Middle School Math Quiz: Use Variables In Real World Problems
20 questions · exam conditions
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Use Variables In Real World ProblemsQuestion 1 of 20

A taxi company charges a base fee plus an additional amount per mile. The total cost for a 12-mile trip is $18, and the total cost for a 20-mile trip is $26. If $bb representsthebasefeeandrepresents the base fee and mm representsthecostpermile,whichexpressioncouldrepresentthetotalcostforanytripofrepresents the cost per mile, which expression could represent the total cost for any trip of dd $ miles?

1+d1 + d
d+6d + 6
b+mdb + md
6+d6 + d
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Middle School Math Quiz

Middle School Math Quiz: Use Variables In Real World Problems

Practice Use Variables In Real World Problems in Middle School Math 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 Use Variables In Real World Problems, giving you a quick way to practice the rules, question types, and explanations that matter most for Middle School Math.

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.

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Question 1

A taxi company charges a base fee plus an additional amount per mile. The total cost for a 12-mile trip is $18, and the total cost for a 20-mile trip is $26. If $bb representsthebasefeeandrepresents the base fee and mm representsthecostpermile,whichexpressioncouldrepresentthetotalcostforanytripofrepresents the cost per mile, which expression could represent the total cost for any trip of dd $ miles?

  1. 1+d1 + d
  2. d+6d + 6
  3. b+mdb + md (correct answer)
  4. 6+d6 + d
Explanation: The general form for taxi fare is base fee plus (cost per mile × number of miles), which is b + md. While we could solve for specific values (b = 6, m = 1), the question asks for an expression in terms of the given variables b and m. Choice A uses specific calculated values but in wrong positions. Choice B uses a calculated base fee but assumes $1 per mile. Choice D is similar to B with the calculated values.

Question 2

A school fundraiser sells boxes of cookies for $5 each and boxes of candy for $3 each. The goal is to raise at least $200. If $cc representsthenumberofcookieboxessoldandrepresents the number of cookie boxes sold and aa $ represents the number of candy boxes sold, which inequality represents the combinations that will meet or exceed the fundraising goal?

  1. 5c+3a≥2005c + 3a \geq 200 (correct answer)
  2. 5c+3a≤2005c + 3a \leq 200
  3. c+a≥200c + a \geq 200
  4. 8(c+a)≥2008(c + a) \geq 200
Explanation: The total money raised is $5 per cookie box times c boxes plus $3 per candy box times a boxes: 5c + 3a. Since they want to raise 'at least $200', this means greater than or equal to 200: 5c + 3a ≥ 200. Choice B uses the wrong inequality direction (less than or equal). Choice C ignores the different prices of the items. Choice D incorrectly adds the prices and multiplies by total items.

Question 3

A parking garage charges different rates for cars and motorcycles. Last Tuesday, 15 cars and 8 motorcycles were parked, generating $92 in revenue. Last Wednesday, 12 cars and 10 motorcycles were parked, generating $86 in revenue. If $cc representstheparkingfeeforonecarandrepresents the parking fee for one car and mm $ represents the parking fee for one motorcycle, which system of equations represents this situation?

  1. 15c+8m=9215c + 8m = 92 and 12c+10m=8612c + 10m = 86 (correct answer)
  2. 15c+12c=9215c + 12c = 92 and 8m+10m=868m + 10m = 86
  3. 15m+8c=9215m + 8c = 92 and 12m+10c=8612m + 10c = 86
  4. 23c+18m=17823c + 18m = 178
Explanation: On Tuesday: 15 cars at ceachplus8motorcyclesatc each plus 8 motorcycles at m each equals $92 total, so 15c + 8m = 92. On Wednesday: 12 cars at $c each plus 10 motorcycles at $m each equals $86 total, so 12c + 10m = 86. Choice B separates cars and motorcycles into different equations incorrectly. Choice C switches the variables for cars and motorcycles. Choice D combines both days into one equation, losing important information.

Question 4

A water bottle holds 750750 milliliters. Someone pours the water equally into pp identical cups. Let pp be the number of cups (a positive whole number). Which expression gives the amount of water in each cup, in milliliters?​

  1. p750\dfrac{p}{750}
  2. 750p750p
  3. 750p\dfrac{750}{p} (correct answer)
  4. 750−p750-p
Explanation: This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "750 ml divided equally into p cups" → expression 750/p (p any positive whole number representing cups, 750/p gives amount per cup for any p). The correct choice is C, $\dfrac{750}{p},becauseprepresentsanypositivewholenumberofcups,andtheexpressioncorrectlydividesthetotalwaterbyptofindthegeneralamountpercup.AcommonerrorischoosingA,, because p represents any positive whole number of cups, and the expression correctly divides the total water by p to find the general amount per cup. A common error is choosing A, 750p,whichmultipliesinsteadofdivides,reversingtheoperation,orB,, which multiplies instead of divides, reversing the operation, or B, 750-p$, which subtracts, misinterpreting division as subtraction in sharing equally. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

Question 5

A bakery sells small cakes for $4 each and large cakes for $7 each. On Saturday, they sold twice as many small cakes as large cakes and earned $132 total. If $ss representsthenumberofsmallcakessoldandrepresents the number of small cakes sold and ℓ\ell $ represents the number of large cakes sold, which equation correctly represents the relationship between the numbers of cakes sold?

  1. 2s=ℓ2s = \ell
  2. ℓ=2s\ell = 2s
  3. s=2ℓs = 2\ell (correct answer)
  4. s+ℓ=2s + \ell = 2
Explanation: When you encounter word problems involving relationships between quantities, the key is to carefully translate the words into mathematical expressions by identifying what each variable represents and how they relate to each other. Let's break down what the problem tells us: ss represents small cakes sold, ℓ\ell represents large cakes sold, and "they sold twice as many small cakes as large cakes." This phrase means the number of small cakes equals two times the number of large cakes. If they sold 10 large cakes, they sold 20 small cakes. If they sold 15 large cakes, they sold 30 small cakes. Mathematically, this translates to s=2ℓs = 2\ell, which is answer choice C. Let's examine why the other choices are incorrect. Choice A (2s=ℓ2s = \ell) says that twice the small cakes equals the large cakes, meaning they sold more large cakes than small cakes—the opposite of what the problem states. Choice B (ℓ=2s\ell = 2s) also incorrectly suggests they sold twice as many large cakes as small cakes. Choice D (s+ℓ=2s + \ell = 2) claims they only sold 2 cakes total, which makes no sense given they earned $132. The correct answer is C: $s=2ℓs = 2\ell $. Study tip: When translating "twice as many A as B," the equation is always A = 2B. The quantity mentioned first equals two times the quantity mentioned second. Practice identifying which variable goes where by asking yourself: "Which quantity is larger?"

Question 6

Jenny is filling a swimming pool with water. The pool already contains some water, and she adds water at a constant rate. After 2 hours, the pool contains 150 gallons. After 5 hours, it contains 240 gallons. If rr represents the rate in gallons per hour and ii represents the initial amount of water, what does the variable rr specifically represent in this context?

  1. The total amount of water added during the entire filling process
  2. The rate at which water is being added to the pool each hour (correct answer)
  3. The final amount of water that will be in the pool when completely filled
  4. The amount of time needed to add each gallon of water to the pool
Explanation: The variable r is defined as the rate in gallons per hour, which means the amount of water added to the pool each hour. This is a rate of change. Choice A describes a total amount, not a rate. Choice C describes a final capacity, not a rate. Choice D describes time per gallon, which would be the reciprocal of the rate.

Question 7

A rectangle has length ll centimeters and width ww centimeters. Which expression represents the perimeter of the rectangle?​

  1. lwlw
  2. 2l+2w2l+2w (correct answer)
  3. 2lw2lw
  4. l+wl+w
Explanation: This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "rectangle with length l and width w" → expression 2l + 2w (l and w any positive numbers representing dimensions, 2l + 2w gives perimeter for any l and w). The correct choice is B, 2l+2w2l+2w, because l and w represent any positive lengths, and the expression correctly adds twice the length and twice the width to give the general perimeter formula. A common error is choosing A, lwlw, which is the area instead of perimeter, confusing the two concepts, or D, 2lw2lw, which doubles the area formula, misapplying the operations for perimeter. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

Question 8

A class is ordering pizzas. Each pizza costs $10. The class has $80 to spend. Let $pbethenumberofpizzastheycanbuy.Whichequationshouldyouwritetofindbe the number of pizzas they can buy. Which equation should you write to findp$?

  1. p=10−80p=10-80
  2. 10+p=8010+p=80
  3. 80p=1080p=10
  4. 10p=8010p=80 (correct answer)
Explanation: This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has 8left:xisspecificunknown,equationx−12=8tosolve),(2)anynumberinspecifiedset(costformula5nfornitems:nrepresentsanynon−negativeinteger0,1,2,...,generalrelationship).Writing:identifywhat′sunknownorgeneral(numberofitems,person′sage,cost),definevariable(letn=items,letx=age),writeexpression(5nforcost)orequation(x−12=8forSarah′sdollars).Context:"3yearsolderthanMaryagem"→ageism+3(mrepresentsMary′sunknownage).Here,theexampleis"8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "10 per pizza, $80 to spend, p pizzas" → equation 10p=80 (p is unknown specific number, solve: p=8). The correct choice is B, 10p=8010p=80, where p is a specific unknown, and the equation multiplies cost per pizza by number to equal budget. A common error is choosing A, 10+p=8010+p=80, which adds instead of multiplying, or D, 80p=1080p=10, which reverses. Defining variables: state what variable represents (let p=number of pizzas—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: p in 10p=80 has one answer p=8, specific). Writing from context: read problem (identifies relationship: cost times number equals total), write equation (10p=80).

Question 9

Jamal is 33 years older than his sister. Let ss be his sister's age in years. Which expression represents Jamal's age?

  1. 3s3s
  2. 3−s3-s
  3. s−3s-3
  4. s+3s+3 (correct answer)
Explanation: This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "Jamal is 3 years older than sister age s" → expression s+3 (s any non-negative number representing sister's age, s+3 gives Jamal's age for any s). The correct choice is C, $s+3,becausesrepresentsanyageforthesister,andadding3correctlyexpressesJamal′sageingeneraltermsrelativetos.AcommonerrorischoosingA,, because s represents any age for the sister, and adding 3 correctly expresses Jamal's age in general terms relative to s. A common error is choosing A, s-3,whichsubtractsinstead,reversingthe′olderthan′relationship,orD,, which subtracts instead, reversing the 'older than' relationship, or D, 3s$, which multiplies, confusing addition with multiplication in age differences. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

Question 10

Maya is mm years old. Her brother is 4 years older than she is. Which expression represents her brother's age?

  1. m−4m-4
  2. $4m$
  3. m+4m+4 (correct answer)
  4. m/4m/4
Explanation: This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). For example, "5 per item for n items" → expression 5n (n any number representing items purchased, 5n gives total cost for any n); or "Sarah has x dollars, spends $12, has $8 remaining" → equation x-12=8 (x is unknown specific starting amount, solve: x=20); or "temperature t between -10°C and 40°C" → variable t represents any number in set [-10,40] (general range). In this case, the correct expression is $m + 4becausemrepresentsMaya′sunknownspecificage,andadding4givesherbrother′sagerelativetohers.Acommonerrorischoosingbecause m represents Maya's unknown specific age, and adding 4 gives her brother's age relative to hers. A common error is choosingm - 4$ (subtracting instead of adding), or $4m$ (multiplying ages), or confusing the variable as representing a general set instead of a specific unknown. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

Question 11

A movie ticket costs $12. Jordan bought one ticket and has $5 left. Let $x$ be the number of dollars Jordan had before buying the ticket. Which equation matches the situation?

  1. 12−x=512-x=5
  2. x−12=5x-12=5 (correct answer)
  3. x−5=12x-5=12
  4. x+12=5x+12=5
Explanation: This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "Jordan had x dollars, spent $12 on ticket, has $5 left" → equation x-12=5 (x is unknown specific starting amount, solve: x=17). The correct choice is C, $x-12=5,wherexisaspecificunknownvalue,andtheequationcorrectlysubtractsthecostfromtheinitialamounttoequaltheremainder.AcommonerrorischoosingA,, where x is a specific unknown value, and the equation correctly subtracts the cost from the initial amount to equal the remainder. A common error is choosing A, 12-x=5,whichreversesthesubtraction,orB,, which reverses the subtraction, or B, x+12=5$, which adds instead of subtracting. Defining variables: state what variable represents (let x=dollars Jordan had before—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=5 has one answer x=17, specific). Mistakes: wrong purpose (treating x as general when it's specific), expression/equation mismatch (using an expression instead of an equation).

Question 12

A bus travels at 4040 miles per hour. Let tt be the time in hours. Which equation shows the relationship between distance dd (in miles) and time tt?

  1. t=40dt=40d
  2. d=40td=40t (correct answer)
  3. d=40+td=40+t
  4. d=40td=\dfrac{40}{t}
Explanation: This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "bus travels 40 mph for t hours" → equation d = 40t (t any non-negative number representing time, d = 40t gives distance for any t). The correct choice is A, $d=40t,becausetrepresentsanytime,andtheequationcorrectlymultipliesspeedbytimetoexpressthegeneraldistanceformula.AcommonerrorischoosingD,, because t represents any time, and the equation correctly multiplies speed by time to express the general distance formula. A common error is choosing D, d=\dfrac{40}{t},whichdividesinstead,confusingdistancewithanotherrate,orB,, which divides instead, confusing distance with another rate, or B, t=40d$, which solves for t incorrectly, swapping variables. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

Question 13

A gym charges a $12 sign-up fee plus $5 per visit. Let vv be the number of visits. Which expression represents the total cost in dollars?​

  1. 5(v+12)5(v+12)
  2. 12v+512v+5
  3. 125v\dfrac{12}{5v}
  4. 12+5v12+5v (correct answer)
Explanation: This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has 8left:xisspecificunknown,equationx−12=8tosolve),(2)anynumberinspecifiedset(costformula5nfornitems:nrepresentsanynon−negativeinteger0,1,2,...,generalrelationship).Writing:identifywhat′sunknownorgeneral(numberofitems,person′sage,cost),definevariable(letn=items,letx=age),writeexpression(5nforcost)orequation(x−12=8forSarah′sdollars).Context:"3yearsolderthanMaryagem"→ageism+3(mrepresentsMary′sunknownage).Here,theexampleis"8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "12 sign-up plus $5 per visit v" → expression 12 + 5v (v any non-negative whole number representing visits, 12 + 5v gives total cost for any v). The correct choice is C, 12+5v12+5v, because v represents any number of visits, and the expression correctly adds the fixed fee to the variable cost per visit for a general total. A common error is choosing A, 5(v+12)5(v+12), which distributes incorrectly, or B, 12v+512v+5, which swaps the coefficients, both misapplying the fixed and variable parts of the cost. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

Question 14

Jamal is 33 years older than his sister. Let ss be his sister's age in years. Which expression represents Jamal's age?​

  1. s−3s-3
  2. 3−s3-s
  3. s+3s+3 (correct answer)
  4. 3s3s
Explanation: This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "Jamal is 3 years older than sister age s" → expression s+3 (s any non-negative number representing sister's age, s+3 gives Jamal's age for any s). The correct choice is C, $s+3,becausesrepresentsanyageforthesister,andadding3correctlyexpressesJamal′sageingeneraltermsrelativetos.AcommonerrorischoosingA,, because s represents any age for the sister, and adding 3 correctly expresses Jamal's age in general terms relative to s. A common error is choosing A, s-3,whichsubtractsinstead,reversingthe′olderthan′relationship,orD,, which subtracts instead, reversing the 'older than' relationship, or D, 3s$, which multiplies, confusing addition with multiplication in age differences. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

Question 15

Compare these two situations:

Situation 1: "A sandwich costs $6. Let $nbethenumberofsandwiches.Totalcostisbe the number of sandwiches. Total cost is6n.”Situation2:“Samboughtonesandwichfor$6andhas$2left.Let$x.” Situation 2: “Sam bought one sandwich for $6 and has $2 left. Let $x be the dollars Sam had before buying it.”

Which choice correctly describes the purpose of the variables nn and xx?

  1. nn is an unknown specific value; xx can be any non-negative integer.
  2. nn can be any non-negative integer; xx is an unknown specific value. (correct answer)
  3. Both nn and xx are unknown specific values you are trying to find.
  4. Both nn and xx can be any real number.
Explanation: This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation $x-12=8tosolve),(2)anynumberinspecifiedset(costformulato solve), (2) any number in specified set (cost formula5nfornitems:nrepresentsanynon−negativeinteger0,1,2,...,generalrelationship).Writing:identifywhat′sunknownorgeneral(numberofitems,person′sage,cost),definevariable(letn=items,letx=age),writeexpression( for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5nforcost)orequation( for cost) or equation (x-12=8forSarah′sdollars).Context:"3yearsolderthanMaryagem"→ageism+3(mrepresentsMary′sunknownage).Here,theexamplesareSituation1:"6persandwichfornsandwiches"→expressionfor Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the examples are Situation 1: "6 per sandwich for n sandwiches" → expression6n(nanynon−negativeinteger),Situation2:"xdollars,boughtsandwichfor6,has2left"→equation(n any non-negative integer), Situation 2: "x dollars, bought sandwich for 6, has 2 left" → equationx-6=2$ (x specific unknown). The correct choice is C, where n is general for any number, and x is specific unknown to solve for. A common error is choosing B, treating both as specific unknowns, or D, allowing any real numbers without context. Defining variables: state what variable represents (let n=number of sandwiches, let x=dollars before—clear definition prevents confusion). Unknown vs general: distinguish purposes (n general, x specific). Mistakes: purpose wrong (confusing unknown specific with general set).

Question 16

A student has $24 to spend on identical water bottles that cost $3 each. Let $bbethenumberofbottlesthestudentcanbuy.Whichequationcouldbeusedtofindbe the number of bottles the student can buy. Which equation could be used to findb$?

  1. 3+b=243+b=24
  2. 24b=324b=3
  3. b−3=24b-3=24
  4. 3b=243b=24 (correct answer)
Explanation: This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). For example, "5 per item for n items" → expression 5n (n any number representing items purchased, 5n gives total cost for any n); or "Sarah has x dollars, spends $12, has $8 remaining" → equation x-12=8 (x is unknown specific starting amount, solve: x=20); or "temperature t between -10°C and 40°C" → variable t represents any number in set [-10,40] (general range). In this case, the correct equation is 3b=243b = 24 because b represents the specific unknown number of bottles, and multiplying by the cost per bottle gives the total money, solvable for b=8. A common error is choosing 3+b=243 + b = 24 (adding instead of multiplying), or 24b=324b = 3 (reversing), or using an expression instead of an equation for this specific unknown. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

Question 17

On a field trip, the bus travels at 50 miles per hour. Let tt be the number of hours the bus travels, and let dd be the distance in miles. Which equation models the relationship between distance and time?

  1. d=t÷50d=t\div 50
  2. d=50td=50t (correct answer)
  3. t=50dt=50d
  4. d=50+td=50+t
Explanation: This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "bus at 50 mph for t hours, distance d" → equation d=50t (t any non-negative number representing time, d gives distance for any t). The correct choice is B, $d=50t,wheretisageneralvariableforanytime,andtheequationmodelsdistanceasspeedtimestime.AcommonerrorischoosingC,, where t is a general variable for any time, and the equation models distance as speed times time. A common error is choosing C, t=50d,whichsolvesfortimeinsteadofdistance,orD,, which solves for time instead of distance, or D, d=t\div 50$, which divides instead of multiplying. Defining variables: state what variable represents (let t=hours traveled, let d=distance in miles—clear definition prevents confusion). Real-world: variables make formulas general (distance d=50t works for any time t, not just specific values).

Question 18

In science class, the temperature in a terrarium can be any value from 18∘C18^\circ\text{C} to 30∘C30^\circ\text{C}. Let tt be the temperature in degrees Celsius. Which statement correctly describes what values tt can take?

  1. tt can be any number such that t≥18t\ge 18 only.
  2. tt can be any whole number greater than 3030.
  3. tt can be any number such that 18≤t≤3018\le t\le 30. (correct answer)
  4. tt must be a negative number.
Explanation: This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the example is "temperature t between 18°C and 30°C" → variable t represents any number in the set [18,30] (general range). The correct choice is B, $tcanbeanynumbersuchthatcan be any number such that18\le t\le 30,becausetrepresentsanyvalueinthatclosedinterval,correctlydescribingthegeneralsetofpossibletemperatureswithoutimplyingspecificsornegatives.AcommonerrorischoosingA,, because t represents any value in that closed interval, correctly describing the general set of possible temperatures without implying specifics or negatives. A common error is choosing A, tcanbeanywholenumbergreaterthan30,whichexceedstheupperlimit,orC,can be any whole number greater than 30, which exceeds the upper limit, or C,t$ must be a negative number, which ignores the positive range, both treating the variable's purpose as unrestricted or wrong in set definition. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

Question 19

Two students write variables for different situations:

Situation 1: "A notebook costs $4 each. Let nn be the number of notebooks."
Situation 2: "Lena had some dollars, spent $6, and has $10 left. Let xx be the dollars Lena had at first."

Which choice correctly describes the purpose of nn and xx?​

  1. nn and xx both represent any number (including negatives) with no restrictions.
  2. nn and xx both represent unknown specific values to solve for.
  3. nn represents any non-negative whole number of notebooks, and xx represents an unknown specific starting amount of money. (correct answer)
  4. nn represents an unknown specific value to solve for, and xx represents any non-negative whole number.
Explanation: This question tests using variables to represent unknowns (specific values to find) or any numbers in sets (general formulas), writing expressions/equations from contexts, understanding variable purpose varies by problem. Variable purposes: (1) unknown specific value (Sarah has x dollars, buys $12 item, has $8 left: x is specific unknown, equation x-12=8 to solve), (2) any number in specified set (cost formula 5n for n items: n represents any non-negative integer {0,1,2,...}, general relationship). Writing: identify what's unknown or general (number of items, person's age, cost), define variable (let n=items, let x=age), write expression (5n for cost) or equation (x-12=8 for Sarah's dollars). Context: "3 years older than Mary age m" → age is m+3 (m represents Mary's unknown age). Here, the examples are Situation 1: n any non-negative whole number for notebooks (general set), Situation 2: x unknown specific starting money (solve for specific value). The correct choice is B, because it accurately distinguishes n as general for any notebooks and x as specific unknown to solve for, matching the purposes in each context. A common error is choosing A, treating both as specific unknowns, or C, swapping the purposes, both confusing general variables with specific ones. Defining variables: state what variable represents (let x=Sarah's starting dollars, let n=number of items, let t=temperature in °C—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: x in x-12=8 has one answer x=20, specific), general set (n in 5n can be any value: n=1 gives 5, n=10 gives 50, formula works for all n). Writing from context: read problem (identifies relationship: cost per item), choose variable (n for number), write expression/equation (5n for cost, or 5n=20 if total given), define (state what n means). Real-world: variables make formulas general (perimeter 2l+2w works for any rectangle dimensions l,w, not just specific values). Mistakes: undefined variables (forgetting to state what represents), wrong purpose (unknown vs general confused), expression/equation mismatch for problem type, not using variables when should (solving only arithmetically without algebra).

Question 20

A rectangular garden has a length that is 8 feet longer than its width. The perimeter of the garden is 56 feet. If ww represents the width of the garden, which equation can be used to find the width?

  1. 2w+16=562w + 16 = 56
  2. 4w+16=564w + 16 = 56 (correct answer)
  3. 2w+8=562w + 8 = 56
  4. w+8=56w + 8 = 56
Explanation: If the width is w, then the length is w + 8. The perimeter of a rectangle is 2(length + width) = 2(w + 8 + w) = 2(2w + 8) = 4w + 16. Setting this equal to 56 gives 4w + 16 = 56. Choice A incorrectly uses 2w instead of 4w in the perimeter calculation. Choice C forgets to double both dimensions in the perimeter formula. Choice D represents only length = perimeter, ignoring the width entirely.