Translate Word Problems - TEAS Mathematics

1. A lab technician mixes $x$ milliliters (mL) of a 10% acid solution with $y$ mL of a 25% acid solution. To this new mixture, the technician adds $z$ mL of a 0% acid solution (pure water). Which of the following expressions represents the final acid concentration of the mixture as a percentage?
   Answer: $\frac{10x + 25y}{x+y+z}$ The final concentration is the total amount of acid divided by the total volume of the solution, multiplied by 100 to express it as a percentage. The amount of acid from the first solution is $0.10x$ mL. The amount of acid from the second solution is $0.25y$ mL. The pure water adds 0 mL of acid. The total amount of acid is $0.10x + 0.25y$ mL. The total volume of the mixture is the sum of the volumes of all three parts: $x + y + z$ mL. The concentration as a decimal is $\frac{0.10x + 0.25y}{x+y+z}$. To express this as a percentage, multiply the expression by 100: $100 \times \frac{0.10x + 0.25y}{x+y+z} = \frac{10x + 25y}{x+y+z}$.
2. A worker is paid $18 per hour for the first 40 hours and $27 for each overtime hour. If $h$ is the number of overtime hours, which expression represents the worker's weekly pay?
   Answer: $720+27h$ This problem tests your ability to build algebraic expressions from word problems involving piecewise pay structures. When you see "first X hours at one rate, then overtime at another rate," you need to separate the fixed portion from the variable portion. Let's break down the worker's pay: For the first 40 hours, they earn $\$18 \times 40 = \$720$ regardless of overtime. This is the fixed base pay. For overtime hours $h$, they earn an additional $27 per hour, which adds $27h$ to their weekly pay. The total weekly pay is therefore $720 + 27h$. Looking at the wrong answers: Choice B ($18h + 27 \times 40$) incorrectly applies the regular hourly rate to overtime hours and the overtime rate to regular hours - this completely reverses the pay structure. Choice C ($27h - 720$) suggests the worker loses their base pay, which makes no sense since they should earn at least $720 for their first 40 hours. Choice D ($$18(40 + h)$$) pays the regular rate for all hours worked, including overtime, completely ignoring the higher overtime rate of $27. Choice A correctly captures both components: the fixed base pay of $720 for the first 40 hours plus the variable overtime pay of $27 per overtime hour. Study tip: For piecewise pay problems, always identify the fixed portion first (regular hours × regular rate), then add the variable portion (overtime hours × overtime rate). Write them as separate terms, then combine.
3. A rental car company charges a flat fee of $55 plus $0.35 for every mile driven. If $m$ represents the miles driven, which expression models the total charge?
   Answer: $55+0.35m$ When you encounter word problems involving costs with both fixed and variable components, you need to identify which part stays constant and which part changes based on usage. This rental car problem has two cost components: a flat fee of $55 that you pay regardless of miles driven, and a variable cost of $0.35 that gets multiplied by each mile you drive. To build the expression, you add the fixed cost to the variable cost: $55 + 0.35m$. The flat fee remains $55, while $0.35 gets multiplied by $m$ (the number of miles) to give you the mileage charges. Choice A, $55 + 0.35m$, correctly represents this structure. Choice B, $0.35 + 55m$, incorrectly makes the small amount ($0.35) the flat fee and multiplies the large amount ($55) by miles driven—this would mean paying $55 per mile, which is unrealistic. Choice C, $55m + 0.35$, makes the same error as B by charging $55 per mile with only a $0.35 flat fee. Choice D, $0.35(55-m)$, creates a bizarre scenario where your total cost decreases as you drive more miles, and you'd actually get money back if you drove more than 55 miles. For TEAS cost problems, always identify the fixed amount first (it stands alone), then find what gets multiplied by the variable. The structure is typically: fixed cost + (rate × variable). Watch for answer choices that flip these components—they're common trap answers.
4. A store mixes $x$ pounds of peanuts costing $6 per pound with $y$ pounds of almonds costing $9 per pound. Which expression represents the total cost of the mixture?
   Answer: $6x+9y$ When you encounter mixture problems involving costs, you're dealing with the fundamental principle that total cost equals the sum of individual costs for each component. To find the total cost of this mixture, you need to calculate the cost of each ingredient separately, then add them together. For the peanuts: $x$ pounds at $6 per pound gives you $6x$ dollars. For the almonds: $y$ pounds at $9 per pound gives you $9y$ dollars. The total cost is simply $6x + 9y$. Choice A ($6x + 9y$) is correct because it properly multiplies each quantity by its respective price per pound and adds the results. Choice B ($(6+9)(x+y)$) incorrectly adds the prices first, then multiplies by the total weight. This would mean both ingredients cost the same blended price of $15 per pound, which isn't how pricing works in reality. Choice C ($9x + 6y$) switches the coefficients, incorrectly pricing peanuts at $9 per pound and almonds at $6 per pound. Choice D ($6xy + 9$) makes two errors: it multiplies the quantities together ($xy$) rather than treating them separately, and it adds a flat $9 regardless of the actual amount of almonds. For mixture problems on the TEAS, always set up your cost calculation as: (price₁ × quantity₁) + (price₂ × quantity₂). Keep each component separate and match each price with its correct ingredient.
5. The formula to convert a temperature of $c$ degrees Celsius to degrees Fahrenheit is represented by which expression?
   Answer: $\dfrac{9}{5}c+32$ Temperature conversion formulas are fundamental relationships you need to memorize for the TEAS. The Celsius-to-Fahrenheit conversion involves two key adjustments: scaling and shifting the temperature scale. The correct formula is $\frac{9}{5}c + 32$, which is choice A. Here's why this works: Celsius and Fahrenheit scales have different-sized degrees and different starting points. The fraction $\frac{9}{5}$ (which equals 1.8) accounts for the fact that Fahrenheit degrees are smaller than Celsius degrees—you need 1.8 Fahrenheit degrees for every 1 Celsius degree. The "+32" shifts the scale because 0°C equals 32°F, not 0°F. Choice B ($\frac{5}{9}c - 32$) uses the wrong fraction and operation. The $\frac{5}{9}$ fraction is used when converting Fahrenheit to Celsius, not the reverse direction shown here. Choice C ($c + 32$) only accounts for the shift between scales but ignores that the degree sizes are different. This would incorrectly suggest that 10°C equals 42°F instead of the actual 50°F. Choice D ($\frac{9c}{5-32}$) incorrectly places the 32 in the denominator, creating a meaningless mathematical expression that doesn't represent any valid temperature relationship. For the TEAS, memorize both conversion formulas: C to F uses $\frac{9}{5}c + 32$, while F to C uses $\frac{5}{9}(f - 32)$. Remember that 9/5 goes with "to Fahrenheit" and 5/9 goes with "to Celsius."
6. Concert tickets cost $12 for adults and $7 for children. If $a$ adults and $c$ children attend, which expression represents the total revenue from ticket sales?
   Answer: $12a+7c$ When you encounter word problems involving multiple categories with different costs or rates, you need to set up expressions that multiply each rate by its corresponding quantity, then add the results together. Here, adults pay $12 each and children pay $7 each. If $a$ adults attend, the revenue from adults is $12a$ (price per adult × number of adults). If $c$ children attend, the revenue from children is $7c$ (price per child × number of children). Total revenue is the sum: $12a + 7c$, making choice A correct. Choice B ($12c + 7a$) switches the coefficients incorrectly—it charges children the adult price ($12) and adults the child price ($7). This is a common trap that tests whether you're carefully matching rates to the right variables. Choice C ($(12+7)(a+c)$) adds the prices together first, then multiplies by the total attendance. This would mean everyone pays $19, which doesn't reflect the different pricing structure. This approach incorrectly assumes a single ticket price. Choice D ($19ac$) multiplies the total price by both variables together, creating a nonsensical expression. The product $ac$ doesn't represent any meaningful quantity in this context—you can't multiply the number of adults by the number of children to get total revenue. Study tip: In multi-category pricing problems, always match each price with its corresponding quantity using multiplication, then add the separate revenue streams. Watch for answer choices that mix up which price goes with which group—this is a frequent TEAS trap.
7. Which algebraic expression stands for half of the sum of a number $n$ and eight?
   Answer: $\dfrac{n+8}{2}$ When translating word problems into algebraic expressions, you need to carefully identify the order of operations described in the phrase. The key phrase here is "half of the sum of a number n and eight." Let's break this down step by step. First, you need to find "the sum of a number n and eight," which translates to $n + 8$. Then, you need to take "half of" that entire sum. Taking half of something means dividing it by 2, so you divide the entire expression $(n + 8)$ by 2, giving you $\dfrac{n+8}{2}$. This matches choice A. Now let's see why the other options are incorrect. Choice B, $\dfrac{n}{2}+8$, represents "half of n, plus eight" rather than "half of the sum." This expression takes half of n first, then adds 8, which changes the entire meaning. Choice C, $\dfrac{8-n}{2}$, represents "half of the difference of eight and n," which involves subtraction rather than addition and reverses the order. Choice D, $2(n+8)$, represents "twice the sum of n and eight," which is the opposite of taking half. The critical strategy here is to work from the inside out when translating nested operations. Always identify what's inside parentheses first (the sum), then apply the outer operation (taking half). Watch for phrases like "half of," "twice," or "double" that apply to entire expressions, not just single terms.
8. A certain antibiotic is prescribed at a dosage of 20 milligrams (mg) per kilogram (kg) of body weight for the first 40 kg. For each kilogram of body weight over 40 kg, the dosage is reduced to 12 mg. Which expression represents the total dosage in mg for a patient weighing $w$ kg, where $w > 40$?
   Answer: $12w + 320$ The dosage calculation is split into two parts. The first part is for the first 40 kg of weight, which is a fixed amount: $20 \text{ mg/kg} \times 40 \text{ kg} = 800 \text{ mg}$. The second part is for the weight over 40 kg. This amount of weight is represented by $(w - 40)$ kg. The dosage for this portion is $12 \text{ mg/kg}$. So, the dosage for the additional weight is $12(w - 40)$ mg. The total dosage is the sum of these two parts: $800 + 12(w - 40)$. Distributing the 12 gives $800 + 12w - 480$. Combining the constant terms gives the final simplified expression: $12w + 320$.
9. A road trip is 220 miles long. After driving $d$ miles, which expression represents the distance (in miles) still left to travel?
   Answer: $220-d$ When you encounter word problems involving remaining quantities, think about the relationship between what you started with, what you've used, and what's left over. This is a fundamental subtraction scenario. You start with a total trip of 220 miles. After driving $d$ miles, you need to find how much distance remains. Since you've completed $d$ miles of the 220-mile journey, the remaining distance equals the total minus what you've already traveled: $220 - d$. This gives you answer choice A. Let's examine why the other options don't work. Choice B gives $d - 220$, which subtracts the total trip from the distance traveled. This would only make sense if you had driven more miles than the trip length, resulting in a negative number for most realistic scenarios. Choice C shows $220 + d$, which adds your traveled distance to the total trip length. This would represent the total of your trip plus extra driving, not the remaining distance. Choice D presents $\dfrac{220}{d}$, which divides the total trip by miles driven. This gives you a ratio or rate, not a distance measurement. Remember this pattern for TEAS math questions: when finding "remaining" or "left over" quantities, you typically subtract what's been used from what you started with. The structure is always "original amount minus amount used equals amount remaining." Watch for this setup in problems involving time remaining, money left to spend, or any scenario where you're tracking what's left of an original quantity.
10. A hospital purchases $n$ cases of sterile gloves at a cost of $c$ dollars per case, where $n > 100$. The supplier offers a $d\%$ discount on the total pre-tax cost for orders over 100 cases. After the discount is applied, a sales tax of $t\%$ is added to the discounted price. Which expression represents the final cost of the purchase?
    Answer: $nc(1 - \frac{d}{100})(1 + \frac{t}{100})$ First, calculate the initial total cost, which is the number of cases $n$ times the cost per case $c$, or $nc$. Next, apply the $d\%$ discount. A discount of $d\%$ means the remaining cost is $(100-d)\%$, or $(1 - \frac{d}{100})$ of the original price. The discounted price is $nc(1 - \frac{d}{100})$. Finally, apply the $t\%$ sales tax to this discounted price. Adding a $t\%$ tax is equivalent to multiplying by $(1 + \frac{t}{100})$. The final cost is $nc(1 - \frac{d}{100})(1 + \frac{t}{100})$.
11. The length of a rectangle is 3 cm more than its width $w$. Which expression gives the area of the rectangle in square centimeters?
    Answer: $w(w+3)$ When you encounter a geometry problem asking for area, always start by identifying what measurements you have and what formula applies. For rectangles, the area formula is length × width. Here, you're told the width is $w$ and the length is "3 cm more than its width." This means the length equals $w + 3$. To find the area, multiply these dimensions: $w \times (w + 3) = w(w + 3)$. Choice A, $w(w + 3)$, correctly represents this calculation and gives you the area in square centimeters. Choice B, $w + 3$, only gives you the length of the rectangle, not the area. This is a common trap for students who identify one dimension but forget to complete the area calculation. Choice C, $2w + 3$, looks like someone tried to add the width and length together. This might come from confusing area with perimeter concepts, but even then, the perimeter would be $2w + 2(w + 3) = 4w + 6$. Choice D, $(w + 3)^2$, represents the area of a square with side length $(w + 3)$. This error occurs when students assume the shape is a square rather than carefully reading that the length and width are different. Remember this pattern: when a problem describes one dimension in terms of another, write out both dimensions clearly, then apply the appropriate formula. For rectangles, always multiply length times width—never add them or square just one dimension.
12. A phone plan includes 5 GB of data for $50 and charges $10 for each additional gigabyte used. If $g$ represents gigabytes beyond the included 5 GB, which expression models the monthly bill?
    Answer: $50+10g$ When you encounter word problems involving costs with both fixed and variable components, focus on identifying the base cost and what changes based on usage. This phone plan has a fixed base cost of $50 that covers the first 5 GB of data. Since $g$ represents gigabytes beyond the included 5 GB, you only pay the additional $10 per gigabyte for those extra gigabytes. The total monthly bill equals the base cost plus the variable cost: $50 + 10g$. This matches choice A. Let's examine why the other options are incorrect. Choice B ($10 + 50g$) reverses the fixed and variable costs, making the base cost only $10 and charging $50 per additional gigabyte—an unrealistic phone plan structure. Choice C ($$50g + 10$$) treats the base 5 GB as if it costs $50 per gigabyte, which contradicts the problem since those 5 GB are included in the flat $50 fee. Choice D ($$50 - 10g$$) suggests the bill decreases as you use more data, which makes no business sense and would result in negative costs for heavy users. Remember that in cost problems, the variable ($g$) should be multiplied by the per-unit rate, while fixed costs stand alone. Always check that your expression makes logical sense—costs should increase with usage, not decrease, and the base cost should reflect what you pay for zero additional usage.
13. Which expression gives the average of three numbers $a$, $b$, and $c$?
    Answer: $\dfrac{a+b+c}{3}$ When you encounter a question about finding the average (also called the arithmetic mean) of numbers, remember that an average represents the equal distribution of a total among all items. To find an average, you always divide the sum of all values by the count of values. For three numbers $a$, $b$, and $c$, you need to add them together and divide by 3 (since there are three numbers). This gives you $\dfrac{a+b+c}{3}$, which is choice A. This formula ensures that if you multiplied the average by 3, you'd get back the original sum $a+b+c$. Let's examine why the other choices are incorrect. Choice B, $\dfrac{abc}{3}$, divides the product of the three numbers by 3, not their sum. This would give you one-third of what you get when you multiply the numbers together, which has no relationship to their average. Choice C, $a+b+c$, is simply the sum of the three numbers without dividing by the count—this gives you the total, not the average. Choice D, $\dfrac{a+b}{c}$, only uses two of the three numbers in the numerator and incorrectly uses the third number as a divisor rather than including it in the sum. Remember this pattern: average equals sum divided by count. Whether you're finding the average of test scores, measurements, or algebraic expressions, this fundamental relationship never changes. On the TEAS, average problems often appear in word problem format, so practice identifying when a question is asking for a mean.
14. Nurse A can prepare $v$ vials of medication in $h$ hours. Nurse B takes 50% longer than Nurse A to prepare the same number of vials. Which of the following expressions represents the total number of vials the two nurses can prepare if they work together for one hour?
    Answer: $\frac{5v}{3h}$ First, find the rate of work for each nurse in vials per hour. Nurse A's rate is $R_A = \frac{v}{h}$. Nurse B takes 50% longer, which is $1.5$ times the hours, so Nurse B's time is $1.5h$. Nurse B's rate is $R_B = \frac{v}{1.5h} = \frac{v}{\frac{3}{2}h} = \frac{2v}{3h}$. To find their combined rate, add their individual rates: $R_{total} = R_A + R_B = \frac{v}{h} + \frac{2v}{3h}$. To add these fractions, find a common denominator, which is $3h$. $R_{total} = \frac{3v}{3h} + \frac{2v}{3h} = \frac{5v}{3h}$. This expression represents the number of vials they prepare together in one hour.
15. The length of a rectangular examination room is 5 feet less than three times its width, $w$. The room is renovated, and its new length is 8 feet shorter than the original length, while its new width is 2 feet greater than the original width. Which expression represents the perimeter of the renovated room?
    Answer: $8w-22$ First, define the original and new dimensions in terms of $w$. Original width = $w$. Original length = $3w - 5$. The new width is 2 feet greater than the original: $w_{new} = w + 2$. The new length is 8 feet shorter than the original: $L_{new} = (3w - 5) - 8 = 3w - 13$. The perimeter of a rectangle is given by the formula $P = 2(L+w)$. Using the new dimensions, the perimeter of the renovated room is $P_{new} = 2(L_{new} + w_{new}) = 2((3w - 13) + (w + 2))$. Simplify the expression inside the parentheses: $P_{new} = 2(4w - 11)$. Distribute the 2 to get the final expression: $P_{new} = 8w - 22$.
16. A plant is 15 cm tall and grows 2.5 cm each day. After $d$ days, which expression represents its height?
    Answer: $15+2.5d$ This is a linear growth problem where you need to build an equation that models how something changes over time. When you see questions about consistent growth or change, think about identifying the starting value and the rate of change. The plant starts at 15 cm tall - this is your initial value. Then it grows at a constant rate of 2.5 cm per day. After $d$ days, the total growth will be $2.5d$ cm (the daily rate times the number of days). To find the plant's total height, you add this growth to the starting height: $15 + 2.5d$. Choice A ($15 + 2.5d$) correctly represents this relationship with the initial height plus the accumulated growth over $d$ days. Choice B ($15d + 2.5$) incorrectly multiplies the initial height by the number of days and adds the daily growth rate just once. This suggests the plant starts at different heights each day, which doesn't make sense. Choice C ($15(2.5d)$) multiplies the initial height by the total growth, creating an unrealistic exponential-type growth pattern rather than the steady linear increase described. Choice D ($2.5(15 - d)$) suggests the plant gets shorter over time since you're subtracting days from the initial height, which contradicts the growth described in the problem. Remember the linear growth formula: final amount = initial amount + (rate × time). This pattern appears frequently on the TEAS, so practice identifying the starting value versus the rate of change in word problems.
17. John's age is six years more than one-third of his sister's age. If his sister is $a$ years old, which expression represents John's age?
    Answer: $\dfrac{a}{3}+6$ When you encounter word problems involving algebraic expressions, the key is to translate each phrase methodically into mathematical symbols, working from the inside out. Let's break down "John's age is six years more than one-third of his sister's age" step by step. Start with the sister's age: $a$ years old. Next, find "one-third of his sister's age," which translates to $\dfrac{a}{3}$. Finally, "six years more than" that amount means you add 6 to get $\dfrac{a}{3} + 6$. This confirms that choice A is correct: $\dfrac{a}{3}+6$ properly represents John's age. Now let's see why the other options are wrong. Choice B, $\dfrac{a+6}{3}$, incorrectly adds 6 to the sister's age first, then takes one-third of that sum. This would mean "one-third of six years more than his sister's age" — the opposite order of operations. Choice C, $3(a+6)$, multiplies by 3 instead of dividing, and again adds 6 to the wrong quantity. This expression would make John much older than his sister, which contradicts the problem setup. Choice D, $\dfrac{a-6}{3}$, subtracts 6 instead of adding it, giving "one-third of six years less than his sister's age." For TEAS word problems, always translate phrases in the exact order given, and watch out for the common trap of performing operations in the wrong sequence. Practice identifying key phrases like "more than," "less than," and fractional expressions to build your translation skills.
18. Including an 8% sales tax, the total price of an item with pre-tax price $p$ dollars can be written as which expression?
    Answer: $1.08p$ When you encounter sales tax problems, you're working with percent increases. The key insight is that sales tax gets added to the original price, so you end up paying more than the pre-tax amount. To find the total price including an 8% sales tax, you need to calculate the original price plus the tax amount. The tax equals 8% of the pre-tax price, which is $0.08p$. So the total becomes $p + 0.08p$. You can factor out $p$ to get $p(1 + 0.08) = 1.08p$. This means you're paying 108% of the original price, which makes sense since you're paying the full original price (100%) plus an additional 8%. Choice A ($1.08p$) correctly represents this relationship. Choice B ($0.92p$) represents a 8% discount, not a tax increase - this would be what you'd pay if the item were marked down by 8%. Choice C ($p - 0.08p$) also represents a decrease in price, giving you only 92% of the original price. Choice D ($p + 8$) incorrectly adds 8 dollars rather than 8 percent, which would only work if the sales tax were a flat fee instead of a percentage. For TEAS math problems involving percentages, always ask yourself whether you're dealing with an increase or decrease. Sales tax, tips, and markups increase the original amount, so your multiplier should be greater than 1. Discounts and markdowns decrease it, giving multipliers less than 1.
19. What expression represents the product of 4 and the difference between a number $x$ and 9?
    Answer: $4(x-9)$ When you encounter word problems asking you to translate English phrases into mathematical expressions, you need to identify the key operations and their order carefully. This question tests your ability to recognize how grouping symbols reflect the structure of written language. The phrase "the product of 4 and the difference between a number x and 9" contains two main operations. First, you need "the difference between a number x and 9," which translates to $x - 9$. Second, you need "the product of 4 and" that difference, meaning you multiply 4 by the entire difference. Since the difference must be calculated first, it needs parentheses: $4(x-9)$. This makes choice A correct. Choice B, $4x - 9$, represents "the product of 4 and x, minus 9" - this misses the grouping entirely. The phrase doesn't ask for 4 times x minus 9; it asks for 4 times the whole difference. Choice C, $4(x+9)$, would represent "the product of 4 and the sum of x and 9," but the problem specifically mentions a difference, not a sum. Choice D, $(x-9)/4$, represents "the difference between x and 9, divided by 4," which describes division rather than multiplication. On TEAS math questions involving word-to-expression translation, always identify the main operation last and work backward through the phrase. Grouping words like "the difference" or "the sum" signal that you'll need parentheses to preserve the order of operations.
20. Each batch of cookies requires $\tfrac34$ cup of sugar. If $b$ batches are made, which expression shows the total cups of sugar needed?
    Answer: $\tfrac34b$ This question tests your ability to translate word problems into algebraic expressions, specifically involving multiplication with fractions. When you need to find a total amount that depends on repeated use of the same quantity, you're looking at a multiplication situation. Here, each batch requires $\frac{3}{4}$ cup of sugar, and you're making $b$ batches. To find the total sugar needed, you multiply the amount per batch by the number of batches: $\frac{3}{4} \times b = \frac{3}{4}b$. Choice A ($\frac{3}{4}b$) correctly represents this multiplication. This expression means "three-fourths times $b$," which gives you the total cups of sugar for $b$ batches. Choice B ($\frac{3}{4}+b$) represents addition instead of multiplication. This would mean you're adding the sugar per batch to the number of batches, which doesn't make sense since you're adding cups to a count of batches. Choice C ($\frac{b}{\frac{3}{4}}$) represents division, which would tell you how many batches you could make with $b$ cups of sugar - the opposite of what you want to find. Choice D ($\frac{4}{3}b$) has the fraction flipped. While it correctly shows multiplication, $\frac{4}{3}$ cups per batch is not the amount given in the problem. Remember: when you see "each" or "per" followed by a quantity, and you need to find a total for multiple items, multiply the unit amount by the number of units. Watch for the correct fraction - don't flip numerators and denominators.

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