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.

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.

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.

Tom's current age is 2s - 3. In 5 years, Tom's age will be (2s - 3) + 5 = 2s + 2. Choice A shows the addition but doesn't simplify the expression. Choice C incorrectly applies the 'twice' relationship to the sister's future age rather than current age. Choice D confuses Tom's age formula with a simple modification of the sister's age.

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.

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).

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?"

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).

First, we need to find the values of s and w. From the given information: s + 3w = 45 and s + 7w = 73. Subtracting the first equation from the second: 4w = 28, so w = 7. Substituting back: s + 3(7) = 45, so s = 24. However, the question asks for the expression in terms of the variables s and w, not the specific values. After 10 weeks, she will have her starting amount plus 10 weeks of savings: s + 10w. Choice B represents only 7 weeks of savings. Choice C substitutes a specific value for s. Choice D incorrectly multiplies s by 10.

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.