Solve simple algebraic equations for one variable - Health Education Systems Admission Assessment HESI

1. A pharmacy's inventory equation is $3(V - 12) + 2V = 4V + 6$, where V represents the number of vials ordered last month. How many vials were ordered?
   Answer: 42 vials This question tests your ability to solve linear equations, a fundamental algebra skill that appears regularly on the HESI exam. When you encounter an equation with variables, your goal is to isolate the variable by using inverse operations systematically. Starting with $3(V - 12) + 2V = 4V + 6$, first distribute the 3: $3V - 36 + 2V = 4V + 6$. Combine like terms on the left side: $5V - 36 = 4V + 6$. Now isolate the variable by subtracting $4V$ from both sides: $V - 36 = 6$. Finally, add 36 to both sides: $V = 42$. Let's verify: $3(42 - 12) + 2(42) = 3(30) + 84 = 90 + 84 = 174$, and $4(42) + 6 = 168 + 6 = 174$. Both sides equal 174, confirming our answer. Choice (A) 40 vials results from incorrectly combining terms or making arithmetic errors during distribution. Choice (B) 48 vials likely comes from sign errors when moving terms across the equal sign—a common mistake when students forget to change signs. Choice (C) 45 vials represents errors in the final addition step, possibly adding incorrectly when isolating the variable. When solving linear equations on the HESI, always work systematically: distribute first, combine like terms, then isolate the variable. Most importantly, substitute your answer back into the original equation to verify—this catches calculation errors and builds confidence in your solution.
2. A solution is being heated in a lab. Its initial temperature is 12°C. The temperature increases by 2.5°C per minute. Another solution starts at 30°C and cools at a rate of 1.5°C per minute. After how many minutes, m, will the two solutions have the same temperature?
   Answer: 4.5 When you encounter problems involving two quantities changing at different rates over time, you're working with linear equations. The key is setting up expressions for each quantity's value at time m, then finding when they're equal. For the heating solution: it starts at 12°C and increases 2.5°C per minute, so its temperature after m minutes is $12 + 2.5m$. For the cooling solution: it starts at 30°C and decreases 1.5°C per minute, so its temperature is $30 - 1.5m$. To find when temperatures are equal, set the expressions equal: $12 + 2.5m = 30 - 1.5m$. Adding 1.5m to both sides: $12 + 4m = 30$. Subtracting 12 from both sides: $4m = 18$. Therefore: $m = 4.5$ minutes. Let's check why the other answers are wrong. Choice A (3.0) gives temperatures of 19.5°C and 25.5°C respectively—not equal. Choice C (7.2) results in 30°C and 19.2°C—the heating solution would actually be hotter than the cooling one. Choice D (10.5) gives 38.25°C and 14.25°C—an even larger temperature difference. For rate problems like this, always write an expression for each changing quantity, set them equal at the target condition, and solve algebraically. Double-check your answer by substituting back into both original expressions—they should give the same result. This type of problem appears frequently on standardized tests, so mastering the setup is crucial for success.
3. Three-fifths of the patients in a clinic have an appointment with a doctor, and the remaining 16 patients have an appointment with a nurse practitioner. What is the total number of patients, P, in the clinic?
   Answer: 40 When you encounter word problems involving fractions and remainders, you need to set up an equation that accounts for all parts of the whole. Here, you know that three-fifths of patients see doctors, and the remaining patients (16) see nurse practitioners. Let P represent the total number of patients. If three-fifths see doctors, then the remaining fraction is $1 - \frac{3}{5} = \frac{2}{5}$. This means $\frac{2}{5}$ of all patients see nurse practitioners. Since 16 patients see nurse practitioners, you can write: $\frac{2}{5} \times P = 16$ Solving for P: $P = 16 \times \frac{5}{2} = 16 \times 2.5 = 40$ You can verify: If there are 40 total patients, then $\frac{3}{5} \times 40 = 24$ see doctors, and $40 - 24 = 16$ see nurse practitioners. ✓ Choice A (24) represents only the number of patients seeing doctors, not the total. This is a common trap where students solve for part of the problem rather than what's actually being asked. Choice B (32) might result from incorrectly assuming that 16 patients represent $\frac{1}{2}$ of the total instead of $\frac{2}{5}$. Choice D (48) could come from misreading the fraction as $\frac{2}{3}$ instead of $\frac{3}{5}$, leading to $16 \times 3 = 48$. Always double-check fraction problems by verifying that your answer makes both parts of the problem work. Set up your equation carefully and make sure you're solving for what the question asks—the total, not just one component.
4. A patient's temperature in Celsius follows the equation $F = \frac{9C}{5} + 32$ when converted to Fahrenheit. If the patient has a fever of 102.2°F, what is their temperature in Celsius?
   Answer: 39.0°C Substitute F = 102.2: 102.2 = 9C/5 + 32. Subtract 32: 70.2 = 9C/5. Multiply by 5: 351 = 9C. Divide by 9: C = 39.0°C. Choice A results from using the wrong conversion factor. Choice B comes from arithmetic errors in the division. Choice D results from forgetting to subtract 32 before solving.
5. If $4(y - 3) - 2y = 3(y + 1)$, what is the value of y?
   Answer: -15 This question tests your ability to solve linear equations with variables on both sides. When you encounter an equation like this, your goal is to isolate the variable by systematically eliminating terms through algebraic operations. Starting with $4(y - 3) - 2y = 3(y + 1)$, first distribute on both sides: $4y - 12 - 2y = 3y + 3$ Combine like terms on the left side: $2y - 12 = 3y + 3$ Now collect all terms with $y$ on one side and constants on the other. Subtract $2y$ from both sides: $-12 = y + 3$ Finally, subtract 3 from both sides: $y = -15$ Let's examine why the other answers are incorrect. Answer B (-6) might result from incorrectly distributing the 4 or making sign errors when moving terms. Answer C (-3) could come from forgetting to subtract the final 3, stopping at $y + 3 = 0$. Answer D (3) likely results from multiple sign errors, particularly when handling the negative terms during distribution or when moving variables between sides. The correct answer is A (-15). For HESI success, always work systematically through multi-step equations: distribute first, combine like terms, then isolate the variable. Double-check your work by substituting your answer back into the original equation. Also, pay careful attention to negative signs—they're the most common source of errors in algebraic manipulation.
6. The amount of medication remaining in a patient's system follows $4(M + 2) - 3M = 18$, where M represents the initial dose in mg. What was the initial dose?
   Answer: 10 mg Expand: 4M + 8 - 3M = 18. Combine like terms: M + 8 = 18. Subtract 8: M = 10 mg. Choice A results from distribution errors. Choice C comes from adding 8 instead of subtracting. Choice D results from incorrectly combining the M terms.
7. The relationship between a patient's heart rate before and after exercise is given by $\frac{H + 20}{4} = \frac{H - 10}{2}$, where H is the resting heart rate in beats per minute. What is the resting heart rate?
   Answer: 100 bpm Cross multiply: 2(H + 20) = 4(H - 10). Expand: 2H + 40 = 4H - 40. Subtract 2H: 40 = 2H - 40. Add 40: 80 = 2H. Divide by 2: H = 100 bpm. Choice A results from sign errors. Choice B comes from arithmetic errors in cross multiplication. Choice C results from incorrectly moving terms between sides.
8. A total of 180 milliliters (mL) of a saline solution needs to be infused over 3 hours. For the first hour, the solution is infused at a rate of 50 mL/hr. To complete the infusion on time, what must be the infusion rate, r, in mL/hr for the remaining 2 hours?
   Answer: 65 When you encounter infusion rate problems, you're working with the fundamental relationship between volume, rate, and time. These calculations are essential for safe medication administration in clinical practice. Let's break down what happens during this 3-hour infusion. In the first hour, 50 mL is infused at 50 mL/hr. This leaves 180 - 50 = 130 mL remaining to be infused over the next 2 hours. To find the required rate for the remaining time, divide the remaining volume by the remaining time: $r = \frac{130 \text{ mL}}{2 \text{ hours}} = 65 \text{ mL/hr}$ Looking at the wrong answers: Choice A (60 mL/hr) would only deliver 120 mL over 2 hours, leaving 10 mL uninfused and failing to complete the order on time. Choice C (70 mL/hr) would deliver 140 mL over 2 hours, resulting in 10 mL too much being infused, which exceeds the prescribed amount. Choice D (130 mL/hr) represents a common error where students use the remaining volume as the rate, confusing units and potentially creating a dangerous overdose situation. The correct answer is B (65 mL/hr), which ensures exactly 130 mL is delivered over the remaining 2 hours. For HESI infusion problems, always identify what's already been given, calculate what remains, then apply the rate formula. Double-check that your final rate will deliver the exact remaining volume in the exact remaining time—no more, no less.
9. A hospital charges a flat fee of $2,200 for a surgery. In addition, there is a charge of $900 for each day the patient is admitted for recovery. If a patient's total bill was $5,800, for how many days, d, was the patient admitted?
   Answer: 4 When you encounter word problems involving total costs with both fixed and variable components, you need to set up an equation that separates these different charges. Here, the hospital has a fixed surgery fee of $2,200 plus a variable daily charge of $900 per day. With a total bill of $5,800, you can write: $2,200 + 900d = 5,800$ To solve for the number of days (d), first subtract the fixed surgery cost: $900d = 5,800 - 2,200 = 3,600$ Then divide by the daily rate: $d = 3,600 ÷ 900 = 4 \text{ days}$ You can verify: $2,200 + (4 × $900) = $2,200 + $3,600 = $5,800 ✓ Looking at the wrong answers: Choice A (3 days) would give you $2,200 + $2,700 = $4,900, which is $900 short of the actual bill. Choice C (6 days) would result in $2,200 + $5,400 = $7,600, exceeding the bill by $1,800. Choice D (9 days) would cost $2,200 + $8,100 = $10,300, more than double the actual charges. For HESI math problems involving costs, always identify what's fixed versus what varies with time or quantity. Set up your equation systematically, separating these components, then solve step by step. Double-check by substituting your answer back into the original scenario to ensure it produces the given total.
10. Solve for x: $\frac{x+5}{3} = 2x - 10$
    Answer: 7 When you encounter an equation with a fraction equal to an algebraic expression, your goal is to isolate the variable by eliminating the fraction and combining like terms systematically. To solve $\frac{x+5}{3} = 2x - 10$, start by eliminating the fraction. Multiply both sides by 3: $x + 5 = 3(2x - 10)$. Distribute the 3 on the right side: $x + 5 = 6x - 30$. Now collect all terms with x on one side and constants on the other. Subtract x from both sides: $5 = 5x - 30$. Add 30 to both sides: $35 = 5x$. Finally, divide by 5: $x = 7$. You can verify this by substituting back: $\frac{7+5}{3} = \frac{12}{3} = 4$, and $2(7) - 10 = 14 - 10 = 4$. Both sides equal 4, confirming our answer. Choice A (-7) results from sign errors when moving terms across the equation. Choice B (3) likely comes from incorrectly distributing or combining terms early in the process. Choice D (9) could result from calculation mistakes when isolating x, particularly in the final division step. The key strategy for linear equations with fractions is to eliminate fractions first by multiplying both sides by the denominator, then systematically move all variable terms to one side and constants to the other. Always verify your answer by substituting back into the original equation—this catches arithmetic errors that are common on timed exams like the HESI.
11. The cost of renting a particular medical device is a one-time fee of $75 plus $40 per day. If a clinic's budget for this device is $595, what is the maximum number of days, d, they can rent it?
    Answer: 13 When you encounter cost problems involving a fixed fee plus a variable rate, you're dealing with a linear equation. Set up the relationship between total cost, fixed costs, and variable costs to find the maximum within budget constraints. The total cost equation is: Fixed fee + (Daily rate × Number of days) ≤ Budget. Here, that's $75 + 40d ≤ 595$, where d represents the number of days. To solve this inequality, subtract the fixed fee from both sides: $40d ≤ 595 - 75$, which gives you $40d ≤ 520$. Dividing both sides by 40 yields $d ≤ 13$. Since the clinic can rent for a maximum of 13 days while staying within their $595 budget, the answer is C. Let's verify why the other options are incorrect. Choice A (11 days) would cost $75 + 40(11) = 515$, which is well under budget, so they're not maximizing their rental period. Choice B (12 days) costs $75 + 40(12) = 555$, also under budget and not the maximum. Choice D (14 days) would cost $75 + 40(14) = 635$, which exceeds their $595 budget by $40. For HESI math problems involving budgets or constraints, always set up an inequality rather than an equation when finding maximum values within limits. Double-check your answer by calculating the actual cost to ensure it doesn't exceed the given budget, and verify that adding one more unit would put you over the limit.
12. The formula to convert Celsius to Fahrenheit is $F = \frac{9}{5}C + 32$. A patient has a temperature of 102.2° F. The doctor orders the temperature to be reduced to a target of 38° C. How many degrees Fahrenheit does the patient's temperature need to decrease to meet the target?
    Answer: 1.8° F Temperature conversion problems on the HESI require careful attention to what's being asked. You're not just converting between scales—you're finding the difference between two temperature values. Start by converting the target temperature from Celsius to Fahrenheit using the given formula: $F = \frac{9}{5}C + 32$. With the target of 38°C: $F = \frac{9}{5}(38) + 32 = \frac{342}{5} + 32 = 68.4 + 32 = 100.4°F$ Now find the difference between the current temperature and the target: $102.2°F - 100.4°F = 1.8°F$ The patient's temperature needs to decrease by 1.8°F to reach the target, making (B) 1.8°F correct. Let's examine the wrong answers: (A) 1.0°F likely comes from rounding errors or miscalculating the Celsius conversion. (C) 2.2°F represents a common trap—this is simply the decimal portion of the original temperature (102.2), which students might mistakenly think relates to the answer. (D) 4.2°F could result from calculation errors in the conversion process or confusing the relationship between the numbers given. Study tip: Always convert to the same temperature scale before finding differences. Write out each step clearly: convert the target temperature, then subtract to find the difference. Don't try to shortcut these multi-step problems—the wrong answer choices are specifically designed to catch common computational errors.
13. A patient's heart rate is measured three times. The second measurement is 5 beats per minute (bpm) higher than the first, and the third measurement is 2 bpm lower than the second. If the sum of the three measurements is 233 bpm, what was the first measurement, x?
    Answer: 75 bpm When you encounter word problems involving sequential measurements or changes, the key is to define your variables clearly and set up equations that represent the relationships described in the problem. Let's call the first measurement $x$. According to the problem, the second measurement is 5 bpm higher than the first, so it equals $x + 5$. The third measurement is 2 bpm lower than the second, making it $(x + 5) - 2 = x + 3$. Since the sum of all three measurements is 233 bpm, you can write: $x + (x + 5) + (x + 3) = 233$ Simplifying: $3x + 8 = 233$ Solving for x: $3x = 225$, so $x = 75$ Let's verify: First measurement = 75, second = 80, third = 78. Sum: 75 + 80 + 78 = 233 ✓ Now examining the wrong answers: Choice B (77 bpm) would give you measurements of 77, 82, and 80, totaling 239 bpm—too high. Choice C (78 bpm) results in 78, 83, and 81, summing to 242 bpm. Choice D (80 bpm) produces 80, 85, and 83, totaling 248 bpm. Each of these exceeds the required sum of 233. For HESI math problems involving sequential relationships, always define your starting variable first, then express each subsequent value in terms of that variable. Set up your equation based on the given constraint (like a total sum), and always verify your answer by substituting back into the original conditions.
14. A medical supply company offers two billing plans. Plan A has a $50 monthly fee plus $15 per box of supplies. Plan B has a $100 monthly fee plus $10 per box of supplies. For how many boxes of supplies, x, would the total monthly cost be the same for both plans?
    Answer: 10 This is a classic linear equation problem where you need to find the point where two different cost structures intersect. When you see questions comparing two pricing plans, set up equations for each plan and solve for where they're equal. To solve this, write an equation for each plan's total monthly cost. Plan A costs $50 + 15x$ dollars, where $x$ is the number of boxes. Plan B costs $100 + 10x$ dollars. Set these equal to find where the costs are the same: $50 + 15x = 100 + 10x$ Subtract $10x$ from both sides: $50 + 5x = 100$ Subtract 50 from both sides: $5x = 50$ Divide by 5: $x = 10$ You can verify this by substituting back: Plan A at 10 boxes costs $50 + 15(10) = 200$ dollars. Plan B at 10 boxes costs $100 + 10(10) = 200$ dollars. Both equal $200, confirming our answer. Choice A (5 boxes) gives you Plan A = $125 and Plan B = $150, so they're not equal. Choice B (8 boxes) gives you Plan A = $170 and Plan B = $180, still not equal. Choice D (15 boxes) gives you Plan A = $275 and Plan B = $250, with Plan B actually cheaper. Remember that for cost comparison problems, always set up your equations carefully and double-check by substituting your answer back into both original equations. The intersection point tells you exactly where the two options become equivalent.
15. The perimeter of a rectangular therapy room is 110 feet. The length of the room is 10 feet longer than twice the width. What is the width, w, of the room in feet?
    Answer: 15 feet When you encounter perimeter problems involving rectangles with algebraic relationships, you need to set up equations using the given constraints and solve systematically. Start by defining your variables and writing the perimeter formula. For a rectangle, $P = 2l + 2w$, where $l$ is length and $w$ is width. You know the perimeter is 110 feet, and the length is 10 feet longer than twice the width, so $l = 2w + 10$. Substitute the length expression into the perimeter formula: $110 = 2(2w + 10) + 2w$. Simplifying: $110 = 4w + 20 + 2w = 6w + 20$. Subtract 20 from both sides: $90 = 6w$. Therefore, $w = 15$ feet. Let's verify: if $w = 15$, then $l = 2(15) + 10 = 40$. The perimeter is $2(40) + 2(15) = 80 + 30 = 110$ ✓ Answer choice A (15 feet) is correct. Answer B (25 feet) would give a length of 60 feet and perimeter of 170 feet—too large. Answer C (33 feet) would result in a length of 76 feet and perimeter of 218 feet—way too large. Answer D (40 feet) would make the length 90 feet with a perimeter of 260 feet—massively oversized. For HESI geometry problems, always define your variables clearly, translate word relationships into algebraic expressions, and check your answer by substituting back into the original constraints. This verification step catches calculation errors and confirms your solution makes sense.
16. The concentration of a drug in a patient's bloodstream, C, in mg/L, is modeled by the equation $C = 15 - 1.25t$, where t is the number of hours after administration. How many hours, t, will it take for the concentration to reach 8.75 mg/L?
    Answer: 5 This question tests your ability to solve linear equations in a healthcare context, where you need to find when a drug concentration reaches a specific level. To find when the concentration reaches 8.75 mg/L, you substitute this value for C in the equation $C = 15 - 1.25t$ and solve for t: $8.75 = 15 - 1.25t$ Subtract 15 from both sides: $8.75 - 15 = -1.25t$ This gives you: $-6.25 = -1.25t$ Divide both sides by -1.25: $t = \frac{-6.25}{-1.25} = 5$ So the answer is B) 5 hours. Let's examine why the other options are incorrect: A) 3 hours would give $C = 15 - 1.25(3) = 15 - 3.75 = 11.25$ mg/L, which is too high. C) 7 hours would give $C = 15 - 1.25(7) = 15 - 8.75 = 6.25$ mg/L, which is too low. D) 19 hours would give $C = 15 - 1.25(19) = 15 - 23.75 = -8.75$ mg/L. This negative concentration is impossible in real life and shows this answer is far too large. Study tip: When working with linear drug concentration models on the HESI, always check that your answer makes biological sense. Drug concentrations decrease over time and cannot be negative. Also, practice isolating variables by performing the same operation on both sides of the equation—this skill appears frequently in dosage calculations.
17. A healthcare facility charges a base fee of $85 plus $12 per hour for equipment rental. If a patient's total bill was $157, how many hours was the equipment rented?
    Answer: 6 hours Set up the equation: 85 + 12h = 157. Subtract 85 from both sides: 12h = 72. Divide by 12: h = 6. Choice B results from adding instead of subtracting the base fee. Choice C comes from using the wrong base fee ($73 instead of $85). Choice D results from calculation errors in the division step.
18. If one-third of a number, decreased by 7, is equal to 2, what is the number?
    Answer: 27 When you encounter word problems involving fractions of unknown numbers, your goal is to translate the words into a mathematical equation, then solve systematically. Let's call the unknown number $x$. The phrase "one-third of a number, decreased by 7, is equal to 2" translates to: $\frac{x}{3} - 7 = 2$ To solve, add 7 to both sides: $\frac{x}{3} = 9$ Then multiply both sides by 3: $x = 27$ Let's verify: one-third of 27 is 9, and 9 decreased by 7 equals 2. ✓ Now let's see why the other options don't work. Choice A (9) seems tempting because 9 appears in our work, but if $x = 9$, then $\frac{9}{3} - 7 = 3 - 7 = -4$, not 2. Choice B (13) gives us $\frac{13}{3} - 7 = 4.33 - 7 = -2.67$, which is negative and nowhere near 2. Choice C (21) yields $\frac{21}{3} - 7 = 7 - 7 = 0$, not 2. Each wrong answer represents a common algebraic mistake: A picks up the intermediate result (9), B might come from incorrectly adding instead of subtracting, and C gives the result if you forget the final step. For HESI math success, always work methodically through word problems: identify the variable, translate to an equation, solve step-by-step, and verify your answer by substituting back into the original problem. This verification step catches most calculation errors.
19. A nurse's schedule shows that she worked $\frac{x}{2} + 3$ hours on Monday and $\frac{x}{4} + 5$ hours on Tuesday, totaling 16 hours for both days. How many hours did she work on Monday?
    Answer: 7 hours Set up: (x/2 + 3) + (x/4 + 5) = 16. Simplify: x/2 + x/4 + 8 = 16. Combine fractions: 3x/4 + 8 = 16. Subtract 8: 3x/4 = 8. Multiply by 4/3: x = 32/3. Monday hours = x/2 + 3 = 16/3 + 3 = 7 hours. Choice B assumes x = 16. Choice C results from calculation errors with fractions. Choice D comes from solving for x incorrectly.
20. Solve for the variable n in the following equation: $\frac{2}{5}n - 6 = 2$
    Answer: 20 When you encounter linear equations with fractions, your goal is to isolate the variable through systematic algebraic operations. Start by eliminating constants, then deal with coefficients. To solve $\frac{2}{5}n - 6 = 2$, first add 6 to both sides to isolate the term containing n: $\frac{2}{5}n = 2 + 6 = 8$. Now you need to solve $\frac{2}{5}n = 8$. To eliminate the fraction coefficient, multiply both sides by the reciprocal $\frac{5}{2}$: $n = 8 \times \frac{5}{2} = \frac{40}{2} = 20$. You can verify this: $\frac{2}{5}(20) - 6 = 8 - 6 = 2$ ✓ Answer A (3.2) likely results from incorrectly dividing 8 by $\frac{2}{5}$ instead of multiplying by its reciprocal, or from calculation errors with decimal conversion. Answer B (10) suggests you may have forgotten to add 6 in the first step, solving $\frac{2}{5}n = 2$ instead of the correct $\frac{2}{5}n = 8$. Answer D (35) probably comes from multiplying 8 by both parts of the fraction separately (8 × 2 = 16, then 16 + something), showing confusion about fraction operations. Remember that dividing by a fraction is the same as multiplying by its reciprocal. When you see $\frac{a}{b}n = c$, multiply both sides by $\frac{b}{a}$ to get $n = c \times \frac{b}{a}$. Always check your answer by substituting back into the original equation.

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