Aimathic – Translate verbal to numerical expressions

Translate verbal to numerical expressions

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Difficulty: 1 – 5

5540595

Write a numerical expression for this description. Do not evaluate it. Seven times the sum of $8$ and $4$.

Hints

- Identify the operation named by “the sum of.” - Decide which whole quantity is multiplied by $7$. - Use grouping symbols to keep that quantity together.

Solution

1. The phrase “the sum of $8$ and $4$” is represented by $8+4$. 2. “Seven times” means multiply that entire sum by $7$, giving $7\times(8+4)$.

Answer

$7\times(8+4)$

5113415

For each description, first write an expression using numbers, operation symbols, and parentheses. Then match your expression to every equivalent expression card. (A) The product of $1.2$ and $\frac{3}{4}$, increased by $0.5$ (B) The sum of $1.2$ and $0.5$, multiplied by $\frac{3}{4}$ (C) The quotient of $1.2$ and the difference of $\frac{3}{4}$ and $0.5$ 1: $(1.2\times\frac{3}{4})+0.5$ 2: $(1.2+0.5)\times\frac{3}{4}$ 3: $1.2\div(\frac{3}{4}-0.5)$ 4: $\frac{3}{4}\times(0.5+1.2)$ 5: $0.5+(1.2\times\frac{3}{4})$

Hints

- Write each verbal description before looking for matching cards. - Pay attention to which quantities must be grouped before multiplication or division. - After writing each expression, look for cards that differ only by a valid reordering of addition or multiplication.

Solution

1. Description A translates to $(1.2\times\frac34)+0.5$. It matches Cards 1 and 5 because addition is commutative. 2. Description B translates to $(1.2+0.5)\times\frac34$. It matches Cards 2 and 4 because addition and multiplication are commutative. 3. Description C translates to $1.2\div(\frac34-0.5)$. It matches Card 3.

Answer

(A) $(1.2\times\frac34)+0.5$; Cards 1 and 5 (B) $(1.2+0.5)\times\frac34$; Cards 2 and 4 (C) $1.2\div(\frac34-0.5)$; Card 3

5122685

Write an expression for each description, then evaluate it efficiently. a) Find the product of $16$ and $27$, and subtract the product of $16$ and $17$. b) Add the product of $0.2$ and $35$ to the product of $0.2$ and $15$.

Hints

- Translate each description into one expression before evaluating. - Keep each stated product grouped as a product in your written expression. - After the expression is written, look for a common factor that can make the arithmetic shorter.

Solution

1. For a), write $16\times27-16\times17$. Factoring the common factor gives $16(27-17)=16\times10=160$. 2. For b), write $0.2\times35+0.2\times15$. Factoring gives $0.2(35+15)=0.2\times50=10$.

Answer

a) Expression: $16\times27-16\times17$; value: $160$ b) Expression: $0.2\times35+0.2\times15$; value: $10$

5179255

Write a numerical expression for each description, then evaluate it. a) Add the sum of $234$ and $416$ to $350$. b) Add the sum of $145$ and $255$ to the sum of $310$ and $190$.

Hints

- Translate the word “sum” into addition. - Use grouping symbols to show which numbers form each stated sum. - Evaluate the grouped sums before combining them.

Solution

1. For part a), the expression is $350+(234+416)$. Evaluate the grouped sum: $234+416=650$, then $350+650=1000$. 2. For part b), the expression is $(310+190)+(145+255)$. The grouped sums are $500$ and $400$, so the value is $500+400=900$.

Answer

a) $350+(234+416)=1000$ b) $(310+190)+(145+255)=900$

5179485

Write a numerical expression for the instruction, then evaluate it: Add the difference of $420$ and $150$ to $230$.

Hints

- Translate “difference” into subtraction. - Use grouping symbols around the stated difference. - Add that difference to $230$.

Solution

1. The difference is represented by $420 - 150$. 2. The expression is $230 + (420 - 150)$. 3. Evaluate: $230 + 270 = 500$.

Answer

Expression: $230 + (420 - 150)$ Value: $500$

5179495

Write a numerical expression for the instruction, then evaluate it: Subtract the sum of $85$ and $115$ from the difference of $740$ and $220$.

Hints

- Pay attention to the order in the phrase “subtract from.” - Use grouping symbols for both the stated sum and difference. - Evaluate those two parts before the final subtraction.

Solution

1. The minuend is $740 - 220$, and the subtrahend is $85 + 115$. 2. The expression is $(740 - 220) - (85 + 115)$. 3. Evaluate: $520 - 200 = 320$.

Answer

Expression: $(740 - 220) - (85 + 115)$ Value: $320$

5181535

For each instruction, write a numerical expression that represents it, then evaluate the expression. a) Find four times $18$. b) Decrease $130$ by $50$. c) How many groups of $6$ are in $54$? d) Increase $37$ by $23$.

Hints

- Decide which operation each phrase describes before calculating. - Pay attention to the order of the numbers in phrases involving subtraction or division. - Write each numerical expression first, and then evaluate it.

Solution

1. For part a), four times $18$ is $18\times4$, and $18\times4=72$. 2. For part b), decreasing $130$ by $50$ is $130-50$, and $130-50=80$. 3. For part c), the number of groups is $54\div6$, and $54\div6=9$. 4. For part d), increasing $37$ by $23$ is $37+23$, and $37+23=60$.

Answer

a) $18\times4=72$ b) $130-50=80$ c) $54\div6=9$ d) $37+23=60$

5181545

For each question, write a numerical expression that represents the operation phrase, then evaluate the expression. a) What is the difference between $100$ and $16$? b) How many groups of $20$ are in $80$? c) What number is $25$ less than $100$? d) What number is one-fourth of $100$?

Hints

- Translate each operation phrase into symbols before evaluating it. - For subtraction and division phrases, check which quantity comes first in the expression. - After writing an expression, make sure its operation matches the meaning of the words.

Solution

1. For part a), the difference is $100-16$, and $100-16=84$. 2. For part b), the number of groups is $80\div20$, and $80\div20=4$. 3. For part c), a number $25$ less than $100$ is $100-25$, and $100-25=75$. 4. For part d), one-fourth of $100$ is $100\div4$, and $100\div4=25$.

Answer

a) $100-16=84$ b) $80\div20=4$ c) $100-25=75$ d) $100\div4=25$

5191605

Write and evaluate each numerical expression. a) Find the product of $240$ and $5$. b) Find the quotient of $800$ and $4$, then subtract $150$ from the result.

Hints

- “Product” indicates multiplication. - “Quotient” indicates division. - Follow the operations in the order stated.

Solution

1. For part a), the product is $240\times5$, and $240\times5=1200$. 2. For part b), the expression is $(800\div4)-150$. The quotient is $200$, so $200-150=50$.

Answer

a) $240\times5=1200$ b) $(800\div4)-150=50$

5193295

Write a numerical expression for each description, and then evaluate it. a) Add the product of $25$ and $14$ to the quotient of $1200$ and $30$. b) Divide $1560$ by the product of $13$ and $4$.

Hints

- Identify the operation named by each math term. - Use grouping symbols when one result must be used as a single quantity. - Write the expression before calculating.

Solution

1. For part a), the expression is $25\times14+1200\div30$. The product is $350$ and the quotient is $40$, so the value is $350+40=390$. 2. For part b), the expression is $1560\div(13\times4)$. The grouped product is $52$, so $1560\div52=30$.

Answer

a) $25\times14+1200\div30=390$ b) $1560\div(13\times4)=30$

5194655

Write a numerical expression for the statement, and then evaluate it: “Multiply the sum of $14$ and $6$ by the difference between $25$ and $15$.”

Hints

- Match each math term to its operation. - Use grouping symbols around the sum and the difference. - Evaluate the grouped quantities before multiplying.

Solution

1. The sum of $14$ and $6$ is represented by $(14 + 6)$, and the difference between $25$ and $15$ is represented by $(25 - 15)$. 2. Multiply the two quantities: $(14 + 6) \times (25 - 15)$. 3. Evaluate: $20 \times 10 = 200$.

Answer

$(14 + 6) \times (25 - 15) = 200$

5195125

Write a numerical expression for the statement, and then evaluate it: “Divide the sum of $64$ and $16$ by the difference between $15$ and $7$.”

Hints

- Match each math term to its operation. - Use grouping symbols around the complete sum and difference. - The sum is the dividend, and the difference is the divisor.

Solution

1. The sum is $(64 + 16)$, and the difference is $(15 - 7)$. 2. The expression is $(64 + 16) \div (15 - 7)$. 3. Evaluate the grouped quantities: $80 \div 8 = 10$.

Answer

$(64 + 16) \div (15 - 7) = 10$

5197245

Compare the two expressions. Insert $<$, $>$, or $=$, and justify your answer with calculations. The quotient of $45$ and $9$ $\square$ the difference between $15$ and $8$

Hints

- Translate and evaluate each operation phrase separately. - “Quotient” indicates division, and “difference” indicates subtraction. - Compare the two resulting numbers.

Solution

1. The quotient is $45 \div 9 = 5$. 2. The difference is $15 - 8 = 7$. 3. Since $5 < 7$, the correct symbol is $<$.

Answer

$<$, because $45 \div 9 = 5$ and $15 - 8 = 7$.

5205375

Determine whether the results described below are equal. 1. The product of the factors $6$ and $4$. 2. The difference between $50$ and $26$. Write an expression for each description and compare the results.

Hints

- Translate “product” into multiplication. - Translate “difference” into subtraction. - Evaluate and compare both expressions.

Solution

1. The product is $6 \times 4 = 24$. 2. The difference is $50 - 26 = 24$. 3. Both expressions equal $24$, so the results are equal.

Answer

Yes. $6 \times 4 = 24$ and $50 - 26 = 24$.

5216195

Write a numerical expression for the statement, and then evaluate it: Subtract the difference between $189$ and $76$ from the sum of $156$ and $234$.

Hints

- Match “sum” and “difference” to their operations. - Use grouping symbols around the two complete quantities. - In the phrase “subtract ... from,” the quantity after “from” comes first.

Solution

1. The expression is $(156 + 234) - (189 - 76)$. 2. Evaluate the grouped quantities: $156 + 234 = 390$ and $189 - 76 = 113$. 3. Subtract: $390 - 113 = 277$.

Answer

$(156 + 234) - (189 - 76) = 277$

5540605

The numerical expression is $48\div(10-4)$. Write one verbal description that translates back to this expression. Do not evaluate it.

Hints

- Describe the grouped subtraction before describing the outer operation. - Pay attention to which quantity is the dividend and which is the divisor. - Use operation words such as “difference” and “quotient.”

Solution

1. The grouped part $10-4$ is the difference of $10$ and $4$. 2. The entire expression divides $48$ by that difference, so one matching description is “the quotient of $48$ and the difference of $10$ and $4$.”

Answer

One possible description: the quotient of $48$ and the difference of $10$ and $4$.

5540615

Nina is asked to translate this description into a numerical expression: “The quotient of $42$ and the sum of $6$ and $1$.” She writes $42\div6+1$. Explain the translation error and write the correct numerical expression. Do not evaluate it.

Hints

- Identify the complete quantity named after the word “and.” - Ask what the whole divisor should be. - Check whether the written symbols preserve the wording’s grouping.

Solution

1. The words “the sum of $6$ and $1$” name one complete quantity, so $6+1$ must be grouped. 2. The quotient uses $42$ as the dividend and that grouped sum as the divisor. The correct expression is $42\div(6+1)$.

Answer

Nina did not group the stated sum. The correct expression is $42\div(6+1)$.

5106335

Write a numerical expression for each description. Do not evaluate the expressions. Expression A: the difference of $10$ and the sum of $2\frac{1}{4}$ and $3\frac{1}{2}$ Expression B: the sum of $2\frac{1}{2}$ and the difference of $5$ and $3\frac{1}{4}$

Hints

- Identify which two numbers belong to each stated sum or difference. - Use grouping symbols to keep each stated sum or difference together. - Pay attention to the order in the phrase “the difference of.”

Solution

1. In Expression A, “the sum of” groups $2\frac{1}{4}$ and $3\frac{1}{2}$, and that sum is subtracted from $10$. So the expression is $10-(2\frac{1}{4}+3\frac{1}{2})$. 2. In Expression B, “the difference of” groups $5$ and $3\frac{1}{4}$, and that difference is added to $2\frac{1}{2}$. So the expression is $2\frac{1}{2}+(5-3\frac{1}{4})$.

Answer

Expression A: $10-(2\frac{1}{4}+3\frac{1}{2})$ Expression B: $2\frac{1}{2}+(5-3\frac{1}{4})$

5113325

Write the expression and find its value. The expression is a difference. The minuend is the quotient of $2.4$ and $0.6$. The subtrahend is the product of $\frac{1}{2}$ and $5$.

Hints

- Identify the minuend and subtrahend before writing the outer subtraction. - Translate “quotient” with division and “product” with multiplication. - Use parentheses so each described component stays grouped. - Evaluate only after the full expression has been written.

Solution

1. Translate the description as $ (2.4\div0.6)-\left(\frac12\times5\right). $ 2. The quotient is $4$, and the product is $2.5$. 3. The difference is $4-2.5=1.5$.

Answer

The expression is $(2.4\div0.6)-\left(\frac12\times5\right)$, and its value is $1.5$.

5113405

Write an expression for the description, and then evaluate it. Subtract the product of $1.2$ and $\frac{5}{6}$ from the difference of $2.25$ and $0.75$.

Hints

- “Subtract A from B” means $B-A$. - Keep the difference and product grouped when you write the expression. - Evaluate the grouped parts before the outer subtraction.

Solution

1. The expression is $(2.25-0.75)-(1.2\times\frac56)$. 2. The first difference is $1.5$. 3. The product is $1.2\times\frac56=1$. 4. The final value is $1.5-1=0.5$.

Answer

The expression is $(2.25-0.75)-(1.2\times\frac56)$, and its value is $0.5$.

5120795

Write an expression for each description, and then evaluate it. a) Subtract the product of $15.5$ and $4$ from the sum of $82.7$ and $17.3$. b) Divide the difference of $100$ and $12.5$ by the product of $5$ and $0.5$.

Hints

- Identify the operation represented by each mathematical term. - In “subtract A from B,” B comes first. - Use parentheses to preserve the described structure.

Solution

1. For a), the expression is $(82.7+17.3)-(15.5\times4)$. The sum is $100$, the product is $62$, and the value is $38$. 2. For b), the expression is $(100-12.5)\div(5\times0.5)$. The difference is $87.5$, the product is $2.5$, and the value is $35$.

Answer

a) $(82.7+17.3)-(15.5\times4)=38$ b) $(100-12.5)\div(5\times0.5)=35$

5122695

Write an expression for each description, then evaluate it efficiently. a) Subtract the difference of $7.8$ and $2.4$ from the difference of $10.2$ and $2.4$. b) Multiply the difference of $25$ and $2.5$ by $4$.

Hints

- Translate “subtract ... from ...” carefully so the order is correct. - Use parentheses to preserve each described difference. - After writing the expressions, look for cancellation or distribution that simplifies the arithmetic.

Solution

1. For a), write $(10.2-2.4)-(7.8-2.4)$. Removing the second parentheses gives $10.2-2.4-7.8+2.4=10.2-7.8=2.4$. 2. For b), write $(25-2.5)\times4$. Distributing gives $25\times4-2.5\times4=100-10=90$.

Answer

a) Expression: $(10.2-2.4)-(7.8-2.4)$; value: $2.4$ b) Expression: $(25-2.5)\times4$; value: $90$

5122935

For each description: 1. Write a numerical expression that preserves the stated operation structure. 2. Evaluate it. Then compare the two values. A: The product of $\frac34$ and $\frac85$. B: Subtract $\frac12$ from the quotient of $\frac{17}{10}$ and $1$.

Hints

- Write each expression before carrying out any arithmetic. - Translate “product,” “quotient,” and “subtract ... from ...” in the order stated. - Compare the evaluated values only after preserving the original structures.

Solution

1. A translates to $\frac34\times\frac85=\frac65=1.2$. 2. B translates to $\left(\frac{17}{10}\div1\right)-\frac12=1.7-0.5=1.2$. 3. The two values are equal.

Answer

A: $\frac34\times\frac85=1.2$ B: $\left(\frac{17}{10}\div1\right)-\frac12=1.2$ The two values are equal.

5122945

A student translates this instruction into an expression: “Divide the difference of $7.5$ and $2.5$ by the product of $0.25$ and $8$.” The student writes $7.5-2.5\div0.25\times8$. Explain why this expression does not match the instruction, write the correct expression, and find its value.

Hints

- Identify the two complete quantities named by “difference” and “product.” - Group each quantity before applying the outer division. - Compare the order of operations in the student's expression with the wording.

Solution

1. The student's expression lacks the grouping required by the words “difference” and “product,” so division and multiplication would be performed before the intended difference is formed. 2. The correct expression is $(7.5-2.5)\div(0.25\times8)$. 3. The difference is $5$, and the product is $2$. 4. The quotient is $5\div2=2.5$.

Answer

Parentheses are required around both described quantities. The correct expression is $(7.5-2.5)\div(0.25\times8)$, and its value is $2.5$.

5128025

Write an expression for each description, and then evaluate it. a) Multiply the difference of $12.5$ and $3.5$ by the sum of $0.4$ and $0.6$. b) Divide the sum of $7.2$ and $4.8$ by the product of $2$ and $1.5$.

Hints

- Translate sum, difference, product, and quotient carefully. - Use parentheses to preserve the described order. - Evaluate inside parentheses first.

Solution

1. For a), $(12.5-3.5)\times(0.4+0.6)=9\times1=9$. 2. For b), $(7.2+4.8)\div(2\times1.5)=12\div3=4$.

Answer

a) $(12.5-3.5)\times(0.4+0.6)=9$ b) $(7.2+4.8)\div(2\times1.5)=4$

5142255

Write an expression and find its value. Subtract the product of $\frac{2}{3}$ and $\frac{3}{4}$ from the sum of $\frac{1}{2}$ and $\frac{1}{6}$.

Hints

- Build the sum and product as separate grouped quantities first. - “Subtract the product from the sum” means sum minus product. - Evaluate each grouped quantity before the final subtraction.

Solution

1. The expression is $\left(\frac12+\frac16\right)-\left(\frac23\times\frac34\right)$. 2. The sum is $\frac12+\frac16=\frac23$. 3. The product is $\frac23\times\frac34=\frac12$. 4. Subtract: $\frac23-\frac12=\frac16$.

Answer

The expression is $\left(\frac12+\frac16\right)-\left(\frac23\times\frac34\right)$, and its value is $\frac16$.

5179265

Write a numerical expression for each description, then evaluate it. a) Subtract $285$ from $1000$. b) Subtract the difference of $92$ and $38$ from $150$. c) Add the sum of $18$ and $32$ to the difference of $120$ and $45$.

Hints

- Translate “sum” as addition and “difference” as subtraction. - Pay attention to the order in phrases such as “subtract from.” - Use grouping symbols around a stated sum or difference.

Solution

1. For part a), “subtract $285$ from $1000$” gives $1000-285=715$. 2. For part b), first represent the stated difference as $92-38$. Subtract that difference from $150$: $150-(92-38)=150-54=96$. 3. For part c), represent the difference and sum as grouped parts: $(120-45)+(18+32)=75+50=125$.

Answer

a) $1000-285=715$ b) $150-(92-38)=96$ c) $(120-45)+(18+32)=125$

5179305

Write a numerical expression for the description, then evaluate it: “Subtract the sum of $450$ and $550$ from $3000$. Add the difference of $800$ and $200$ to that result.”

Hints

- Translate “sum” and “difference” into operations. - Pay attention to what is subtracted from $3000$. - Use grouping symbols for the parts described as a sum or difference.

Solution

1. The expression is $(3000 - (450 + 550)) + (800 - 200)$. 2. Evaluate the grouped expressions: $450 + 550 = 1000$ and $800 - 200 = 600$. 3. Complete the calculation: $3000 - 1000 + 600 = 2600$.

Answer

Expression: $(3000 - (450 + 550)) + (800 - 200)$ Value: $2600$

5179505

Write a numerical expression for the instruction, then evaluate it: Find the difference between the sum of $2500$ and $1350$ and the difference of $1800$ and $650$.

Hints

- A “difference between A and B” means $A - B$. - Replace $A$ and $B$ with the stated sum and difference. - Evaluate inside the grouping symbols first.

Solution

1. The expression is $(2500 + 1350) - (1800 - 650)$. 2. Evaluate the grouped expressions: $2500 + 1350 = 3850$ and $1800 - 650 = 1150$. 3. Evaluate the final difference: $3850 - 1150 = 2700$.

Answer

Expression: $(2500 + 1350) - (1800 - 650)$ Value: $2700$

5190595

Start with the number $60$. Write a numerical expression for each part, then evaluate it. a) Decrease $60$ by $6$. b) Divide $60$ by $6$. c) Write and evaluate a numerical expression that shows how much greater the result from part a is than the result from part b.

Hints

- Translate each instruction into symbols before calculating. - Keep the two meanings of “decrease by” and “divide by” distinct. - For the comparison, think about which result must be subtracted from which.

Solution

1. For part a), decreasing $60$ by $6$ gives $60-6=54$. 2. For part b), dividing $60$ by $6$ gives $60\div6=10$. 3. For part c), subtract the part b result from the part a result: $(60-6)-(60\div6)=54-10=44$.

Answer

a) $60-6=54$ b) $60\div6=10$ c) $(60-6)-(60\div6)=44$

5191625

Write a numerical expression for this description, then evaluate it: Multiply the difference of $45$ and $37$ by the quotient of $120$ and $3$.

Hints

- Identify the two quantities named in the description before combining them. - Use grouping symbols so each stated quantity remains a single part of the expression. - After writing the complete expression, evaluate the grouped parts first.

Solution

The difference is $45 - 37$, and the quotient is $120 \div 3$. The numerical expression is $(45 - 37) \times (120 \div 3)$. Evaluate the grouped parts: $45 - 37 = 8$ and $120 \div 3 = 40$. Then $8 \times 40 = 320$.

Answer

$(45 - 37) \times (120 \div 3) = 320$

5193305

Write a numerical expression for each description, and then evaluate it. a) Subtract the product of $48$ and $75$ from $5000$. b) Multiply the sum of $145$ and $55$ by the quotient of $168$ and $14$.

Hints

- In the phrase “subtract ... from,” the number after “from” comes first. - Use grouping symbols around a sum or quotient that acts as one factor. - Evaluate the grouped parts before combining them.

Solution

1. For part a), the expression is $5000-48\times75$. Since $48\times75=3600$, the value is $5000-3600=1400$. 2. For part b), the expression is $(145+55)\times(168\div14)$. The grouped quantities are $200$ and $12$, so the value is $200\times12=2400$.

Answer

a) $5000-48\times75=1400$ b) $(145+55)\times(168\div14)=2400$

5193315

Write a numerical expression for each description, and then evaluate it. a) Divide the product of $32$ and $15$ by the difference between $94$ and $78$. b) Subtract the sum of $347$ and $253$ from the product of $125$ and $8$.

Hints

- Use grouping symbols to show the complete dividend and divisor. - In the phrase “subtract ... from,” place the quantity after “from” first. - Evaluate each grouped quantity separately.

Solution

1. For part a), the expression is $(32\times15)\div(94-78)$. The grouped quantities are $480$ and $16$, so $480\div16=30$. 2. For part b), the expression is $125\times8-(347+253)$. The product is $1000$ and the grouped sum is $600$, so $1000-600=400$.

Answer

a) $(32\times15)\div(94-78)=30$ b) $125\times8-(347+253)=400$

5194485

A school receives the same fruit delivery on each of the five school days. One delivery contains $12$ boxes of apples weighing $15\,\text{kg}$ each, $8$ boxes of pears weighing $12\,\text{kg}$ each, and $10$ boxes of oranges weighing $20\,\text{kg}$ each. Write one numerical expression for the total mass of fruit delivered during the week, and evaluate it.

Hints

- Represent the mass of each type of fruit in one delivery. - Use grouping symbols so the daily total is found before multiplying by the number of days. - Evaluate multiplication before addition inside the parentheses.

Solution

1. The daily masses are represented by $12\times15$, $8\times12$, and $10\times20$. 2. Add the three daily amounts inside grouping symbols and multiply by the $5$ delivery days: $(12\times15+8\times12+10\times20)\times5$. 3. Evaluate inside the parentheses: $180+96+200=476$. 4. Multiply by $5$: $476\times5=2380$.

Answer

Expression: $(12\times15+8\times12+10\times20)\times5$ Total mass: $2380\,\text{kg}$

5194675

Determine whether the two statements lead to the same value. Write and evaluate a numerical expression for each statement. Statement 1: “Four times the difference between $20$ and $5$.” Statement 2: “Subtract $5$ from four times $20$.”

Hints

- Decide what “four times” applies to in each statement. - Determine whether grouping symbols are needed. - Compare the evaluated expressions.

Solution

1. Statement 1 means $4 \times (20 - 5)$. Its value is $4 \times 15 = 60$. 2. Statement 2 means $4 \times 20 - 5$. Its value is $80 - 5 = 75$. 3. Since $60 \ne 75$, the statements do not lead to the same value.

Answer

No. Statement 1: $4 \times (20 - 5) = 60$ Statement 2: $4 \times 20 - 5 = 75$

5196485

A school-supply store packs $8$ identical classroom kits. Each kit contains $12$ thick pencils and $18$ thin pencils. The store also prepares $5$ packages with $6$ erasers in each package. Write a numerical expression for the total number of pencils and erasers, and evaluate it.

Hints

- Find the number of pencils in one kit first. - Use grouping symbols to show that the two pencil amounts are combined before multiplying by $8$. - Add the separate eraser total.

Solution

1. Each classroom kit contains $12+18=30$ pencils. 2. The $8$ kits contain $8\times30=240$ pencils. 3. The eraser packages contain $5\times6=30$ erasers. 4. The complete expression is $8\times(12+18)+5\times6$, and its value is $240+30=270$.

Answer

Expression: $8\times(12+18)+5\times6$ Total: $270$ items

5199755

A group of $12$ hikers rents a cabin for one night. The plan is for each person to pay $\$20$. Shortly before the trip, $4$ people cancel, but each of them still contributes $\$6$. By how many dollars does the cost increase for each remaining hiker? Write one numerical expression and evaluate it.

Hints

- Identify which amounts in the situation contribute to the total cabin cost and which reduce what the remaining hikers must cover. - Think about how the number of people sharing the remaining cost changes after the cancellations. - Build one expression that produces the new cost per remaining hiker before comparing it with the original $\$20$.

Solution

1. A single expression for the increase is $(12 \times 20 - 4 \times 6) \div (12 - 4) - 20$. 2. The amount the remaining hikers must cover is $240 - 24 = 216$, and there are $8$ remaining hikers. 3. Evaluate: $216 \div 8 - 20 = 27 - 20 = 7$.

Answer

The cost increases by $\$7$ for each remaining hiker. One expression is $(12 \times 20 - 4 \times 6) \div (12 - 4) - 20 = 7$.

5199775

A sports club orders a buffet that costs $\$600$. At first, $15$ members plan to split the cost equally. Then $5$ more members decide to attend, but the total cost stays the same. By how many dollars does each person's share decrease? Write one numerical expression and evaluate it.

Hints

- Represent the original share and the new share as two quotients. - Account for the change in the number of people before writing the second quotient. - Combine the two shares in one expression that measures the decrease.

Solution

1. A single expression for the decrease is $600 \div 15 - 600 \div (15 + 5)$. 2. The original share is $600 \div 15 = 40$, and the new share is $600 \div 20 = 30$. 3. Evaluate the difference: $40 - 30 = 10$.

Answer

Each person's share decreases by $\$10$. One expression is $600 \div 15 - 600 \div (15 + 5) = 10$.

5203916

Maya makes a chain of triangles with craft sticks. The first triangle needs $3$ sticks. Each additional triangle needs only $2$ more sticks because neighboring triangles share one side. a) How many sticks are needed for a chain of $25$ triangles? b) Maya uses all $61$ sticks in a box. How many triangles are in the chain? c) Let $n$ be the number of triangles and $s$ be the number of sticks. Write an equation that relates $s$ to $n$.

Hints

- Compare what changes when one more triangle is added to the chain. - For the reverse question, undo the same relationship that connects the number of triangles to the number of sticks. - In the equation, make the number of sticks depend on the number of triangles.

Solution

a) The first triangle uses $3$ sticks and the other $24$ triangles add $2$ sticks each, so $3 + 24 \times 2 = 51$ sticks. b) The relationship can be reversed: $(61 - 1) \div 2 = 30$, so the chain has $30$ triangles. c) Each triangle contributes two sticks to the repeating pattern, with one additional stick at the end. The equation is $s = 2n + 1$.

Answer

a) $51$ sticks b) $30$ triangles c) $s = 2n + 1$

5206885

A cargo ship can carry at most $80{,}000\,\text{kg}$ of cargo. It already carries $52{,}650\,\text{kg}$. Workers add $45$ containers with a mass of $420\,\text{kg}$ each, $18$ machines with a mass of $315\,\text{kg}$ each, and $950\,\text{kg}$ of other equipment. Write and evaluate one numerical expression to find how much more cargo mass the ship can carry. Your answer must include both the expression and its value.

Hints

- Represent all cargo already on the ship as one complete quantity before subtracting it from the capacity. - Use multiplication for each group of equal-mass items. - Include the complete numerical expression in your response, not only the remaining mass.

Solution

1. Represent the remaining capacity by subtracting the entire loaded mass from the maximum: $80{,}000-(52{,}650+45\times420+18\times315+950)$. 2. Find the repeated-item masses: $45\times420=18{,}900$ and $18\times315=5670$. 3. The total loaded mass is $52{,}650+18{,}900+5670+950=78{,}170\,\text{kg}$. 4. The remaining capacity is $80{,}000-78{,}170=1830\,\text{kg}$.

Answer

$80{,}000-(52{,}650+45\times420+18\times315+950)=1830$ The ship can carry $1830\,\text{kg}$ more cargo.

5209035

Two books cost $\$28.40$ altogether. A nonfiction book costs exactly $\$4.20$ more than a novel. Write and evaluate numerical expressions to find the price of each book, and briefly explain your reasoning.

Hints

- Represent the two prices as a lower amount and that same amount with an extra $\$4.20$. - Look for a way to adjust the total so it represents two equal amounts. - Check that your two prices have both the required total and the required difference.

Solution

1. If the extra $\$4.20$ is removed from the total, the remaining amount is twice the novel's price. Thus the novel's price is $(\$28.40 - \$4.20) \div 2 = \$12.10$. 2. The nonfiction book's price is $(\$28.40 - \$4.20) \div 2 + \$4.20 = \$16.30$. 3. Check: $\$12.10 + \$16.30 = \$28.40$, and the difference is $\$4.20$.

Answer

Novel: $(\$28.40 - \$4.20) \div 2 = \$12.10$ Nonfiction book: $(\$28.40 - \$4.20) \div 2 + \$4.20 = \$16.30$

5212445

For each part, write a numerical expression that represents the operation phrase, then evaluate it. a) What number is $5$ greater than $15$? b) What number is $3$ times as large as $8$? c) An unknown number is decreased by $2$, and the result is $8$. What is the result if the original unknown number is instead multiplied by $10$?

Hints

- Translate each phrase into symbols before evaluating it. - In part c), first represent the operation that recovers the original number. - Use grouping symbols when the recovered number becomes one factor in a new calculation.

Solution

1. For part a), $5$ greater than $15$ is represented by $15+5$, and $15+5=20$. 2. For part b), $3$ times as large as $8$ is represented by $8\times3$, and $8\times3=24$. 3. For part c), recover the original number with $8+2$. Then multiply that number by $10$: $(8+2)\times10=100$.

Answer

a) $15+5=20$ b) $8\times3=24$ c) $(8+2)\times10=100$

5216205

Write a numerical expression for the statement, and then evaluate it: Add to the difference between $412$ and $255$ the sum of $63$ and the difference between $124$ and $88$.

Hints

- Separate the statement into its two main quantities. - The second quantity contains a difference inside a sum, so nested grouping symbols are useful. - Evaluate from the innermost grouping symbols outward.

Solution

1. The expression is $(412 - 255) + [63 + (124 - 88)]$. 2. Evaluate the first difference: $412 - 255 = 157$. 3. Evaluate the nested difference and sum: $124 - 88 = 36$, and $63 + 36 = 99$. 4. Add: $157 + 99 = 256$.

Answer

$(412 - 255) + [63 + (124 - 88)] = 256$

5216215

Write a numerical expression for the statement, and then evaluate it: Subtract from the difference between $800$ and $345$ the difference between $150$ and the sum of $45$ and $23$.

Hints

- Translate the statement one complete quantity at a time. - The second difference contains a sum, so nested grouping symbols are needed. - In the phrase “subtract ... from,” the quantity after “from” comes first.

Solution

1. The expression is $(800 - 345) - [150 - (45 + 23)]$. 2. Evaluate the first difference: $800 - 345 = 455$. 3. Evaluate the nested sum and difference: $45 + 23 = 68$, and $150 - 68 = 82$. 4. Subtract: $455 - 82 = 373$.

Answer

$(800 - 345) - [150 - (45 + 23)] = 373$

5186275

A youth sports club is taking $22$ children and $4$ chaperones on a ferry. One group pass covers up to $6$ adult fare units and costs $\$38$. Two children count as one adult fare unit. A single adult fare costs $\$12$. For each choice of $0$, $1$, $2$, or $3$ group passes, write one numerical expression for the total fare cost and evaluate it. Then identify the least possible cost. Your answer must include all four expressions and values.

Hints

- First convert the children to adult fare units and combine them with the chaperones. - For each pass choice, determine how many fare units are still uncovered before writing the cost expression. - Include the numerical expression for every option before comparing their values.

Solution

1. The $22$ children count as $22\div2=11$ adult fare units, so the club needs $11+4=15$ adult fare units altogether. 2. With no group passes, the cost expression is $15\times12=180$. 3. With one group pass, $9$ fare units remain, so the cost expression is $38+9\times12=146$. 4. With two group passes, $3$ fare units remain, so the cost expression is $2\times38+3\times12=112$. 5. With three group passes, the cost expression is $3\times38=114$. 6. The least of the four evaluated expressions is $112$, so the least cost is $\$112$.

Answer

$0$ passes: $15\times12=180$ $1$ pass: $38+9\times12=146$ $2$ passes: $2\times38+3\times12=112$ $3$ passes: $3\times38=114$ Least cost: $\$112$

5186295

An adventure park offers a group pass for up to $2$ adults and $3$ children for $\$45$. Anyone not covered by a group pass pays $\$15$ per adult and $\$8$ per child. A hiking group has $5$ adults and $8$ children. For each choice of $0$, $1$, $2$, or $3$ group passes, write one numerical expression for the total admission cost and evaluate it. Then identify the least possible cost. Your answer must include all four expressions and values.

Hints

- Determine how many adults and children remain uncovered for each number of group passes. - Translate each option into one total-cost expression before evaluating it. - Compare the four evaluated expressions only after all four models are written.

Solution

1. With no group passes, the cost expression is $5\times15+8\times8=139$. 2. With one group pass, $3$ adults and $5$ children remain, so the expression is $45+3\times15+5\times8=130$. 3. With two group passes, $1$ adult and $2$ children remain, so the expression is $2\times45+15+2\times8=121$. 4. Three group passes cover the entire group, so the expression is $3\times45=135$. 5. The least of the four evaluated expressions is $121$, so the least cost is $\$121$.

Answer

$0$ passes: $5\times15+8\times8=139$ $1$ pass: $45+3\times15+5\times8=130$ $2$ passes: $2\times45+15+2\times8=121$ $3$ passes: $3\times45=135$ Least cost: $\$121$

5203926

A square flower bed is surrounded by one row of square paving stones. The side length of the bed is measured in stone lengths. <table> <tr><td>Side length of flower bed</td><td>$1$</td><td>$2$</td><td>$3$</td></tr> <tr><td>Number of paving stones</td><td>$8$</td><td>$12$</td><td>$16$</td></tr> </table> a) How many paving stones are needed when the side length is $12$? b) Let $n$ be the side length and $p$ be the number of paving stones. Write an equation that relates $p$ to $n$.

Hints

- Compare consecutive rows of the table to identify how the paving-stone count changes. - Relate the changing paving-stone count to the changing side length. - In part b), write the paving-stone count as the dependent quantity.

Solution

a) The number of paving stones increases by $4$ whenever the side length increases by $1$. For side length $12$, $4 \times 12 + 4 = 52$, so $52$ paving stones are needed. b) Four side lengths contribute $4n$, with $4$ additional corner stones. The equation is $p = 4n + 4$.

Answer

a) $52$ paving stones b) $p = 4n + 4$

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