TACHS Quiz: Translating Word Problems
15 questions · exam conditions
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Translating Word ProblemsQuestion 1 of 15

Which algebraic expression best represents the phrase "seven more than twice a number x"?

x+72\frac{x+7}{2} (average of the two parts)
2(x+7)2(x+7) (product followed by addition)
7x+27x+2 (terms reversed in order)
2x+72x+7 (twice the number, then add seven)
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TACHS Quiz

TACHS Quiz: Translating Word Problems

Practice Translating Word Problems in TACHS with focused quiz questions that help you check what you know, review explanations, and build confidence with test-style prompts.

What this quiz covers

This quiz focuses on Translating Word Problems, giving you a quick way to practice the rules, question types, and explanations that matter most for TACHS.

How to use this quiz

Try each quiz question before looking at the correct answer. Use the explanations to review missed ideas, then come back to similar questions until the pattern feels familiar.

All questions

Question 1

Which algebraic expression best represents the phrase "seven more than twice a number x"?

  1. x+72\frac{x+7}{2} (average of the two parts)
  2. 2(x+7)2(x+7) (product followed by addition)
  3. 7x+27x+2 (terms reversed in order)
  4. 2x+72x+7 (twice the number, then add seven) (correct answer)
Explanation: When translating word problems into algebraic expressions, you need to carefully identify the mathematical operations and their order. The phrase "seven more than twice a number x" contains two key operations that must be performed in the correct sequence. Let's break down the phrase systematically. "Twice a number x" means you multiply x by 2, giving you 2x2x. Then "seven more than" this result means you add 7 to what you already have. So you get 2x+72x + 7, which is choice D. Now let's examine why the other options are incorrect. Choice A, x+72\frac{x+7}{2}, represents "half of seven more than x" - this involves division, which doesn't appear in our original phrase. Choice B, 2(x+7)2(x+7), means "twice the quantity of seven more than x" - this would be the correct translation if the phrase were "twice seven more than a number x," but that's not what we have. Choice C, 7x+27x+2, reverses the coefficients entirely, giving us "two more than seven times x," which is completely different from our target phrase. The key insight is recognizing that "more than" indicates addition, and you must identify what you're adding to what. The phrase structure "A more than B" translates to "B + A" in algebra. Strategy tip: When translating word problems, work from the inside out. Identify the base quantity first (twice x), then apply additional operations (add seven) in the order the sentence presents them.

Question 2

Kia bought several pens at $1.25 each and one notebook for $3.50. She spent a total of $15. Which equation could be used to find the number of pens p she purchased?

  1. 1.25p+3.50=151.25p+3.50=15 (sum of pen cost and notebook cost) (correct answer)
  2. 1.25+3.50p=151.25+3.50p=15 (prices attached to wrong items)
  3. 1.25p−3.50=151.25p-3.50=15 (subtracts notebook cost)
  4. 3.50p+1.25=153.50p+1.25=15 (notebook cost multiplied by p)
Explanation: When you encounter word problems involving multiple items with different costs, your goal is to translate the written information into a mathematical equation by identifying what's fixed versus what varies. In this problem, Kia bought an unknown number of pens at $1.25 each, plus one notebook for $3.50, spending 15total.Sincethenumberofpensvaries,youmultiplythecostperpen(15 total. Since the number of pens varies, you multiply the cost per pen (1.25) by the variable p. The notebook cost stays fixed at $3.50 since she bought exactly one. Adding these costs together gives you the total spent: $1.25p+3.50=151.25p + 3.50 = 15 $. Choice A is correct because it properly represents pen cost ( 1.25p ) plus notebook cost ( 3.50 ) equaling the total spent. Choice B switches the variable and constant incorrectly, suggesting she bought multiple notebooks at $3.50 each and only one pen. This contradicts the problem statement. Choice C uses subtraction instead of addition, which doesn't make sense since both items contribute to her total spending. You can't subtract a purchase from your total cost. Choice D multiplies the notebook price by p, implying she bought p notebooks instead of p pens. This reverses which item she bought multiples of. Study tip: In cost problems, always attach the variable to the item you're asked to find the quantity of. Read carefully to identify which quantities are fixed (bought once) versus variable (bought multiple times), then multiply unit prices by their respective quantities before adding all costs together.

Question 3

A taxi ride costs a flat fee of $3 plus $2 for every mile driven. Which expression gives the total cost, in dollars, for m miles?

  1. 2m+32m+3 (variable charge plus flat fee) (correct answer)
  2. 3m+23m+2 (flat fee treated as rate)
  3. 2(m+3)2(m+3) (distributes to $2m + $6)
  4. 3+2m3+\frac{2}{m} (rate divided by miles)
Explanation: When you encounter word problems about costs with both fixed and variable components, you need to identify what stays constant versus what changes with usage. Here, you have a flat fee (stays the same regardless of distance) and a per-mile charge (varies with distance). The total cost equals the fixed flat fee plus the variable charge based on miles driven. The flat fee is $3 - this doesn't change whether you go 1 mile or 10 miles. The variable charge is $2 per mile, so for m miles, you pay $2×m=2m2 \times m = 2m dollars.Addingthesecomponentsgivesyoudollars. Adding these components gives you 3+2m3 + 2m ,whichequals, which equals 2m+32m + 3 $. Choice A ( 2m + 3 ) correctly represents variable charge ( 2m ) plus flat fee ( 3 ). Choice B ( 3m + 2 ) incorrectly treats the $3 flat fee as if it's charged per mile, making the variable rate $3 per mile and the flat fee 2 - backwards from what the problem states. Choice C ($$2(m + 3) = 2m + 6$$) doubles the flat fee to $6 instead of keeping it at $3, as if the 3 is also charged per mile. Choice D ( 3 + \frac{2}{m} ) makes the per-mile charge decrease as miles increase, which defies logic - longer trips should cost more, not less per mile. For cost problems, always separate fixed costs (add once) from variable costs (multiply by the quantity). Write the expression as: fixed amount + (rate × quantity).

Question 4

A temperature F in degrees Fahrenheit is described as "nine less than twice the Celsius reading C." Which equation matches this description?

  1. F=2(C−9)F=2(C-9) (subtract first, then double)
  2. F=2C−9F=2C-9 (double then subtract nine) (correct answer)
  3. F=9−2CF=9-2C (nine decreased by twice C)
  4. F=C−92F=\dfrac{C-9}{2} (half the difference of C and nine)
Explanation: When you encounter word problems involving algebraic expressions, the key is translating English phrases into mathematical operations in the correct order. Let's break down "nine less than twice the Celsius reading C" step by step. The phrase "twice the Celsius reading" means 2C2C. Then "nine less than" this amount means we subtract 9 from 2C2C, giving us 2C−92C - 9. Therefore, F=2C−9F = 2C - 9, which matches choice B. Now let's see why the other options miss the mark. Choice A gives F=2(C−9)F = 2(C - 9), which represents "twice the quantity of C minus 9" or "double the difference." This would be the translation if the problem said "twice nine less than the Celsius reading." Choice C shows F=9−2CF = 9 - 2C, which means "nine minus twice C" or "nine decreased by twice C" — this reverses the subtraction order entirely. Choice D presents F=C−92F = \frac{C-9}{2}, which translates to "half the difference of C and nine," involving division rather than the multiplication described in the problem. The key strategy here is to identify the main operation first, then apply any modifying operations. "Twice the Celsius reading" establishes 2C2C as your foundation, and "nine less than" tells you to subtract 9 from that result. Watch for order-dependent phrases like "less than" — they often reverse the intuitive order of operations and are common sources of errors on standardized tests.

Question 5

"Five fewer than three times a number equals 16." Which equation correctly models this sentence for the number n?

  1. 3(n−5)=163(n-5)=16 (subtract before multiplying)
  2. 3n+5=163n+5=16 (adds instead of subtracts)
  3. 5−3n=165-3n=16 (order of subtraction reversed)
  4. 3n−5=163n-5=16 (times three, then subtract five) (correct answer)
Explanation: When translating word problems into algebraic equations, you need to carefully identify the operations and their order. This sentence contains two key mathematical operations: multiplication ("three times a number") and subtraction ("five fewer than"). Let's break down "Five fewer than three times a number equals 16" step by step. First, identify "three times a number" as 3n3n. Next, "five fewer than" means you subtract 5 from that result, giving you 3n−53n - 5. Finally, this expression "equals 16," so 3n−5=163n - 5 = 16. Choice A represents 3(n−5)=163(n-5) = 16, which translates to "three times the quantity of a number minus five." This changes the order of operations by subtracting 5 from n first, then multiplying by 3, rather than multiplying by 3 first. Choice B shows 3n+5=163n + 5 = 16, which means "five more than three times a number." This incorrectly adds 5 instead of subtracting it, representing the opposite of what the problem states. Choice C gives us 5−3n=165 - 3n = 16, which translates to "five minus three times a number." This reverses the order of the subtraction, making 5 the starting value rather than the amount being subtracted. Choice D correctly represents 3n−5=163n - 5 = 16: multiply the number by 3, then subtract 5. Remember this pattern: "fewer than" or "less than" means subtraction, and the number being reduced comes first in your expression. Always multiply or divide before adding or subtracting unless parentheses change the order.

Question 6

Maria's age is m years. "In 4 years, she will be three times as old as she was 2 years ago." Which equation models this situation?

  1. m−2=3m+4m-2=3m+4 (signs reversed on both sides)
  2. m−2=3(m+4)m-2=3(m+4) (past age equals triple future age)
  3. m+4=3m−2m+4=3m-2 (incorrect placement of 3 and 2)
  4. m+4=3(m−2)m+4=3(m-2) (future age equals triple past age) (correct answer)
Explanation: When you encounter age problems with algebraic expressions, the key is to carefully translate each time reference into mathematical terms. Maria's current age is m years, so you need to express her age at different times relative to this baseline. Let's break down the sentence: "In 4 years, she will be three times as old as she was 2 years ago." The future age (in 4 years) is m+4m + 4. The past age (2 years ago) is m−2m - 2. The relationship states that her future age equals three times her past age, giving us m+4=3(m−2)m + 4 = 3(m - 2). Choice A (m−2=3m+4m - 2 = 3m + 4) reverses the relationship entirely, setting her past age equal to something involving her future age, plus it incorrectly adds 4 to 3m3m instead of subtracting 6. Choice B (m−2=3(m+4)m - 2 = 3(m + 4)) also reverses the relationship by making her past age equal to three times her future age, which contradicts the given statement. Choice C (m+4=3m−2m + 4 = 3m - 2) has the correct left side but fails to properly group the past age calculation—it should be 3(m−2)=3m−63(m - 2) = 3m - 6, not 3m−23m - 2. Choice D correctly models the situation: her future age (m+4m + 4) equals three times her past age (3(m−2)3(m - 2)). For age problems, always identify the baseline age first, then carefully add or subtract years to express other time periods. Pay special attention to which age equals which—the order matters significantly in setting up your equation.

Question 7

The sum of two consecutive integers is 55. If n represents the smaller integer, which equation correctly models the situation?

  1. n+n+2=55n+n+2=55 (distance of two, not one)
  2. n+(n+1)=55n+(n+1)=55 (shows consecutive integers) (correct answer)
  3. 2n=552n=55 (omits the +1 term)
  4. n(n+1)=55n(n+1)=55 (product, not sum, of integers)
Explanation: When you encounter word problems about consecutive integers, you need to carefully translate the relationship into mathematical language. Consecutive integers are numbers that follow each other in order, like 3 and 4, or 27 and 28 – they differ by exactly 1. If nn represents the smaller integer, then the next consecutive integer must be n+1n + 1. Since we want their sum to equal 55, we write: n+(n+1)=55n + (n + 1) = 55. This equation correctly captures both integers and their addition, making choice B correct. Let's examine why the other options miss the mark. Choice A uses n+n+2=55n + n + 2 = 55, which represents integers that are 2 apart (like 5 and 7) – these would be consecutive odd or even integers, not simply consecutive integers. Choice C gives us 2n=552n = 55, which completely ignores the fact that consecutive integers differ by 1; this equation suggests both integers are the same value. Choice D shows n(n+1)=55n(n + 1) = 55, which represents the product (multiplication) of consecutive integers rather than their sum. You can verify choice B works by solving: n+(n+1)=55n + (n + 1) = 55 becomes 2n+1=552n + 1 = 55, so 2n=542n = 54 and n=27n = 27. The consecutive integers are 27 and 28, and indeed 27+28=5527 + 28 = 55. Remember: consecutive integers always differ by 1, so if the smaller is nn, the larger is n+1n + 1. Watch out for problems asking about consecutive even or odd integers – those differ by 2.

Question 8

The length of a garden is described as "four feet less than three times its width." If w is the width, which expression represents the length?

  1. 3(w−4)3(w-4) (subtract first, then triple)
  2. 3w−43w-4 (triple width, then subtract four) (correct answer)
  3. 4−3w4-3w (order of subtraction reversed)
  4. (w−4)/3(w-4)/3 (one-third after subtraction)
Explanation: When you encounter word problems involving algebraic expressions, the key is translating English phrases into mathematical operations in the correct order. Let's break down "four feet less than three times its width" step by step. The phrase "three times its width" translates to 3w3w. Then "four feet less than" that amount means we subtract 4 from 3w3w, giving us 3w−43w - 4. This matches choice B. Think of it like following a recipe: first you triple the width (3w3w), then you subtract four feet from that result. Now let's see why the other choices don't work. Choice A gives us 3(w−4)3(w-4), which represents "three times the quantity of width minus four." This reverses the order of operations—it subtracts first, then multiplies by 3. Choice C shows 4−3w4-3w, which would mean "four minus three times the width"—this completely flips the subtraction around. Choice D gives us (w−4)/3(w-4)/3, which represents "width minus four, divided by three," introducing division that wasn't mentioned in the problem. The trap here is that English doesn't always follow the same order as mathematical operations. "Four less than three times the width" means you calculate "three times the width" first, then subtract four from that result. Study tip: When translating "A less than B" phrases, remember that B comes first in your expression, then you subtract A. So "4 less than 3w3w" becomes "3w−43w - 4."

Question 9

Bryan has q quarters and d dimes in his pocket. Which expression represents the total value of his coins in cents?

  1. 10q+25d10q+25d (values of coins reversed)
  2. 25q+10d25q+10d (value per coin multiplied appropriately) (correct answer)
  3. 0.25q+0.10d0.25q+0.10d (values given in dollars, not cents)
  4. (25+10)(q+d)(25+10)(q+d) (adds values before multiplying)
Explanation: When you encounter coin value problems, remember that you need to multiply the number of each coin type by its individual value, then add those products together. Let's work through this systematically. Bryan has q quarters and d dimes. A quarter is worth 25 cents, and a dime is worth 10 cents. To find the total value in cents, you multiply the number of quarters by 25 and the number of dimes by 10, then add these amounts: 25q+10d25q + 10d. Choice A gives us 10q+25d10q + 25d, which reverses the coin values—it treats quarters as worth 10 cents and dimes as worth 25 cents. This is a common error when students mix up which coin has which value. Choice B correctly gives us 25q+10d25q + 10d, properly matching each coin with its value in cents. Choice C shows 0.25q+0.10d0.25q + 0.10d, which uses the dollar values of the coins (a quarter is $0.25, a dime is $0.10). While mathematically this would give the correct dollar amount, the question specifically asks for the value in cents, making this incorrect. Choice D presents $(25+10)(q+d)(25 + 10)(q + d) $, which incorrectly adds the coin values first, then multiplies by the total number of coins. This assumes all coins have the same value of 35 cents, which doesn't make sense. Study tip: In coin problems, always match each coin type with its correct value, then multiply before adding. Double-check whether the answer should be in cents or dollars—the TACHS often includes both as trap answers.

Question 10

An item with original price p dollars is put on sale at a 20% discount. Which algebraic expression gives the sale price?

  1. 0.2p0.2p (gives the amount of the discount, not price)
  2. p−0.02pp-0.02p (subtracts only 2% of p)
  3. p+0.2pp+0.2p (adds 20% instead of subtracting)
  4. 0.8p0.8p (keeps 80% of the price) (correct answer)
Explanation: When you encounter discount problems, focus on what percentage of the original price the customer actually pays. A 20% discount means the customer saves 20% and pays the remaining 80%. To find the sale price, you can think about it two ways. First, you could calculate the discount amount and subtract it from the original price: p−0.20p=0.80pp - 0.20p = 0.80p. Second, you could directly calculate what percentage remains after the discount: if 20% is removed, then 80% remains, so the sale price is 0.80p0.80p or 0.8p0.8p. Both methods give the same result, making choice D correct. Let's examine why the other options are wrong. Choice A gives 0.2p0.2p, which represents the dollar amount of the discount itself, not the sale price. If you paid this amount, you'd only be paying 20% of the original price—that's not a 20% discount, that's an 80% discount! Choice B subtracts 0.02p0.02p, which only removes 2% from the price instead of 20%—this confuses the decimal 0.02 with 0.20. Choice C adds 0.2p0.2p to the original price, which would increase the price by 20% rather than decrease it. Remember this pattern: for any discount percentage, subtract that percentage from 100% to find what the customer pays. A 20% discount means paying 80%, a 30% discount means paying 70%, and so on. This direct approach often saves time and reduces calculation errors.

Question 11

Lena has m marbles. Her sister has "five fewer than half as many marbles as Lena." Which expression gives the sister's number of marbles?

  1. m2+5\dfrac{m}{2}+5 (half, then add five)
  2. 12(m−5)\dfrac{1}{2}(m-5) (subtract first, then halve)
  3. m−55\dfrac{m-5}{5} (difference divided by five)
  4. 12m−5\dfrac{1}{2}m-5 (half of m, then subtract) (correct answer)
Explanation: When translating word problems into algebraic expressions, you need to carefully parse each phrase and convert it step-by-step into mathematical operations. Let's break down "five fewer than half as many marbles as Lena." Start with the core phrase: "half as many marbles as Lena." Since Lena has m marbles, half of that is 12m\frac{1}{2}m or m2\frac{m}{2}. Now apply "five fewer than" this amount. "Five fewer" means subtract 5, giving us 12m−5\frac{1}{2}m - 5. Choice A gives m2+5\frac{m}{2} + 5, which represents "five more than half," not "five fewer." This reverses the subtraction into addition—a common error when students rush through the translation. Choice B shows 12(m−5)\frac{1}{2}(m-5), which means "half of the quantity (m minus 5)." This incorrectly subtracts 5 from Lena's total first, then takes half. The order of operations doesn't match the phrase structure. Choice C presents m−55\frac{m-5}{5}, which divides by 5 instead of taking half (dividing by 2). This misinterprets "five fewer" as somehow involving division by 5, while also incorrectly placing the subtraction first. Choice D correctly gives 12m−5\frac{1}{2}m - 5, which takes half of Lena's marbles first, then subtracts 5. Strategy tip: When translating complex phrases, identify the main quantity first ("half as many"), then apply any modifiers ("five fewer"). Work from the inside out, and always double-check that your mathematical order matches the English structure.

Question 12

A rectangle has perimeter 50 cm. Its length is "three centimeters more than twice the width." Which equation expresses this relationship if w is the width (in cm)?

  1. 2w+3=502w+3=50 (treats 50 as the length, not perimeter)
  2. 2(2w−3)+2w=502(2w-3)+2w=50 (uses "less than" instead of "more than")
  3. (2w+3)+w=50(2w+3)+w=50 (missing factor of 2 for perimeter)
  4. 2(2w+3)+2w=502(2w+3)+2w=50 (perimeter using given length) (correct answer)
Explanation: When you encounter perimeter problems with algebraic relationships, you need to translate the word description into mathematical expressions, then use the perimeter formula. Start by defining your variable: let w = width. The length is described as "three centimeters more than twice the width," which translates to 2w+32w + 3. This means you take twice the width (2w2w) and add 3 to it. For any rectangle, the perimeter formula is P=2(length)+2(width)P = 2(\text{length}) + 2(\text{width}). Substituting our expressions: 50=2(2w+3)+2w50 = 2(2w + 3) + 2w. This is exactly what choice D shows: 2(2w+3)+2w=502(2w+3)+2w=50. Choice A treats the given measurement (50) as the length rather than the perimeter, completely ignoring the perimeter relationship. Choice B uses 2w−32w - 3 instead of 2w+32w + 3, which would mean the length is "three less than twice the width" rather than "three more than." Choice C has the right length expression (2w+32w + 3) but incorrectly writes the perimeter as (2w+3)+w=50(2w+3)+w=50, which only adds the length and width once instead of using the proper perimeter formula that accounts for all four sides. Study tip: Always write out the perimeter formula first: P=2l+2wP = 2l + 2w. Then carefully translate word relationships into algebraic expressions, paying close attention to "more than" versus "less than." Finally, substitute everything into your perimeter equation.

Question 13

Which algebraic expression represents "a number y decreased by the square of itself"?

  1. y2+yy^{2}+y (square added to number)
  2. y2−yy^{2}-y (square minus the number)
  3. (y−1)2(y-1)^{2} (square of the difference)
  4. y−y2y-y^{2} (number minus its square) (correct answer)
Explanation: When translating word problems into algebraic expressions, you need to carefully identify the operations and the order in which they're performed. The phrase "a number y decreased by the square of itself" tells you to start with the number y, then subtract (decrease by) the square of that number. Let's break this down step by step. "A number y" is simply yy. "The square of itself" means y2y^2. "Decreased by" means subtraction, where you subtract the second quantity from the first. So you have: the number y minus the square of y, which gives you y−y2y - y^2. This matches choice D. Now let's see why the other options are incorrect. Choice A gives y2+yy^2 + y, which would represent "the square of a number plus the number itself" – this uses addition instead of subtraction. Choice B shows y2−yy^2 - y, which represents "the square of a number decreased by the number itself" – this reverses the order of subtraction. Choice C presents (y−1)2(y-1)^2, which means "the square of a number decreased by 1" – this changes both the structure and introduces an unrelated constant. The key strategy here is to translate word problems piece by piece while paying close attention to the order of operations. "Decreased by" always means the first quantity minus the second quantity. Watch out for reversed order in the answer choices – this is a common trap on algebra problems involving subtraction.

Question 14

Kia bought several pens at $1.25 each and one notebook for $3.50. She spent a total of $15. Which equation could be used to find the number of pens p she purchased?

  1. 3.50p+1.25=153.50p+1.25=15 (notebook cost multiplied by p)
  2. 1.25+3.50p=151.25+3.50p=15 (prices attached to wrong items)
  3. 1.25p−3.50=151.25p-3.50=15 (subtracts notebook cost)
  4. 1.25p+3.50=151.25p+3.50=15 (sum of pen cost and notebook cost) (correct answer)
Explanation: When you see word problems involving costs and quantities, start by identifying what varies and what stays constant. Here, the number of pens varies (that's what we're solving for), while the price per pen and notebook cost are fixed. Let's build the equation step by step. Kia bought pp pens at $1.25 each, so the total cost of pens is $1.25p1.25p .Shealsoboughtonenotebookfor$3.50.Hertotalspendingwas$15,soweneed:(costofpens)+(costofnotebook)=(totalspent),whichgivesus$. She also bought one notebook for $3.50. Her total spending was $15, so we need: (cost of pens) + (cost of notebook) = (total spent), which gives us $1.25p + 3.50 = 15$$. This matches choice D. Now let's see why the other options are wrong. Choice A writes 3.50p+1.25=153.50p + 1.25 = 15, which incorrectly suggests she bought pp notebooks at $3.50 each, plus something costing $1.25. Choice B gives us $1.25 + 3.50p = 15$$, implying she bought one pen for $1.25 and $p notebooks at $3.50 each—the opposite of what actually happened. Choice C shows 1.25p−3.50=151.25p - 3.50 = 15, which subtracts the notebook cost instead of adding it, suggesting the notebook somehow reduced her total spending. Remember this pattern: in cost problems, multiply the variable quantity by its unit price, then add any fixed costs. The variable (pp) always goes with the item she bought multiple units of, not the single-purchase item.

Question 15

A taxi ride costs a flat fee of $3 plus $2 for every mile driven. Which expression gives the total cost, in dollars, for m miles?

  1. 2(m+3)2(m+3) (distributes to $2m + $6)
  2. 3m+23m+2 (flat fee treated as rate)
  3. 2m+32m+3 (variable charge plus flat fee) (correct answer)
  4. 3+2m3+\frac{2}{m} (rate divided by miles)
Explanation: When you encounter word problems involving costs with both fixed and variable components, you need to identify what stays constant versus what changes based on usage or quantity. In this taxi problem, there are two cost components: a flat fee of $3 that you pay regardless of distance, and a variable charge of $2 per mile that depends on how far you travel. For m miles, the variable cost is $2 × m = $2m. The total cost is the sum of these parts: fixed cost + variable cost = $3 + $2m = $2m + 3. Choice C (2m+32m + 3) correctly represents this relationship, with $2m as the variable charge based on miles driven and $3 as the flat fee. Choice A (2(m+3)2(m + 3)) makes the error of grouping miles and the flat fee together, then multiplying by the per-mile rate. This gives $2m + 6,incorrectlydoublingtheflatfee.ChoiceB(6, incorrectly doubling the flat fee. Choice B (3m + 2)reversestheroles,treatingthe$3flatfeeasifit′smultipliedbymilesandthe) reverses the roles, treating the $3 flat fee as if it's multiplied by miles and the 2 per-mile rate as a flat fee. Choice D (3+2m3 + \frac{2}{m}) creates an inverse relationship where the per-mile cost decreases as you travel more miles, which contradicts the problem setup. Strategy tip: In cost problems, always identify fixed costs (constants) and variable costs (multiplied by the changing quantity) separately, then add them together. Watch for answer choices that incorrectly combine or reverse these components.