Writing Algebraic Expressions - Pre Algebra

A cell phone plan charges $25 per month plus $0.15 for each text message over $200$ . If $t$ represents the total number of text messages sent in a month, which expression represents the monthly cost when $t > 200$ ?
Answer: $25 + 0.15(t - 200)$ When you encounter word problems involving tiered pricing or charges that apply only above a certain threshold, you need to carefully identify what triggers the additional cost and how to calculate it. This cell phone plan has two components: a fixed monthly fee of $25 and an overage charge of $0.15 per text beyond the 200-message allowance. The key insight is that you only pay extra for messages that exceed 200, not for all $t$ messages. If you send $t$ messages where $t > 200$ , the number of messages you pay extra for is $(t - 200)$ . For example, if you send 250 messages, you pay extra for $250 - 200 = 50$ messages. The total cost becomes: fixed fee plus overage charge, or $25 + 0.15(t - 200)$ . This matches answer choice B. Looking at the wrong answers: Choice A ( $25 + 0.15t$ ) incorrectly charges $0.15 for every single text message, ignoring the 200-message allowance. Choice C ($ $25 + 0.15(t + 200)$ $) mistakenly adds 200 to your total messages, which would mean paying for 200 extra messages you never sent. Choice D ($ $25(t - 200) + 0.15$ $) confuses the structure entirely, multiplying the fixed fee by the overage amount and treating the per-message rate as a fixed charge. Study tip: In tiered pricing problems, always identify the threshold carefully. The phrase "over 200" means you subtract 200 from the total to find the amount subject to extra charges. Look for this subtraction pattern in similar problems.
A recipe calls for $\frac{1}{3}$ cup of flour for every $2$ servings. Sarah wants to make enough for $s$ servings, but she only has $\frac{2}{3}$ the amount of flour the recipe requires. Which expression represents the amount of flour Sarah has, in cups?
Answer: $\frac{1}{3} \cdot \frac{s}{2} \cdot \frac{2}{3}$ For $s$ servings, the recipe requires $\frac{1}{3} \cdot \frac{s}{2} = \frac{s}{6}$ cups of flour. Sarah has $\frac{2}{3}$ of this amount, so she has $\frac{2}{3} \cdot \frac{s}{6} = \frac{s}{9}$ cups. This can be written as $\frac{1}{3} \cdot \frac{s}{2} \cdot \frac{2}{3}$ (order of multiplication doesn't matter). Choice A has an extra $\frac{1}{3}$ factor. Choice B is equivalent to the correct answer but not in the same form as choice C. Choice D subtracts $\frac{2}{3}$ instead of multiplying.
Maya has $x$ stickers. She gives away $5$ stickers to each of her $4$ friends, then receives $12$ more stickers from her teacher. Which expression represents the number of stickers Maya has now?
Answer: $x - 5 \cdot 4 + 12$ Maya starts with $x$ stickers, gives away $5$ stickers to each of $4$ friends ( $5 \cdot 4 = 20$ stickers), then receives $12$ more. This gives $x - 5 \cdot 4 + 12 = x - 20 + 12$ . Choice B incorrectly adds the number of friends instead of multiplying. Choice C has the wrong signs. Choice D adds the stickers per friend and number of friends instead of multiplying.
A parking garage charges a flat fee plus an additional amount for each hour parked. If the total cost is $8 more than $3$ times the number of hours parked, which expression represents the total cost in dollars for parking $h$ hours?
Answer: $3h + 8$ The phrase '$8 more than $3$ times the number of hours' means we start with $3$ times the hours ( $3h$ ) and add $8$ to it, giving us $3h + 8$ . Choice B incorrectly reverses the coefficients. Choice C distributes incorrectly, representing $3h + 24$ . Choice D represents $8h + 24$ , which misinterprets both parts of the problem.
The number of bacteria in a petri dish triples every hour. If there are initially $b$ bacteria, which expression represents the number of bacteria after the population has tripled $n$ times?
Answer: $b \cdot 3^n$ When a quantity triples $n$ times, it's multiplied by $3^n$ . Starting with $b$ bacteria and tripling $n$ times gives $b \cdot 3^n$ . Choice A represents linear growth ( $3$ times $n$ times $b$ ). Choice B represents adding $3n$ to the original amount. Choice D has the exponent on the wrong factor, representing $3$ times $b$ raised to the $n$ th power.
A delivery service charges $3.50 for the first pound and $1.25 for each additional pound or fraction thereof. If a package weighs $w$ pounds where $w > 1$ , which expression represents the total shipping cost in dollars?
Answer: $3.50 + 1.25(w - 1)$ When you encounter a shipping cost problem with a base fee plus additional charges, break down the pricing structure carefully. This delivery service has two components: a fixed charge for the first pound and a variable charge for everything beyond that. The cost structure is $3.50 for the first pound, then $1.25 for each additional pound. If a package weighs $w$ pounds where $w > 1$ , you need to determine how many pounds are charged at the additional rate. Since the first pound is already covered by the $3.50 fee, the remaining $(w - 1)$ pounds are charged at $1.25 each. The total cost is: Base fee + (Additional pounds × Additional rate per pound) = $3.50 + 1.25(w - 1)$ . This is answer choice D. Looking at the incorrect options: Choice A, $1.25w + 3.50$ , charges the additional rate for all $w$ pounds plus adds the base fee, essentially double-charging. Choice B, $3.50 + 1.25w$ , also charges the additional rate for all pounds instead of just the additional ones beyond the first. Choice C, $3.50w + 1.25(w - 1)$ , incorrectly applies the base fee to every pound rather than just once. Remember this pattern for multi-tier pricing problems: identify what's included in the base fee, then only apply additional charges to quantities beyond that base amount. Always subtract the base quantity from the total when calculating additional charges.
The perimeter of a rectangle is $6$ units less than $4$ times its width. If the width is $w$ units and the length is $3$ units more than twice the width, which expression represents the perimeter?
Answer: $4w - 6$ The problem states that the perimeter IS $6$ units less than $4$ times the width, so perimeter = $4w - 6$ . Choice B represents the actual perimeter formula using the given dimensions ( $2w + 2(2w + 3) = 6w + 6$ ), but this doesn't match the given relationship. Choice C equals $6w + 6$ , which would be $6$ more than $4$ times the width plus $6$ . Choice D represents $4$ times the width plus $3$ .
In a science experiment, the temperature of a solution decreases by $2°C$ every $5$ minutes. If the initial temperature is $T°C$ and $m$ represents the number of $5$ -minute intervals that have passed, which expression represents the current temperature after $30$ minutes?
Answer: $T - 12$ After $30$ minutes, there have been $\frac{30}{5} = 6$ five-minute intervals. The temperature decreases by $2°C$ per interval, so the total decrease is $2 \times 6 = 12°C$ . The current temperature is $T - 12$ . Choice A gives the general formula for $m$ intervals but doesn't substitute $m = 6$ . Choice C incorrectly uses $30$ instead of $6$ intervals. Choice D multiplies the rate by the total time instead of the number of intervals.
In a basketball game, Alex scored $2$ fewer three-point shots than twice the number of two-point shots he made. If he made $f$ two-point shots and $r$ three-point shots, which equation correctly represents this relationship?
Answer: $r = 2f - 2$ When you encounter word problems involving relationships between variables, the key is to translate the English description into mathematical symbols step by step, staying true to the order and meaning of the words. Let's break down the given information: Alex scored "2 fewer three-point shots than twice the number of two-point shots." Start with "twice the number of two-point shots" - since he made $f$ two-point shots, this gives us $2f$ . Then the phrase "2 fewer than" means we subtract 2 from that amount. So the number of three-point shots $r$ equals $2f - 2$ , giving us $r = 2f - 2$ . Looking at the wrong answers: Choice A gives $r = 2f + 2$ , which would mean Alex scored 2 more three-point shots than twice his two-point shots - the opposite of what the problem states. Choice B has $f = 2r - 2$ , which incorrectly makes the number of two-point shots depend on three-point shots, reversing the relationship. Choice C gives $r = 2(f - 2)$ , which means "twice the quantity of 2 fewer two-point shots" - this changes the meaning entirely by putting the subtraction inside the parentheses first. The correct answer is D: $r = 2f - 2$ . Study tip: When translating word problems, work through the phrase in the exact order given, and watch out for answers that flip the relationship between variables or change the order of operations with parentheses.

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