Use Variables in Real-World Problems - 6th Grade Math

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?
Answer: $x-12=5$ 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, $x-12=5$ , 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, $12-x=5$ , 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).
Maya had some money. She spent $15 on a hoodie and now has $8 left. Let $x$ be the amount of money Maya had at first. Which equation matches the situation?
Answer: $x-15=8$ 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 "Maya had x dollars, spent $15, has $8 left" → equation x-15=8 (x is unknown specific starting amount, solve: x=23). The correct choice is C, $x-15=8$ , because x represents a specific unknown value, and the equation correctly subtracts the spent amount from x to equal the remaining $8, allowing us to solve for that specific x. A common error is choosing A, $x+15=8$ , which adds instead of subtracts, confusing the operation, or D, $x=15-8$ , which solves arithmetically without a variable equation, missing the algebraic representation. 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).
A rope is 18 feet long and is cut into 6 equal pieces. Let $p$ be the length (in feet) of each piece. Which equation represents this situation?
Answer: $6p=18$ 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 "18 feet rope cut into 6 equal pieces, p length per piece" → equation 6p=18 (p is unknown specific value, solve: p=3). The correct choice is C, $6p=18$ , where p is a specific unknown, and the equation multiplies the number of pieces by length to equal total. A common error is choosing B, $18p=6$ , which reverses the multiplication, or D, $p\div 6=18$ , which divides incorrectly. Defining variables: state what variable represents (let p=length of each piece in feet—clear definition prevents confusion). Unknown vs general: unknown specific value (solve for: p in 6p=18 has one answer p=3, specific). Writing from context: read problem (identifies relationship: total divided by pieces), choose variable (p for length), write equation (6p=18).
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 $c$ represents the number of cookie boxes sold and $a$ represents the number of candy boxes sold, which inequality represents the combinations that will meet or exceed the fundraising goal?
Answer: $5c + 3a \geq 200$ 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.
A movie theater charges $9 for each student ticket. Let $n$ be the number of student tickets bought. Which expression represents the total cost in dollars?
Answer: $9n$ 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 "$9 per ticket for n tickets" → expression 9n (n any non-negative whole number representing tickets bought, 9n gives total cost for any n). The correct choice is B, $9n$ , because n represents any number of tickets, and the expression 9n correctly multiplies the cost per ticket by the variable n to find the total cost for any value of n. A common error is choosing A, $9+n$ , which would incorrectly add instead of multiply, confusing addition with multiplication in the context of per-item costs, or selecting D, $9/n$ , which divides instead, misrepresenting the total cost relationship. 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).
A rope is $24$ feet long and is cut into $6$ equal pieces. Let $p$ be the length of one piece in feet. Which equation represents this situation?
Answer: $6p=24$ 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 "24 ft rope cut into 6 equal pieces, p length per piece" → equation 6p = 24 (p is unknown specific length, solve: p=4). The correct choice is C, $6p=24$ , because p represents a specific unknown length, and the equation correctly multiplies pieces by p to equal total, allowing solution for p. A common error is choosing B, $\dfrac{p}{6}=24$ , which divides incorrectly, or A, $p-6=24$ , which subtracts, both using wrong operations for equal division. 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).
In science class, the temperature in a terrarium can be any value from $18^\circ\text{C}$ to $30^\circ\text{C}$ . Let $t$ be the temperature in degrees Celsius. Which statement correctly describes what values $t$ can take?
Answer: $t$ can be any number such that $18\le t\le 30$ . 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, $t$ can be any number such that $18\le t\le 30$ , 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, $t$ 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).
A water bottle holds $750$ milliliters. Someone pours the water equally into $p$ identical cups. Let $p$ be the number of cups (a positive whole number). Which expression gives the amount of water in each cup, in milliliters?
Answer: $\dfrac{750}{p}$ 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, $\dfrac{750}{p}$ , 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, $750p$ , 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).
A student has $24 to spend on identical water bottles that cost $3 each. Let $b$ be the number of bottles the student can buy. Which equation could be used to find $b$ ?
Answer: $3b=24$ 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 $3b = 24$ 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 $3 + b = 24$ (adding instead of multiplying), or $24b = 3$ (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).
On a field trip, the bus travels at 50 miles per hour. Let $t$ be the number of hours the bus travels, and let $d$ be the distance in miles. Which equation models the relationship between distance and time?
Answer: $d=50t$ 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, $d=50t$ , where t is a general variable for any time, and the equation models distance as speed times time. A common error is choosing C, $t=50d$ , 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).
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 $s$ represents the number of small cakes sold and $\ell$ represents the number of large cakes sold, which equation correctly represents the relationship between the numbers of cakes sold?
Answer: $s = 2\ell$ 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: $s$ represents small cakes sold, $\ell$ 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 $s = 2\ell$ , which is answer choice C. Let's examine why the other choices are incorrect. Choice A ( $2s = \ell$ ) 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 ( $\ell = 2s$ ) also incorrectly suggests they sold twice as many large cakes as small cakes. Choice D ( $s + \ell = 2$ ) claims they only sold 2 cakes total, which makes no sense given they earned $132. The correct answer is C: $s = 2\ell$ . 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?"
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 $c$ represents the parking fee for one car and $m$ represents the parking fee for one motorcycle, which system of equations represents this situation?
Answer: $15c + 8m = 92$ and $12c + 10m = 86$ On Tuesday: 15 cars at $c 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.
A movie theater charges $9 for each student ticket. Let $n$ be the number of student tickets bought. Which expression represents the total cost in dollars?
Answer: $9n$ 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 "$9 per ticket for n tickets" → expression 9n (n any non-negative whole number representing tickets bought, 9n gives total cost for any n). The correct choice is B, $9n$ , because n represents any number of tickets, and the expression 9n correctly multiplies the cost per ticket by the variable n to find the total cost for any value of n. A common error is choosing A, $9+n$ , which would incorrectly add instead of multiply, confusing addition with multiplication in the context of per-item costs, or selecting D, $9/n$ , which divides instead, misrepresenting the total cost relationship. 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).
Two students write variables for different situations: Situation 1: "A notebook costs $4 each. Let $n$ be the number of notebooks." Situation 2: "Lena had some dollars, spent $6, and has $10 left. Let $x$ be the dollars Lena had at first." Which choice correctly describes the purpose of $n$ and $x$ ?
Answer: $n$ represents any non-negative whole number of notebooks, and $x$ represents an unknown specific starting amount of money. 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).
Two students write variables for different situations: Situation 1: "A notebook costs $4 each. Let $n$ be the number of notebooks." Situation 2: "Lena had some dollars, spent $6, and has $10 left. Let $x$ be the dollars Lena had at first." Which choice correctly describes the purpose of $n$ and $x$ ?
Answer: $n$ represents any non-negative whole number of notebooks, and $x$ represents an unknown specific starting amount of money. 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).
A rectangle has length $l$ centimeters and width $w$ centimeters. Which expression represents the perimeter of the rectangle?
Answer: $2l+2w$ Perimeter means the distance all the way around a shape. For a rectangle, you add up the lengths of all four sides. A rectangle has two lengths and two widths. So you have $l$ , $l$ , $w$ , and $w$ . Adding them gives: $l + l + w + w$ Since $l + l = 2l$ and $w + w = 2w$ , this becomes $2l + 2w$ . That matches choice B! Think of walking around a rectangular garden. You walk one length, one width, the other length, and the other width before you're back where you started. Two of each side, every time. Try this at home: Measure your bedroom or a notebook. Find the length and width, then use $2l + 2w$ to figure out the perimeter. Walk it out with your finger to check that you really touched all four sides!
A class is ordering pizzas. Each pizza costs $10. The class has $80 to spend. Let $p$ be the number of pizzas they can buy. Which equation should you write to find $p$ ?
Answer: $10p=80$ 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 "$10 per pizza, $80 to spend, p pizzas" → equation 10p=80 (p is unknown specific number, solve: p=8). The correct choice is B, $10p=80$ , where p is a specific unknown, and the equation multiplies cost per pizza by number to equal budget. A common error is choosing A, $10+p=80$ , which adds instead of multiplying, or D, $80p=10$ , 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).
A bus travels at $40$ miles per hour. Let $t$ be the time in hours. Which equation shows the relationship between distance $d$ (in miles) and time $t$ ?
Answer: $d=40t$ 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, $d=40t$ , 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, $d=\dfrac{40}{t}$ , 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).
Mia is $m$ years old. Her brother is 4 years older than Mia. Which expression represents her brother's age?
Answer: $m+4$ 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 "Mia age m, brother 4 years older" → expression m+4 (m any positive number representing Mia's age, m+4 gives brother's age for any m). The correct choice is C, $m+4$ , where m is a general variable for Mia's age, and the expression adds 4 to represent the brother's age. A common error is choosing A, $4m$, which multiplies instead of adding, or B, $m-4$ , which subtracts. Defining variables: state what variable represents (let m=Mia's age in years—clear definition prevents confusion). Real-world: variables make formulas general (age m+4 works for any Mia's age m, not just specific values).
A science lab keeps the room temperature between $18^\circ$ C and $24^\circ$ C. Let $T$ be the room temperature in degrees Celsius. Which statement correctly describes the set of possible values for $T$ ?
Answer: $18\le T\le 24$ 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 -10°C and 40°C" → variable t represents any number in set [-10,40] (general range); similarly, T between 18°C and 24°C → 18 ≤ T ≤ 24 (T any number in that interval). The correct choice is B, $18\le T\le 24$ , which describes T as any value in the specified range, a general set. A common error is choosing C, $T\le 18$ or $T\ge 24$ , which is outside the range, or D, only extremes. Defining variables: state what variable represents (let T=temperature in °C—clear definition prevents confusion). Unknown vs general: general set (T in 18 ≤ T ≤ 24 can be any value in range: T=20, T=22, etc., works for all in set). Mistakes: purpose wrong (treating general set as specific value).

Start this quiz