Aimathic – Translate verbal to numerical expressions

Translate verbal to numerical expressions

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Difficulty: 1 – 5

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 a), the expression is $350 + (234 + 416)$. Its value is $350 + 650 = 1000$. 2. For b), the expression is $(310 + 190) + (145 + 255)$. Its 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

Carry out each instruction. 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

- Match each phrase with an operation. - “Four times” indicates multiplication. - “Decrease by” indicates subtraction, and “increase by” indicates addition.

Solution

1. Part a: Four times $18$ means $18 \times 4 = 72$. 2. Part b: Decrease $130$ by $50$: $130 - 50 = 80$. 3. Part c: Find the number of groups by dividing: $54 \div 6 = 9$. 4. Part d: Increase $37$ by $23$: $37 + 23 = 60$.

Answer

a) $72$ b) $80$ c) $9$ d) $60$

5181545

Answer each question using the meaning of the operation phrase. 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

- “Difference” and “less than” indicate subtraction. - “How many groups” indicates division. - “One-fourth of” means divide by $4$.

Solution

1. Part a: The difference is found by subtraction: $100 - 16 = 84$. 2. Part b: The number of groups is found by division: $80 \div 20 = 4$. 3. Part c: A number $25$ less than $100$ is $100 - 25 = 75$. 4. Part d: One-fourth of $100$ is $100 \div 4 = 25$.

Answer

a) $84$ b) $4$ c) $75$ d) $25$

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. The product is $240 \times 5 = 1200$. 2. The quotient is $800 \div 4 = 200$. Then $200 - 150 = 50$.

Answer

a) $240 \times 5 = 1200$ b) $(800 \div 4) - 150 = 50$

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 $15{,}600$ 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. a) The expression is $25 \times 14 + 1200 \div 30$. 2. Evaluate the product and quotient: $25 \times 14 = 350$ and $1200 \div 30 = 40$. 3. Add: $350 + 40 = 390$. 4. b) The expression is $15{,}600 \div (13 \times 4)$. 5. Evaluate the product, then divide: $13 \times 4 = 52$ and $15{,}600 \div 52 = 300$.

Answer

a) $25 \times 14 + 1200 \div 30 = 390$ b) $15{,}600 \div (13 \times 4) = 300$

5194395

Write this statement as an equation using numbers and operation symbols: The quotient of $48$ and $6$ equals the difference between $15$ and $7$.

Hints

- Translate “quotient” into a division expression. - Translate “difference” into a subtraction expression. - Use an equals sign to connect expressions with the same value.

Solution

1. The quotient of $48$ and $6$ is written $48 \div 6$. 2. The difference between $15$ and $7$ is written $15 - 7$. 3. Both expressions equal $8$, so the equation is $48 \div 6 = 15 - 7$.

Answer

$48 \div 6 = 15 - 7$

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 $15 - 8 = 7$.

5200325

Write an equation for each sentence. a) The product of $6$ and $9$ is $54$. b) The difference between $80$ and $15$ is $65$. c) The quotient of $32$ and $4$ is $8$.

Hints

- Match product, difference, and quotient with their operation symbols. - Translate “is” as an equals sign. - Check that each equation is true.

Solution

1. “Product” indicates multiplication: $6 \times 9 = 54$. 2. “Difference” indicates subtraction: $80 - 15 = 65$. 3. “Quotient” indicates division: $32 \div 4 = 8$.

Answer

a) $6 \times 9 = 54$ b) $80 - 15 = 65$ c) $32 \div 4 = 8$

5201045

Write this sentence as an equation using numbers and operation symbols: The sum of $23$ and $17$ equals the product of $4$ and $10$.

Hints

- Translate “sum” into addition. - Translate “product” into multiplication. - Translate “equals” into an equals sign.

Solution

1. The sum of $23$ and $17$ is written $23 + 17$. 2. The product of $4$ and $10$ is written $4 \times 10$. 3. Connect the expressions with an equals sign: $23 + 17 = 4 \times 10$. Both sides equal $40$.

Answer

$23 + 17 = 4 \times 10$

5204605

Determine whether each statement is mathematically correct. Write the equation with the correct result. a) The product of the factors $6$ and $8$ is $48$. b) The quotient of $54$ and $6$ is $6$.

Hints

- Translate “product” into multiplication. - Translate “quotient” into division. - Calculate each expression before judging the statement.

Solution

1. Part a: A product is the result of multiplication. Since $6 \times 8 = 48$, the statement is correct. 2. Part b: A quotient is the result of division. Since $54 \div 6 = 9$, not $6$, the statement is incorrect.

Answer

a) Correct: $6 \times 8 = 48$ b) Incorrect: $54 \div 6 = 9$

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

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

5106335

Compare the two expressions. Which has the greater value? Show your calculations. 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

- Translate each verbal description into an expression with parentheses. - Evaluate the grouped operation first. - Compare the final values.

Solution

1. Expression A is $10-(2\frac{1}{4}+3\frac{1}{2})$. The sum in parentheses is $5\frac{3}{4}$, so the value is $4\frac{1}{4}$. 2. Expression B is $2\frac{1}{2}+(5-3\frac{1}{4})$. The difference in parentheses is $1\frac{3}{4}$, so the value is $4\frac{1}{4}$. 3. The expressions have equal values.

Answer

Neither is greater. Both expressions equal $4\frac{1}{4}$.

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 a), $1000 - 285 = 715$. 2. For b), $150 - (92 - 38) = 150 - 54 = 96$. 3. For c), $(120 - 45) + (18 + 32) = 75 + 50 = 125$.

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 $60$. a) Decrease $60$ by $6$. b) Divide $60$ by $6$. How much greater is the result from part a than the result from part b?

Hints

- “Decrease by” indicates subtraction. - “Divide by” indicates division. - Use subtraction to find how much greater one result is.

Solution

1. Part a: Decrease by $6$: $60 - 6 = 54$. 2. Part b: Divide by $6$: $60 \div 6 = 10$. 3. Compare the results: $54 - 10 = 44$.

Answer

The result from part a is $44$ greater than the result from part b.

5191625

Multiply the difference of $45$ and $37$ by the quotient of $120$ and $3$.

Hints

- “Difference” indicates subtraction. - “Quotient” indicates division. - Evaluate the two parts before multiplying their results.

Solution

1. Find the difference: $45 - 37 = 8$. 2. Find the quotient: $120 \div 3 = 40$. 3. Multiply the two results: $8 \times 40 = 320$.

Answer

$320$

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. a) The expression is $5000 - 48 \times 75$. 2. Evaluate: $48 \times 75 = 3600$, so $5000 - 3600 = 1400$. 3. b) The expression is $(145 + 55) \times (168 \div 14)$. 4. Evaluate the grouped quantities: $145 + 55 = 200$ and $168 \div 14 = 12$. 5. Multiply: $200 \times 12 = 2400$.

Answer

a) $5000 - 48 \times 75 = 1400$ b) $(145 + 55) \times (168 \div 14) = 2400$

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. a) The expression is $(32 \times 15) \div (94 - 78)$. 2. Evaluate the grouped quantities: $32 \times 15 = 480$ and $94 - 78 = 16$. 3. Divide: $480 \div 16 = 30$. 4. b) The expression is $125 \times 8 - (347 + 253)$. 5. Evaluate: $125 \times 8 = 1000$ and $347 + 253 = 600$. 6. Subtract: $1000 - 600 = 400$.

Answer

a) $(32 \times 15) \div (94 - 78) = 30$ b) $125 \times 8 - (347 + 253) = 400$

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

- First find the total cost of the cabin. - How much money is contributed by the people who cancel? - How much of the total cost is left for the remaining hikers? - How many hikers share that amount? - Compare the new cost per person with the original cost.

Solution

1. The total cabin cost is $12\times\$20=\$240$. 2. The people who cancel contribute $4\times\$6=\$24$. 3. The remaining hikers must pay $\$240-\$24=\$216$ altogether. 4. There are $12-4=8$ remaining hikers. 5. Each remaining hiker pays $\$216\div8=\$27$. 6. The increase is $\$27-\$20=\$7$. One expression is $(12\times20-4\times6)\div(12-4)-20=7$.

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

- How much would each person have paid at first? - How many people share the cost after the additional members join? - Find the new cost per person. - Compare the original and new amounts.

Solution

1. The original cost per person is $\$600\div15=\$40$. 2. The new number of people is $15+5=20$. 3. The new cost per person is $\$600\div20=\$30$. 4. The decrease is $\$40-\$30=\$10$. One expression is $600\div15-600\div(15+5)=10$.

Answer

Each person's share decreases by $\$10$. One expression is $600\div15-600\div(15+5)=10$.

5200335

Find the missing number and write the complete equation. a) The first factor is $4$, and the product is $28$. b) The dividend is $36$, and the quotient is $6$. c) The number being subtracted is $12$, and the difference is $40$.

Hints

- Identify the operation named by each set of terms. - Use an inverse operation to find a missing factor or divisor. - In subtraction, the number being subtracted comes after the minus sign.

Solution

1. Part a: Find the second factor with $28 \div 4 = 7$. The equation is $4 \times 7 = 28$. 2. Part b: Find the divisor with $36 \div 6 = 6$. The equation is $36 \div 6 = 6$. 3. Part c: Find the starting number with $40 + 12 = 52$. The equation is $52 - 12 = 40$.

Answer

a) $4 \times 7 = 28$ b) $36 \div 6 = 6$ c) $52 - 12 = 40$

5203525

Represent the statement with mathematical symbols, then calculate to determine whether it is true: The product of $5$ and $8$ equals the quotient of $80$ and $2$.

Hints

- Match “product” with multiplication and “quotient” with division. - Evaluate both sides separately. - Compare the values.

Solution

1. Write the product as $5 \times 8$. 2. Write the quotient as $80 \div 2$. 3. The equation is $5 \times 8 = 80 \div 2$. 4. Both sides equal $40$, so the statement is true.

Answer

$5 \times 8 = 80 \div 2$. The statement is true because both sides equal $40$.

5203915

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) Write a rule for the number of sticks needed for $n$ triangles.

Hints

- Separate the first triangle from the additional triangles. - Look for a rule that doubles the number of triangles and then adjusts by $1$. - Reverse the operations when the total number of sticks is known.

Solution

1. a) The first triangle uses $3$ sticks, and the other $24$ triangles use $2$ sticks each: $3+24\times2=51$. Equivalently, $25\times2+1=51$. 2. b) Reverse the rule: $(61-1)\div2=30$. The chain has $30$ triangles. 3. c) For $n$ triangles, the number of sticks is $2\times n+1$.

Answer

a) $51$ sticks b) $30$ triangles c) $2\times n+1$ sticks

5204815

Is the following statement true? First write it as an equation, then check both sides. The quotient of $72$ and $8$ equals the product of $3$ and $3$.

Hints

- Translate “quotient” into division. - Translate “product” into multiplication. - Evaluate and compare both sides.

Solution

1. The quotient of $72$ and $8$ is $72 \div 8 = 9$. 2. The product of $3$ and $3$ is $3 \times 3 = 9$. 3. Since both expressions equal $9$, the statement is true: $72 \div 8 = 3 \times 3$.

Answer

Yes. $72 \div 8 = 3 \times 3$, and both sides equal $9$.

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 $50 - 26 = 24$.

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

- Think about what the total would be if both books had the lower price. - Subtract the price difference from the total first. - Divide the remaining amount into two equal parts. - Add the difference back to find the higher price.

Solution

1. Remove the extra $\$4.20$ from the total: $\$28.40 - \$4.20 = \$24.20$. 2. The remaining amount represents two copies of the novel’s price. Divide by $2$: $\$24.20 \div 2 = \$12.10$. 3. Add the price difference to find the nonfiction book’s price: $\$12.10 + \$4.20 = \$16.30$. 4. Check: $\$12.10 + \$16.30 = \$28.40$.

Answer

The novel costs $\$12.10$, and the nonfiction book costs $\$16.30$.

5209055

Two weight plates have a total mass of $15.3\,\text{kg}$. One plate has $2.7\,\text{kg}$ more mass than the other. Write and evaluate a numerical expression to find the mass of each plate.

Hints

- Subtract the difference from the total first. - Divide the remaining amount equally between the two plates. - Add the difference to find the heavier plate.

Solution

1. The lighter plate’s mass is $(15.3 - 2.7) \div 2$. 2. Evaluate: $(15.3 - 2.7) \div 2 = 12.6 \div 2 = 6.3$. The lighter plate has a mass of $6.3\,\text{kg}$. 3. Add the difference: $6.3\,\text{kg} + 2.7\,\text{kg} = 9.0\,\text{kg}$.

Answer

The lighter plate has a mass of $6.3\,\text{kg}$, and the heavier plate has a mass of $9.0\,\text{kg}$.

5212445

Solve each problem by interpreting the operation phrase. 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 “greater than” into addition and “times as large as” into multiplication. - In part c, use the inverse operation to find the original number first. - Then follow the new multiplication instruction.

Solution

1. Part a: $15 + 5 = 20$. 2. Part b: $8 \times 3 = 24$. 3. Part c: Find the original number with $8 + 2 = 10$. Then multiply it by $10$: $10 \times 10 = 100$.

Answer

a) $20$ b) $24$ c) $100$

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 $\$12$. Write and evaluate numerical expressions to find the least the club can pay for all the fares.

Hints

- First convert the children to adult fare units. - Write a cost expression for each possible number of group passes. - Compare the values of the expressions.

Solution

1. Convert the children to adult fare units: $22 \div 2 = 11$. Including the chaperones, the group needs $11 + 4 = 15$ adult fare units. 2. With no group passes, the cost is $15 \times \$12 = \$180$. 3. With one group pass, the cost is $\$38 + 9 \times \$12 = \$146$. 4. With two group passes, $12$ fare units are covered and $3$ single fares are needed: $2 \times \$38 + 3 \times \$12 = \$112$. 5. With three group passes, the cost is $3 \times \$38 = \$114$. 6. The least of these costs is $\$112$.

Answer

The club must pay at least $\$112$.

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. Write and evaluate numerical expressions to find the least total admission cost.

Hints

- Determine how many adults and children each number of group passes covers. - Write a total-cost expression for each option. - Compare the values of the expressions.

Solution

1. With no group passes, the cost is $5 \times \$15 + 8 \times \$8 = \$139$. 2. With one group pass, $3$ adults and $5$ children still need tickets: $\$45 + 3 \times \$15 + 5 \times \$8 = \$130$. 3. With two group passes, $1$ adult and $2$ children still need tickets: $2 \times \$45 + \$15 + 2 \times \$8 = \$121$. 4. Three group passes cover the entire group: $3 \times \$45 = \$135$. 5. The least of these costs is $\$121$.

Answer

The least total admission cost is $\$121$.

5203925

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) Write a rule for the number of paving stones needed for a side length of $n$.

Hints

- Find the change from one table value to the next. - Think of four equal sides and four corner stones. - Use the side length in a rule that starts with multiplying by $4$.

Solution

1. The number of stones increases by $4$ whenever the side length increases by $1$. 2. For side length $12$, compute $4\times12+4=48+4=52$. 3. In general, the four sides contribute $4\times n$ stones, and the four corner stones contribute $4$ more. The rule is $4\times n+4$.

Answer

a) $52$ paving stones b) $4\times n+4$ paving stones

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.

Hints

- Find the total mass on the ship after loading. - Use multiplication for groups of equal-mass items. - Put the entire loaded mass in parentheses before subtracting from capacity. - Follow the order of operations.

Solution

1. Write the expression: $80{,}000 - (52{,}650 + 45 \times 420 + 18 \times 315 + 950)$. 2. Find the repeated-item masses: $45 \times 420 = 18{,}900$ and $18 \times 315 = 5670$. 3. Find the total loaded mass: $52{,}650 + 18{,}900 + 5670 + 950 = 78{,}170$. 4. Subtract from the capacity: $80{,}000 - 78{,}170 = 1830$.

Answer

The ship can carry $1830\,\text{kg}$ more cargo.

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