounding to compatible numbers. Then calculate the exact result using a standard algorithm. Check each division with multiplication. a) $423 \times 56$ b) $12{,}648 \div 12$ c) $205 \times 403$ d) $9435 \div 15$

Hints

- Round to nearby compatible numbers that are easy to calculate mentally. - Align partial products by place value in multiplication. - Check a quotient by multiplying it by the divisor.

Solution

1. Estimates: a) $423 \times 56 \approx 400 \times 60 = 24{,}000$; b) $12{,}648 \div 12 \approx 12{,}000 \div 12 = 1000$; c) $205 \times 403 \approx 200 \times 400 = 80{,}000$; d) $9435 \div 15 \approx 9000 \div 15 = 600$. 2. Exact results: a) $423 \times 56 = 23{,}688$; b) $12{,}648 \div 12 = 1054$; c) $205 \times 403 = 82{,}615$; d) $9435 \div 15 = 629$. 3. Checks: $1054 \times 12 = 12{,}648$ and $629 \times 15 = 9435$.

Answer

a) Estimate: $24{,}000$; exact result: $23{,}688$ b) Estimate: $1000$; exact result: $1054$; check: $1054 \times 12 = 12{,}648$ c) Estimate: $80{,}000$; exact result: $82{,}615$ d) Estimate: $600$; exact result: $629$; check: $629 \times 15 = 9435$

5193375

Evaluate and compare each pair of expressions. Write $<$, $>$, or $=$. a) $400 \times 30 \quad \square \quad 120{,}000 \div 100$ b) $15 \times 4000 \quad \square \quad 300 \times 20$ c) $81{,}000 \div 90 \quad \square \quad 90 \times 10$ d) $2500 \times 40 \quad \square \quad 500{,}000 \div 5$

Hints

- Evaluate both sides before comparing. - Use place-value patterns when multiplying or dividing by multiples of $10$. - Record intermediate values to avoid comparing the original expressions by appearance alone.

Solution

1. $400 \times 30 = 12{,}000$ and $120{,}000 \div 100 = 1200$, so the left side is greater. 2. $15 \times 4000 = 60{,}000$ and $300 \times 20 = 6000$, so the left side is greater. 3. $81{,}000 \div 90 = 900$ and $90 \times 10 = 900$, so the values are equal. 4. $2500 \times 40 = 100{,}000$ and $500{,}000 \div 5 = 100{,}000$, so the values are equal.

Answer

a) $>$ b) $>$ c) $=$ d) $=$

5193385

Complete the sequence of operations. Start with $1200$. 1. Divide by $6$. 2. Multiply by $50$. 3. Divide by $400$. 4. Multiply by $12$. What is the final result?

Hints

- Use each result as the input for the next step. - Simplify $10{,}000 \div 400$ by dividing both numbers by $100$. - For multiplying by $12$, use $10 + 2$.

Solution

1. $1200 \div 6 = 200$. 2. $200 \times 50 = 10{,}000$. 3. $10{,}000 \div 400 = 25$. 4. $25 \times 12 = 300$.

Answer

$300$

5194235

Evaluate each expression and classify the entire expression as a sum, difference, product, or quotient. Then identify which expression has the greater value. a) $9 \times 6 - 4 \times 3$ b) $9 \times (6 - 4) \times 3$

Hints

- Apply the order of operations one expression at a time. - Notice how the grouping symbols in b change the calculation. - Classify the entire expression by its outermost operation.

Solution

1. a) Evaluate both products: $9 \times 6 = 54$ and $4 \times 3 = 12$. Then $54 - 12 = 42$. The outermost operation is subtraction, so the expression is a difference. 2. b) Evaluate the grouping symbols: $6 - 4 = 2$. Then $9 \times 2 \times 3 = 54$. The outermost operation is multiplication, so the expression is a product. 3. Since $54 > 42$, expression b has the greater value.

Answer

a) Value: $42$; classification: difference b) Value: $54$; classification: product Expression b has the greater value.

5194365

Evaluate the expression and classify it as a sum, difference, product, or quotient according to its outermost operation. $200 - 50 \times 3 + 12$

Hints

- Perform multiplication before addition or subtraction. - When only addition and subtraction remain, work from left to right. - Classify the expression by the operation performed last.

Solution

1. Multiply first: $50 \times 3 = 150$. The expression becomes $200 - 150 + 12$. 2. Evaluate addition and subtraction from left to right: $200 - 150 = 50$, and $50 + 12 = 62$. 3. The outermost operation is addition, so the expression is a sum.

Answer

Value: $62$; classification: sum

5194375

Evaluate the expression and classify it as a sum, difference, product, or quotient according to its outermost operation. $(24 + 36) \div (15 - 3 \times 3)$

Hints

- Evaluate both sets of grouping symbols first. - Use the order of operations inside the second set. - Identify the operation that connects the two grouped quantities.

Solution

1. Evaluate the first set of grouping symbols: $24 + 36 = 60$. 2. In the second set, multiply before subtracting: $15 - 3 \times 3 = 15 - 9 = 6$. 3. Divide: $60 \div 6 = 10$. 4. The outermost operation is division, so the expression is a quotient.

Answer

Value: $10$; classification: quotient

5194945

Evaluate the expression and classify it as a sum, difference, product, or quotient according to its outermost operation. $125 - 5 \times 15 + 32 \div 4$

Hints

- Perform multiplication and division before addition and subtraction. - When only addition and subtraction remain, work from left to right. - Classify the expression by the operation performed last.

Solution

1. Multiply and divide first: $5 \times 15 = 75$ and $32 \div 4 = 8$. The expression becomes $125 - 75 + 8$. 2. Evaluate from left to right: $125 - 75 = 50$, and $50 + 8 = 58$. 3. The outermost operation is addition, so the expression is a sum.

Answer

Value: $58$; classification: sum

5194955

Evaluate the expression and classify it as a sum, difference, product, or quotient according to its outermost operation. $6 \times (48 - 6 \times 7)$

Hints

- Evaluate the grouping symbols before the operation outside them. - Apply the order of operations inside the grouping symbols. - Classify the expression by its outermost operation.

Solution

1. Inside the grouping symbols, multiply first: $6 \times 7 = 42$. 2. Subtract: $48 - 42 = 6$. 3. Multiply: $6 \times 6 = 36$. 4. The outermost operation is multiplication, so the expression is a product.

Answer

Value: $36$; classification: product

5194965

Evaluate the expression and classify it as a sum, difference, product, or quotient according to its outermost operation. $(14 + 6 \times 11) \div (2 \times 12 - 4)$

Hints

- Evaluate the two grouped quantities separately. - Apply the order of operations inside each set of grouping symbols. - Identify the operation that connects the two grouped quantities.

Solution

1. Evaluate the first set of grouping symbols: $14 + 6 \times 11 = 14 + 66 = 80$. 2. Evaluate the second set: $2 \times 12 - 4 = 24 - 4 = 20$. 3. Divide: $80 \div 20 = 4$. 4. The outermost operation is division, so the expression is a quotient.

Answer

Value: $4$; classification: quotient

5195115

Describe each numerical expression in words, and then evaluate it. a) $(45 + 15) \div (20 - 10)$ b) $8 \times 7 - 3 \times 9$

Hints

- Identify the outermost operation in each expression. - Describe each grouped or multiplied quantity as one complete part. - Evaluate after writing the description.

Solution

1. a) One description is: “Divide the sum of $45$ and $15$ by the difference between $20$ and $10$.” Evaluate: $60 \div 10 = 6$. 2. b) One description is: “Subtract the product of $3$ and $9$ from the product of $8$ and $7$.” Evaluate: $56 - 27 = 29$.

Answer

a) Divide the sum of $45$ and $15$ by the difference between $20$ and $10$. Value: $6$ b) Subtract the product of $3$ and $9$ from the product of $8$ and $7$. Value: $29$

5195135

Describe the numerical expression in words, and then evaluate it. $150 - (40 + 5 \times 8)$

Hints

- Identify the complete quantity being subtracted from $150$. - Describe the product inside the grouped sum. - Apply the order of operations when evaluating.

Solution

1. One description is: “Subtract the sum of $40$ and the product of $5$ and $8$ from $150$.” 2. Inside the grouping symbols, multiply first: $5 \times 8 = 40$. 3. Add: $40 + 40 = 80$. 4. Subtract: $150 - 80 = 70$.

Answer

Subtract the sum of $40$ and the product of $5$ and $8$ from $150$. Value: $70$

5317145

Two expression trees use the same numbers: $12$, $8$, and $5$. a) Write the numerical expression represented by each tree. b) Evaluate both expressions. c) Explain why the values are different.

Hints

- Identify which two numbers are combined first in each tree. - Use grouping symbols when addition must occur before multiplication. - Compare what is multiplied by $5$ in the two expressions.

Solution

1. Tree a first adds $12$ and $8$, then multiplies by $5$. Its expression is $(12 + 8) \times 5$, and its value is $20 \times 5 = 100$. 2. Tree b first multiplies $8$ and $5$, then adds $12$. Its expression is $12 + 8 \times 5$, and its value is $12 + 40 = 52$. 3. The values differ because the grouping symbols in expression a make the addition happen first. In expression b, multiplication is performed before addition.

Answer

a) Tree a: $(12 + 8) \times 5$; tree b: $12 + 8 \times 5$ b) Tree a: $100$; tree b: $52$ c) The expressions group the numbers differently, so the operations are performed in a different order.

5317455

Jordan buys supplies for the class: - $5$ boxes of sidewalk chalk at $\$8.00$ each - $3$ paintbrush sets at $\$12.00$ each Jordan pays with $\$100.00$. a) Which expression tree correctly represents the amount of change Jordan receives, tree A or tree B? Explain. b) Use the correct tree to calculate the change. c) Write and evaluate the corresponding numerical expression.

Hints

- Describe what each branch of the trees represents in the shopping situation. - Find the total cost before calculating the change. - Check whether the final operation in each tree matches the situation.

Solution

1. a) Tree A is correct. The costs of the chalk and paintbrush sets must be added to find the total cost, and then the total cost must be subtracted from $100$. Tree B incorrectly adds the paintbrush cost after subtracting the chalk cost. 2. b) The chalk costs $5 \times 8 = 40$ dollars, and the paintbrush sets cost $3 \times 12 = 36$ dollars. The total cost is $40 + 36 = 76$ dollars. The change is $100 - 76 = 24$ dollars. 3. c) The expression is $100 - (5 \times 8 + 3 \times 12)$, and its value is $24$.

Answer

a) Tree A, because it subtracts the total cost from the amount paid. b) $\$24.00$ c) $100 - (5 \times 8 + 3 \times 12) = 24$

5192475

For each target value $2$, $4$, $6$, and $8$, write two different expressions that each contain exactly one operation symbol. Across your eight expressions, use each operation—addition, subtraction, multiplication, and division—exactly twice.

Hints

- Begin by writing one simple expression for each target value. - Keep a tally of how many times you use each operation. - Once an operation has been used twice, choose different operations for the remaining expressions. - Many answers are possible.

Solution

One possible set is: 1. Target $2$: $1 + 1 = 2$ and $4 \div 2 = 2$. 2. Target $4$: $2 + 2 = 4$ and $8 \div 2 = 4$. 3. Target $6$: $10 - 4 = 6$ and $2 \times 3 = 6$. 4. Target $8$: $10 - 2 = 8$ and $2 \times 4 = 8$. Addition, subtraction, multiplication, and division are each used exactly twice.

Answer

One possible answer is: Target $2$: $1 + 1$ and $4 \div 2$ Target $4$: $2 + 2$ and $8 \div 2$ Target $6$: $10 - 4$ and $2 \times 3$ Target $8$: $10 - 2$ and $2 \times 4$

5193145

Estimate each result, then calculate exactly using a standard algorithm. a) $5020 \times 3040$ b) $12{,}345 \times 67$ c) $36{,}072 \div 12$

Hints

- Track every place value when multiplying numbers with zeros. - You may first multiply without ending zeros, then account for their place values. - In division, write a zero in the quotient when the divisor does not fit into a partial dividend.

Solution

1. Estimates: a) $5020 \times 3040 \approx 5000 \times 3000 = 15{,}000{,}000$; b) $12{,}345 \times 67 \approx 10{,}000 \times 70 = 700{,}000$; c) $36{,}072 \div 12 \approx 36{,}000 \div 12 = 3000$. 2. Exact results: a) $5020 \times 3040 = 15{,}260{,}800$; b) $12{,}345 \times 67 = 827{,}115$; c) $36{,}072 \div 12 = 3006$.

Answer

a) Estimate: $15{,}000{,}000$; exact result: $15{,}260{,}800$ b) Estimate: $700{,}000$; exact result: $827{,}115$ c) Estimate: $3000$; exact result: $3006$

5193635

Use standard algorithms to solve both problems. Estimate each result first, and check the division with multiplication. a) $12{,}045 \times 308$ b) $14{,}976 \div 48$

Hints

- Account for the zero in the tens place of $308$. - Use estimates to check the number of digits in each exact result. - Check division with the inverse operation.

Solution

1. For a), estimate $12{,}045 \times 308 \approx 12{,}000 \times 300 = 3{,}600{,}000$. The partial products are $12{,}045 \times 300 = 3{,}613{,}500$ and $12{,}045 \times 8 = 96{,}360$. Their sum is $3{,}709{,}860$. 2. For b), estimate $14{,}976 \div 48 \approx 15{,}000 \div 50 = 300$. The exact quotient is $312$. 3. Check: $312 \times 48 = 14{,}976$.

Answer

a) Estimate: $3{,}600{,}000$; exact result: $3{,}709{,}860$ b) Estimate: $300$; exact result: $312$; check: $312 \times 48 = 14{,}976$

5213195

Explain why this expression does not represent a valid quantity: $35\,\text{kg} \div 5\,\text{kg} + 12\,\text{g}$.

Hints

- What happens to the units when one mass is divided by another mass in the same unit? - What kind of quantity does the quotient represent? - Evaluate the division before considering the addition.

Solution

1. Perform the division first: $35\,\text{kg} \div 5\,\text{kg} = 7$. The kilograms cancel, so $7$ is a number with no unit. 2. The next operation would be $7 + 12\,\text{g}$. 3. A number with no unit cannot be added to a mass measurement. Therefore, the expression does not represent a valid quantity.

Answer

The division equals the unitless number $7$. Adding $7$ to $12\,\text{g}$ is not meaningful because the two addends represent different kinds of quantities.

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