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Write expressions from verbal phrases

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5222696
Translate each verbal description into an algebraic expression. a) The sum of a number \(x\) and \(14\) b) The product of the variables \(c\) and \(d\) c) The difference between \(20\) and a variable \(y\) d) Eight times a number \(z\)

Hints

- Match the words sum, product, and difference with their operations. - In a difference, preserve the stated order. - “Eight times” means multiplication by \(8\).

Solution

1. The sum of \(x\) and \(14\) is \(x+14\). 2. The product of \(c\) and \(d\) is \(cd\). 3. Subtracting \(y\) from \(20\) gives \(20-y\). 4. Eight times \(z\) is \(8z\).

Answer

a) \(x+14\) b) \(cd\) c) \(20-y\) d) \(8z\)
5223616
Write an algebraic expression for each described number. 1. A number made of \(x\) hundreds and \(y\) tens 2. A number made of \(a\) thousands, \(b\) hundreds, and \(c\) ones 3. Three times a number made of \(n\) tens and \(k\) ones

Hints

- Use the value of each place: hundreds, tens, and ones. - Think of a number such as \(345\) as \(300+40+5\). - Apply “three times” to the entire original number.

Solution

1. Each hundred contributes \(100\), and each ten contributes \(10\), so the expression is \(100x+10y\). 2. The place-value expression is \(1000a+100b+c\). 3. The original number is \(10n+k\). Three times that number is \(3(10n+k)\), which is equivalent to \(30n+3k\).

Answer

1. \(100x+10y\) 2. \(1000a+100b+c\) 3. \(3(10n+k)\), or \(30n+3k\)
5124936
A quadrilateral has side lengths \(x\,\text{cm}\), \(x+2\,\text{cm}\), \(2x\,\text{cm}\), and \(5\,\text{cm}\). a) Write and simplify an expression for its perimeter \(P\). b) Find the perimeter when \(x=3.5\,\text{cm}\).

Hints

- Perimeter is the sum of all side lengths. - Translate each verbal side description into an expression. - Combine like terms before substituting the value of \(x\).

Solution

1. Add the four side lengths: \(P=x+(x+2)+2x+5\). 2. Combine like terms: \(P=4x+7\). 3. Substitute \(x=3.5\): \(P=4\times3.5+7=21\,\text{cm}\).

Answer

a) \(P=4x+7\) b) \(21\,\text{cm}\)
5222096
At a movie theater, an order of popcorn costs \(\$4.50\) and a soft drink costs \(\$3.20\). 1) Find the total cost of \(2\) orders of popcorn and \(3\) soft drinks. 2) Write an expression for the total cost of \(p\) orders of popcorn and \(d\) soft drinks. 3) What purchase is represented by \(3\times 4.50+3.20\)? 4) A customer buys \(x\) orders of popcorn for no more than \(\$20\) and pays with a \(\$20\) bill. Write an expression for the change.

Hints

- Multiply each item's price by the number purchased. - Add the costs of the two types of items. - Variables can represent the number of items in the same way that specific numbers do. - Change equals the amount paid minus the purchase cost.

Solution

1. The total is \(2\times 4.50+3\times 3.20=9.00+9.60=18.60\). 2. The cost expression is \(4.50p+3.20d\). 3. The expression \(3\times 4.50+3.20\) represents \(3\) orders of popcorn and \(1\) soft drink. 4. The popcorn costs \(4.50x\), so the change is \(20-4.50x\).

Answer

1) \(\$18.60\) 2) \(4.50p+3.20d\) 3) The cost of \(3\) orders of popcorn and \(1\) soft drink 4) \(20-4.50x\) dollars
5222176
A model assumes that apples lose \(\frac{4}{5}\) of their original weight when dried. a) Write an expression for the weight of the dried apples when the fresh apples weigh \(x\) pounds. b) Use the expression to find the dried weight of \(15\) pounds of fresh apples.

Hints

- Subtract the fraction lost from the whole. - Express the remaining fraction as a fraction or decimal. - Multiply the remaining fraction by the original amount.

Solution

1. The weight lost is \(\frac{4}{5}x\). The remaining weight is \(x-\frac{4}{5}x=\frac{1}{5}x\), which is also \(0.2x\). 2. For \(x=15\), \(\frac{1}{5}\times 15=3\).

Answer

a) \(\frac{1}{5}x\), or \(0.2x\) b) \(3\) pounds
5222616
Translate each verbal description into an algebraic expression using the given variables. a) The product of \(x\) and the sum of \(y\) and \(z\) b) The difference between five times \(a\) and \(12\) c) The quotient whose dividend is the difference of \(p\) and \(q\) and whose divisor is \(3\)

Hints

- Match the words sum, difference, product, and quotient with their operations. - Use parentheses when an entire sum or difference is part of another operation. - Identify the dividend and divisor in the quotient description.

Solution

1. The sum of \(y\) and \(z\) is \(y+z\). Multiplying that entire sum by \(x\) gives \(x(y+z)\). 2. Five times \(a\) is \(5a\). Subtracting \(12\) gives \(5a-12\). 3. The difference of \(p\) and \(q\) is \(p-q\). Dividing that entire difference by \(3\) gives \(\frac{p-q}{3}\).

Answer

a) \(x(y+z)\) b) \(5a-12\) c) \(\frac{p-q}{3}\)
5222706
Write an algebraic expression for each instruction. a) Five times the difference of \(x\) and \(y\) b) The sum of the product of \(3\) and \(a\) and the quotient of \(b\) and \(4\) c) Subtract twice \(z\) from \(50\)

Hints

- Use parentheses when an entire difference is multiplied. - “Twice” means multiplication by \(2\). - Pay attention to which quantity is subtracted from which.

Solution

1. The difference is \(x-y\), and five times that entire difference is \(5(x-y)\). 2. The product is \(3a\), and the quotient is \(\frac{b}{4}\). Their sum is \(3a+\frac{b}{4}\). 3. Twice \(z\) is \(2z\). Subtracting it from \(50\) gives \(50-2z\).

Answer

a) \(5(x-y)\) b) \(3a+\frac{b}{4}\) c) \(50-2z\)
5222716
Write an algebraic expression for each description. a) The difference between twice a number \(x\) and \(7\) b) The quotient of the sum of \(a\) and \(b\) and \(3\) c) The product of an integer \(y\) and the next integer

Hints

- Match sum, difference, product, and quotient with their operations. - “Twice” means multiplication by \(2\). - Consecutive integers differ by \(1\). - Use parentheses when an entire sum is divided or multiplied.

Solution

1. Twice \(x\) is \(2x\), so the difference is \(2x-7\). 2. The sum is \(a+b\). Dividing the entire sum by \(3\) gives \(\frac{a+b}{3}\). 3. The integer after \(y\) is \(y+1\), so the product is \(y(y+1)\).

Answer

a) \(2x-7\) b) \(\frac{a+b}{3}\) c) \(y(y+1)\)
5223276
A hiking trail is \(w\) miles long. A family hikes \(\frac{1}{5}\) of the trail on the first day and \(\frac{2}{5}\) of the trail on the second day. Write an expression in terms of \(w\) for the total number of miles they hike in the two days.

Hints

- Add the fractions of the trail completed on the two days. - The whole trail has length \(w\). - A fraction of an unknown quantity can be written as a product.

Solution

1. Add the fractions of the trail: \(\frac{1}{5}+\frac{2}{5}=\frac{3}{5}\). 2. Multiply the total fraction by the full trail length: \(\frac{3}{5}w\).

Answer

\(\frac{3}{5}w\) miles
5223356
The variables \(x\), \(y\), and \(z\) represent nonnegative measurement values. 1. How many milliliters are in \(x\) liters? 2. How many grams are in \(y\) kilograms? 3. How many seconds are in \(z\) minutes?

Hints

- Identify how many smaller units make one larger unit. - Try a specific value such as \(2\) before writing the expression with a variable. - Converting to a smaller unit requires multiplication.

Solution

1. Since \(1\,\text{L}=1000\,\text{mL}\), \(x\) liters is \(1000x\,\text{mL}\). 2. Since \(1\,\text{kg}=1000\,\text{g}\), \(y\) kilograms is \(1000y\,\text{g}\). 3. Since \(1\,\text{min}=60\,\text{s}\), \(z\) minutes is \(60z\,\text{s}\).

Answer

1. \(1000x\,\text{mL}\) 2. \(1000y\,\text{g}\) 3. \(60z\,\text{s}\)
5223366
The variables represent nonnegative measurement values. 1. How many centimeters are in a total of \(a\) meters and \(b\) centimeters? 2. How many cents are in an amount of \(d\) dollars and \(c\) cents?

Hints

- Convert each part to the same smaller unit. - Use \(100\,\text{cm}=1\,\text{m}\). - Use \(100\) cents for one dollar.

Solution

1. \(a\) meters is \(100a\) centimeters. Adding \(b\) centimeters gives \((100a+b)\,\text{cm}\). 2. \(d\) dollars is \(100d\) cents. Adding \(c\) cents gives \((100d+c)\) cents.

Answer

1. \((100a+b)\,\text{cm}\) 2. \((100d+c)\) cents
5223566
Algebraic expressions can describe whole groups of numbers. a) Explain why \(2n+1\) is odd for every whole number \(n\). b) Write an expression that represents all positive integers divisible by both \(3\) and \(4\). State the allowed values of your variable.

Hints

- What kind of number results when a whole number is multiplied by \(2\)? - What is the least positive integer divisible by both \(3\) and \(4\)? - How can a variable represent all multiples of one number?

Solution

1. For every whole number \(n\), \(2n\) is even. Adding \(1\) produces an odd number, so \(2n+1\) is always odd. 2. A number divisible by both \(3\) and \(4\) must be a multiple of their least common multiple. The least common multiple of \(3\) and \(4\) is \(12\). 3. Therefore, \(12k\), where \(k\) is any positive integer, represents all positive integers divisible by both \(3\) and \(4\).

Answer

a) \(2n\) is even, so \(2n+1\) is odd. b) \(12k\), where \(k\) is a positive integer.
5223576
Write an algebraic expression for each description using \(x\), \(y\), and \(z\). 1) Five times \(x\) 2) The sum of twice \(y\) and \(z\) 3) The product of \(x\) and the difference of \(y\) and \(z\) 4) The quotient of three times \(z\) and the product of \(x\) and \(y\)

Hints

- Match sum, product, and quotient with their operations. - Use parentheses when a difference must be found before multiplication. - “Twice” and “three times” indicate multiplication by a number.

Solution

1. Five times \(x\) is \(5x\). 2. Twice \(y\) is \(2y\), so the sum is \(2y+z\). 3. The difference is \(y-z\), so the product is \(x(y-z)\). 4. Three times \(z\) is \(3z\), and the product of \(x\) and \(y\) is \(xy\). The quotient is \(\frac{3z}{xy}\).

Answer

1) \(5x\) 2) \(2y+z\) 3) \(x(y-z)\) 4) \(\frac{3z}{xy}\)
5223996
Write an algebraic expression for each description. 1. Three times the sum of \(x\) and \(5\) 2. The difference between the square of \(a\) and twice \(b\) 3. The square of the difference of \(x\) and \(y\) 4. The sum of half of \(z\) and \(9\)

Hints

- Identify which operation must happen first. - Parentheses show that an entire expression is treated as one quantity. - Match sum, difference, product, and quotient with their operations. - “The square of” means raising the entire named quantity to the second power.

Solution

1. The sum is \(x+5\), so three times the sum is \(3(x+5)\). 2. The square of \(a\) is \(a^2\), and twice \(b\) is \(2b\), so the expression is \(a^2-2b\). 3. The difference is \(x-y\), and squaring the entire difference gives \((x-y)^2\). 4. Half of \(z\) is \(\frac{z}{2}\), so the sum is \(\frac{z}{2}+9\).

Answer

1. \(3(x+5)\) 2. \(a^2-2b\) 3. \((x-y)^2\) 4. \(\frac{z}{2}+9\)
5224056
Translate each description into an algebraic expression. a) Six times the sum of \(x\) and \(y\) b) The quotient of the difference of \(a\) and \(b\) and \(5\) c) Add three times \(m\) to four times \(n\) d) The product of the sum of \(p\) and \(q\) and the difference of the same two numbers

Hints

- Use parentheses when an entire sum or difference is part of another operation. - Words such as “of” can help identify which quantities belong together. - Match sum, difference, product, and quotient with their operations. - In a quotient, identify the dividend and divisor in the stated order.

Solution

1. The sum is \(x+y\), so six times the sum is \(6(x+y)\). 2. The difference is \(a-b\), and dividing the entire difference by \(5\) gives \(\frac{a-b}{5}\). 3. Three times \(m\) is \(3m\), and four times \(n\) is \(4n\). Their sum is \(3m+4n\). 4. The sum is \(p+q\), and the difference is \(p-q\). Their product is \((p+q)(p-q)\).

Answer

a) \(6(x+y)\) b) \(\frac{a-b}{5}\) c) \(3m+4n\) d) \((p+q)(p-q)\)
5224176
Write an algebraic expression for each description. 1) The sum of the square of \(x\) and four times \(y\) 2) The product of the difference of \(a\) and \(b\) and \(7\) 3) The quotient of the sum of \(p\) and \(q\) and their product, where \(pq\ne 0\)

Hints

- Identify the final operation in each description. - Put a compound sum or difference in parentheses. - A denominator cannot equal \(0\).

Solution

1. The square of \(x\) is \(x^2\), and four times \(y\) is \(4y\), so the expression is \(x^2+4y\). 2. The difference is \(a-b\), so the product with \(7\) is \(7(a-b)\). 3. The sum is \(p+q\), and the product is \(pq\). The quotient is \(\frac{p+q}{pq}\). The condition \(pq\ne 0\) keeps the denominator nonzero.

Answer

1) \(x^2+4y\) 2) \(7(a-b)\) 3) \(\frac{p+q}{pq}\), where \(pq\ne 0\)
5224196
Write an algebraic expression for each description. 1) The sum of three times a number \(x\) and \(7\) 2) Three times the sum of a number \(x\) and \(7\) 3) The square of the difference of \(a\) and \(b\) 4) The difference of the squares of \(a\) and \(b\)

Hints

- Identify which operation is applied to the entire expression. - Use parentheses when the usual order of operations must be changed. - “The square of” applies to the complete quantity named. - “The difference of the squares” means square first and subtract afterward.

Solution

1. Three times \(x\) is \(3x\), so the sum is \(3x+7\). 2. The sum is \(x+7\), so three times the entire sum is \(3(x+7)\). 3. The difference is \(a-b\), and squaring the entire difference gives \((a-b)^2\). 4. Square each number first, then subtract: \(a^2-b^2\).

Answer

1) \(3x+7\) 2) \(3(x+7)\) 3) \((a-b)^2\) 4) \(a^2-b^2\)
5244606
Write an expression with the variable \(k\) for each description. In each part, \(k\) is any integer. a) All integers divisible by \(8\) b) All integers that leave a remainder of \(5\) when divided by \(8\) c) All odd integers d) All integers that are \(2\) less than a multiple of \(3\)

Hints

- A multiple of a number can be written as that number times an integer. - Add the remainder to a multiple of the divisor. - Start with an even integer and change it by \(1\) to create an odd integer. - The phrase “\(2\) less than” indicates subtraction.

Solution

1. A multiple of \(8\) has the form \(8k\). 2. A number that is \(5\) more than a multiple of \(8\) has the form \(8k+5\). 3. An odd integer can be written as \(2k+1\). The equivalent form \(2k-1\) also generates all odd integers. 4. A multiple of \(3\) has the form \(3k\), so a number \(2\) less is \(3k-2\).

Answer

a) \(8k\) b) \(8k+5\) c) \(2k+1\), or \(2k-1\) d) \(3k-2\)
5222626
Investigate how parentheses affect an expression. a) Write an expression for: “Subtract the sum of \(b\) and \(c\) from \(a\).” b) Evaluate your expression for \(a=50\), \(b=15\), and \(c=5\). c) Evaluate \(a-b+c\) for the same values. Compare the results and explain the difference.

Hints

- Which entire quantity is being subtracted from \(a\)? - Substitute the values and follow the order of operations. - Compare whether \(c\) is subtracted or added in each expression.

Solution

1. The sum of \(b\) and \(c\) is \(b+c\), so subtracting that sum from \(a\) gives \(a-(b+c)\). 2. Substitution gives \(50-(15+5)=50-20=30\). 3. The comparison expression gives \(50-15+5=35+5=40\). 4. The expressions are not equivalent. In \(a-(b+c)\), both \(b\) and \(c\) are subtracted. In \(a-b+c\), \(c\) is added.

Answer

a) \(a-(b+c)\) b) \(30\) c) \(40\). The results differ because the first expression subtracts both \(b\) and \(c\), while the second adds \(c\).
5223586
The order of operations matters when translating verbal descriptions. a) Write an expression for “three times the difference of \(a\) and \(b\).” b) Write an expression for “the difference between three times \(a\) and \(b\).” c) Explain the structural difference between the expressions. d) Evaluate both expressions when \(a=12\) and \(b=4\).

Hints

- Decide whether “three times” applies to one variable or to an entire difference. - Use parentheses to override the usual order of operations. - Substitute the values into each expression separately.

Solution

1. In part a), find the difference first, then multiply: \(3(a-b)\). 2. In part b), only \(a\) is multiplied by \(3\), then \(b\) is subtracted: \(3a-b\). 3. Parentheses in the first expression require subtraction before multiplication. In the second expression, multiplication is performed before subtraction. 4. Substitute \(a=12\) and \(b=4\): \(3\times(12-4)=24\), while \(3\times 12-4=32\).

Answer

a) \(3(a-b)\) b) \(3a-b\) c) In part a), the entire difference is tripled. In part b), only \(a\) is tripled before \(b\) is subtracted. d) Part a): \(24\); Part b): \(32\)
5224066
Complete the following tasks with algebraic expressions. a) Subtract the sum of \(x\) and \(y\) from twice their product. b) Explain the difference between “the sum of twice \(a\) and \(b\)” and “twice the sum of \(a\) and \(b\).” Use expressions in your explanation. c) Evaluate the expression from part a) for \(x=5\) and \(y=2\).

Hints

- The phrase “subtract A from B” means \(B-A\). - Decide which operation must happen first and use parentheses when needed. - When evaluating, follow the order of operations.

Solution

1. The product of \(x\) and \(y\) is \(xy\), so twice the product is \(2xy\). Subtracting the sum \(x+y\) gives \(2xy-(x+y)\). 2. “The sum of twice \(a\) and \(b\)” is \(2a+b\). “Twice the sum of \(a\) and \(b\)” is \(2(a+b)\). In the first expression, only \(a\) is doubled; in the second, the entire sum is doubled. 3. Substitute the values: \(2\times 5\times 2-(5+2)=20-7=13\).

Answer

a) \(2xy-(x+y)\) b) \(2a+b\) doubles only \(a\), while \(2(a+b)\) doubles the entire sum. c) \(13\)
5224186
Translate each verbal description into an algebraic expression. 1) The square of half the sum of \(a\) and \(b\) 2) The difference between the sum of the squares of \(x\) and \(y\) and the square of their sum 3) The quotient of the sum of the cubes of \(m\) and \(n\) and the difference of those cubes, where \(m\ne n\)

Hints

- Read each description from the inside operation outward. - Distinguish “sum of the squares” from “square of the sum.” - A cube has exponent \(3\). - A denominator cannot equal \(0\).

Solution

1. The sum is \(a+b\), half the sum is \(\frac{a+b}{2}\), and its square is \(\left(\frac{a+b}{2}\right)^2\). 2. The sum of the squares is \(x^2+y^2\), and the square of the sum is \((x+y)^2\). Their difference is \((x^2+y^2)-(x+y)^2\). 3. The sum of the cubes is \(m^3+n^3\), and their difference is \(m^3-n^3\). The quotient is \(\frac{m^3+n^3}{m^3-n^3}\). Since \(m\ne n\), the denominator is nonzero.

Answer

1) \(\left(\frac{a+b}{2}\right)^2\) 2) \((x^2+y^2)-(x+y)^2\) 3) \(\frac{m^3+n^3}{m^3-n^3}\), where \(m\ne n\)
5224206
Translate each verbal description into an algebraic expression. 1) The product of the sum of \(x\) and \(4\) and the difference of \(x\) and \(4\) 2) Half the sum of a number \(x\) and the square of a number \(y\) 3) The square of half the sum of \(x\) and \(y\)

Hints

- Identify the entire expression that is multiplied, halved, or squared. - Put sums and differences in parentheses. - Decide whether the square applies to one term or to a complete expression.

Solution

1. The sum is \(x+4\), and the difference is \(x-4\). Their product is \((x+4)(x-4)\). 2. The sum is \(x+y^2\), so half the sum is \(\frac{x+y^2}{2}\). 3. Half the sum is \(\frac{x+y}{2}\), and its square is \(\left(\frac{x+y}{2}\right)^2\).

Answer

1) \((x+4)(x-4)\) 2) \(\frac{x+y^2}{2}\) 3) \(\left(\frac{x+y}{2}\right)^2\)
5279286
Three whole numbers have a sum of \(150\). Two of the numbers are \(a\) and \(b\). a) Write an expression for the third number. b) If \(a\) increases by \(15\), how must the third number change so that the total remains \(150\)? Explain.

Hints

- Subtract the two known parts from the total. - If one addend increases while the sum stays fixed, another addend must compensate. - Test the change with a simple numerical example.

Solution

1. Let the third number be \(z\). Then \(a + b + z = 150\), so \(z = 150 - a - b\). 2. If \(a\) increases by \(15\), the sum would increase by \(15\) unless another addend changes. 3. To keep the total at \(150\), the third number must decrease by \(15\).

Answer

a) \(150 - a - b\) b) The third number must decrease by \(15\).
5279296
Two numbers have a sum of \(s\). One addend is \(15\). 1. Write an expression for the other addend. 2. Find the other addend when \(s = 42\). 3. State a rule for finding an unknown addend when the sum and the other addend are known.

Hints

- Write an equation for the two addends and their sum. - Which operation undoes addition? - Substitute \(s = 42\) into your expression.

Solution

1. If the unknown addend is \(x\), then \(15 + x = s\), so \(x = s - 15\). 2. When \(s = 42\), \(x = 42 - 15 = 27\). 3. Subtract the known addend from the sum.

Answer

1. \(s - 15\) 2. \(27\) 3. Subtract the known addend from the sum.
5279306
A division has a quotient of \(7\) and a divisor of \(d\). 1. Write an expression for the dividend \(x\). 2. Find \(x\) when \(d = 13\). 3. State a rule for finding the dividend from the quotient and divisor.

Hints

- Begin with “dividend divided by divisor equals quotient.” - Use multiplication to undo division. - Substitute \(d = 13\) into your expression.

Solution

1. The division equation is \(x \div d = 7\). Multiplying by \(d\) gives \(x = 7d\). 2. When \(d = 13\), \(x = 7 \times 13 = 91\). 3. Multiply the quotient by the divisor to find the dividend.

Answer

1. \(x = 7d\) 2. \(x = 91\) 3. \(\text{dividend} = \text{quotient} \times \text{divisor}\)

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