Words to Expressions - SHSAT Math

What algebraic expression represents "the number of minutes $m$ , reduced by 12, divided by 4"?
Answer: $\dfrac{m-12}{4}$ When translating word problems into algebraic expressions, you need to carefully follow the order of operations described in the language. The key is identifying what happens first, second, and so on. Let's break down "the number of minutes $m$ , reduced by 12, divided by 4" step by step. First, you start with $m$ minutes. Then you "reduce by 12," which means subtract 12, giving you $(m-12)$ . Finally, you divide this entire result by 4, which gives you $\dfrac{m-12}{4}$ . Choice A, $\dfrac{m-12}{4}$ , correctly represents this sequence: subtract 12 from $m$ , then divide the whole thing by 4. Choice B, $4(m-12)$ , multiplies $(m-12)$ by 4 instead of dividing by 4. This represents "the number of minutes reduced by 12, then multiplied by 4." Choice C, $\dfrac{m}{4}-12$ , changes the order of operations. It divides $m$ by 4 first, then subtracts 12. This would represent "the number of minutes divided by 4, then reduced by 12." Choice D, $\dfrac{12-m}{4}$ , reverses the subtraction order, giving you "12 reduced by the number of minutes, divided by 4" instead of "the number of minutes reduced by 12." Remember that word problems follow a specific sequence, and parentheses in algebra help preserve that order. When you see "reduced by" or "decreased by," subtract that amount from what came before. When you see "divided by" at the end, it typically applies to everything that came before it.
A student with $s$ stickers gives away 40% of them. Which expression represents the number of stickers the student still has?
Answer: $0.6s$ When you encounter percentage problems, remember that "giving away" or "losing" a percentage means you keep the remaining percentage. If someone gives away 40%, they retain 60% of what they originally had. Let's work through this step by step. The student starts with $s$ stickers and gives away 40% of them. To find how many stickers remain, you need to calculate what percentage they still have: 100% - 40% = 60%. Converting 60% to decimal form gives you 0.6, so the student has $0.6s$ stickers left. Looking at the wrong answers: Choice B ( $0.4s$ ) represents the number of stickers given away, not the number remaining. This is a common trap - students sometimes calculate what was lost instead of what remains. Choice C ( $s - 40$ ) incorrectly subtracts 40 stickers rather than 40% of the stickers. This only works if the student originally had exactly 100 stickers, but the expression must work for any value of $s$ . Choice D ( $s + 0.4s$ ) actually gives you 140% of the original amount, which doesn't make sense since the student gave stickers away rather than gaining more. The correct answer is A: $0.6s$ . Study tip: For percentage problems involving "keeping" or "remaining," always subtract the given percentage from 100% first. If someone loses, gives away, or uses 30%, they keep 70%. Convert that remaining percentage to a decimal and multiply by the original amount.
The temperature in degrees Fahrenheit is $32$ more than $\frac{9}{5}$ times the temperature in degrees Celsius. On a day when the high temperature exceeds the low temperature by $d$ degrees Celsius, by how many degrees Fahrenheit does the high temperature exceed the low temperature?
Answer: $\frac{9d}{5}$ If high temp in Celsius is $C_h$ and low temp is $C_l$ , then $C_h - C_l = d$ . In Fahrenheit: high = $\frac{9C_h}{5} + 32$ , low = $\frac{9C_l}{5} + 32$ . Difference = $(\frac{9C_h}{5} + 32) - (\frac{9C_l}{5} + 32) = \frac{9(C_h - C_l)}{5} = \frac{9d}{5}$ . Choice B incorrectly adds 32. Choice C incorrectly distributes. Choice D incorrectly adds 32 before multiplying by 9/5.
Which algebraic expression represents "five less than the product of 4 and $m$ "?
Answer: $4m-5$ When you encounter word problems asking you to translate English phrases into algebraic expressions, work from the inside out and pay careful attention to the order of operations indicated by the words. Let's break down "five less than the product of 4 and $m$ " step by step. First, identify the main operation: "less than" tells you this will be a subtraction problem. Next, find what you're subtracting from: "the product of 4 and $m$ " means $4m$ . Finally, determine what you're subtracting: "five less than" means you subtract 5. So you get $4m - 5$ . The phrase structure "X less than Y" always translates to $Y - X$ , not $X - Y$ . This makes choice A) $4m-5$ correct. Looking at the wrong answers: Choice B) $5-4m$ reverses the subtraction order—this would represent "the product of 4 and $m$ less than five." Choice C) $4(m-5)$ incorrectly groups the subtraction with $m$ , giving you "four times the quantity $m$ minus five." Choice D) $5(4-m)$ makes two errors: it reverses the subtraction and incorrectly applies multiplication to the wrong terms. Strategy tip: When translating "less than" phrases, always write down what comes after "than" first, then subtract what comes before "less." Practice identifying the key phrase patterns like "the product of," "less than," and "more than"—they appear frequently on the SHSAT and have specific mathematical meanings you need to recognize instantly.
Which algebraic expression represents "7 more than twice a number $n$ "?
Answer: $2n+7$ When you encounter word problems asking you to translate English phrases into algebraic expressions, work systematically by identifying the key operations and their order. Let's break down "7 more than twice a number $n$ " step by step. First, identify "twice a number $n$ " — this means $2n$ . Next, "7 more than" means you add 7 to whatever comes after that phrase. So you're adding 7 to $2n$ , giving you $2n + 7$ . Choice A ( $2n + 7$ ) correctly represents this translation. You start with twice the number ( $2n$ ) and then add 7. Choice B ( $7n + 2$ ) reverses the coefficients — this would represent "2 more than 7 times a number," not what we want. Choice C ( $2(n - 7)$ ) means "twice the quantity of a number minus 7," which expands to $2n - 14$ . This represents a completely different relationship. Choice D ( $7 - 2n$ ) means "7 minus twice a number," which gives you a result that decreases as $n$ increases — the opposite of what "more than" should produce. The key strategy for these translation problems is to identify the base expression first (here, "twice a number"), then apply the operation described in relation to it ("7 more than"). Watch out for reversed coefficients and incorrect operations — these are the most common traps in algebraic translation questions on the SHSAT.
Which expression matches the phrase "the quotient of $x$ decreased by 5 and 3"?
Answer: $\dfrac{x-5}{3}$ When you encounter word problems asking you to translate phrases into algebraic expressions, you need to carefully parse each part of the phrase and understand the order of operations. Let's break down "the quotient of $x$ decreased by 5 and 3" step by step. The key word "quotient" tells us we're looking for a division problem. In mathematical language, "the quotient of A and B" means $\frac{A}{B}$ . So we need to identify what A and B are in this phrase. The phrase structure is "the quotient of [ $x$ decreased by 5] and [3]." The first part of our fraction is " $x$ decreased by 5," which translates to $x - 5$ . The second part is simply 3. Therefore, our expression is $\frac{x-5}{3}$ , which is choice A. Let's examine why the other choices are incorrect. Choice B gives us $\frac{3}{x-5}$ , which would be "the quotient of 3 and $x$ decreased by 5" — the numerator and denominator are flipped. Choice C shows $\frac{x}{3} - 5$ , which represents " $x$ divided by 3, then decreased by 5" — this changes the order of operations entirely. Choice D gives us $\frac{x}{3-5}$ , which would mean "the quotient of $x$ and 3 decreased by 5," incorrectly applying the decrease to the 3 instead of to $x$ . Remember: when translating "quotient" problems, identify what comes before "and" as your numerator, and what comes after "and" as your denominator. Always read carefully to see which quantity is being modified by operations like "decreased by."
What expression represents the perimeter of a square with side length $x$ centimeters?
Answer: $4x$ This question tests your understanding of perimeter formulas for geometric shapes. When you encounter perimeter problems, remember that perimeter means the total distance around the outside of a shape. For a square, all four sides have equal length. If each side is $x$ centimeters long, you need to add up all four sides: $x + x + x + x = 4x$ centimeters. This gives you the total distance around the square's border. Choice A ( $4x$ ) is correct because it represents adding all four equal sides of the square. Choice B ( $x^2$ ) represents area, not perimeter. This is the most common trap students fall into—confusing area and perimeter formulas. Area measures the space inside a shape (length × width), while perimeter measures the distance around it. Choice C ( $2x$ ) would only account for two sides of the square, perhaps representing half the perimeter or the perimeter of a rectangle where you're only counting length and width once. Choice D ( $x + 4$ ) incorrectly adds the variable side length to the constant number 4, which doesn't make geometric sense. This might tempt students who remember "4" is important for squares but mix up how it's used. Study tip: Always distinguish between area and perimeter. Area formulas typically involve multiplication (like $x^2$ for a square's area), while perimeter formulas involve addition of all sides. For any polygon, perimeter equals the sum of all side lengths.
A rectangular garden has length that is $3$ feet more than twice its width. If the width is increased by $x$ feet and the length is decreased by $x$ feet, which expression represents the new perimeter?
Answer: $6w + 6$ Original dimensions: width = $w$ , length = $2w + 3$ . New dimensions: width = $w + x$ , length = $(2w + 3) - x = 2w + 3 - x$ . New perimeter = $2(w + x) + 2(2w + 3 - x) = 2w + 2x + 4w + 6 - 2x = 6w + 6$ . Choice B incorrectly adds $2x$ . Choice C incorrectly subtracts $2x$ . Choice D incorrectly adds $4x$ .
A phone plan charges $40 per month plus $0.05 per text message over $500$ messages. In a month when Sarah sends $t$ text messages, where $t > 500$ , which expression represents her total bill in dollars?
Answer: $40 + 0.05(t - 500)$ Sarah pays $40 base fee plus $0.05 for each message over 500. If she sends $t$ messages (where $t > 500$ ), she pays extra for $(t - 500)$ messages. Total: $40 + 0.05(t - 500)$ . Choice A charges for all messages. Choice C incorrectly adds an extra $25 fee. Choice D uses $(500 - t)$ instead of $(t - 500)$ .
A company charges a flat fee for shipping plus an additional amount per pound. If the total shipping cost is $12 for a $3$ -pound package and $18 for a $6$ -pound package, which expression represents the shipping cost for a package weighing $w$ pounds?
Answer: $6 + 2w$ Find the cost per pound first. Going from $3$ pounds to $6$ pounds adds $3$ pounds and raises the cost from $12 to $18, an increase of $6, so the rate is $6 ÷ 3 = $2 per pound. Next find the flat fee: for the $3$ -pound package, $12 = flat fee + $2(3)$ , so the flat fee is $12 − $6 = $6. The cost is $6 + 2w$ . Check: $6 + 2(6) = 18$ . The expression $4w$ fits the 3-pound package but gives $24 for 6 pounds, because it ignores the flat fee. The expression $2 + 6w$ swaps the fee and the rate. The expression $12 + 2(w + 3)$ makes a sign error when shifting from the 3-pound reference point; it gives $24 for a 3-pound package instead of $12.
Which expression matches "triple a number $n$ decreased by four times the same number"?
Answer: $3n-4n$ Translate the phrase one piece at a time. "Triple a number $n$ " is $3n$ . "Decreased by" means subtract whatever comes next, and "four times the same number" is $4n$ . Putting the pieces together gives $3n-4n$ . The expression $4n-3n$ reverses the order of the subtraction, so it describes four times the number decreased by triple the number. The expression $3(n-4)$ subtracts 4 from $n$ before tripling, which equals $3n-12$ . The expression $3n-4$ subtracts the number 4 instead of four times the number, dropping the $n$ from "four times the same number." The expression that matches the phrase is $3n-4n$ .
Marcus has $d$ dollars. He spends one-third of his money on lunch, then spends $5 more than half of what remains on a book. Which expression represents how much money Marcus has left?
Answer: $\frac{d}{3} - 5$ After lunch Marcus has $d - \frac{d}{3} = \frac{2d}{3}$ left. The book costs $5 more than half of that amount: $\frac{1}{2} \cdot \frac{2d}{3} + 5 = \frac{d}{3} + 5$ . Money left: $\frac{2d}{3} - \left(\frac{d}{3} + 5\right) = \frac{d}{3} - 5$ . $\frac{2d}{3} - 5$ subtracts the extra $5 but forgets the half of the remaining money spent on the book. $\frac{d - 15}{6}$ equals $\frac{d}{6} - \frac{5}{2}$ , which is not the same as $\frac{d}{3} - 5$ ; it comes from wrongly dividing the whole result by 2. $\frac{d}{6} - 5$ halves the money term one extra time.
Translate " $c$ pennies, converted to dollars" into an expression.
Answer: $\dfrac{c}{100}$ Unit conversion problems require you to think carefully about the relationship between different units and how to move from one to another using multiplication or division. To convert pennies to dollars, you need to remember that 100 pennies equal 1 dollar. When you have $c$ pennies, you're looking for how many groups of 100 pennies you have, since each group represents one dollar. This means you divide the number of pennies by 100, giving you $\frac{c}{100}$ dollars. Let's verify this makes sense: if you have 250 pennies, then $\frac{250}{100} = 2.5$ dollars, which is correct. Choice A, $\frac{c}{100}$ , correctly performs this conversion by dividing the number of pennies by 100. Choice B, $100c$ , multiplies instead of divides, which would convert dollars to pennies rather than pennies to dollars. If you had 3 dollars, this would incorrectly suggest you have 300 dollars instead of the correct 300 pennies. Choice C, $c - 100$ , uses subtraction, which doesn't make sense for unit conversion. This would mean 150 pennies somehow equals 50 dollars. Choice D, $\frac{100}{c}$ , has the conversion factor upside down. This would give you a fraction that gets smaller as you have more pennies, which is backwards. Study tip: For unit conversions, always ask yourself "what do I multiply or divide by to get from the given unit to the target unit?" Then check your answer with simple numbers to verify it makes sense.
What is the expression for "half the square of $k$ decreased by 4"?
Answer: $\dfrac{k^{2}}{2}-4$ When translating words into mathematical expressions, you need to carefully parse each phrase and convert it step by step. Let's break down "half the square of $k$ decreased by 4." Start with "the square of $k$ " — this gives us $k^2$ . Next, "half the square of $k$ " means we take half of that entire squared term, which is $\frac{k^2}{2}$ . Finally, "decreased by 4" means we subtract 4 from this expression, giving us $\frac{k^2}{2} - 4$ . Choice A, $\frac{k^2}{2} - 4$ , correctly follows this translation process and is our answer. Choice B, $\left(\frac{k}{2}\right)^2 - 4$ , represents "the square of half of $k$ decreased by 4." This squares the result of dividing $k$ by 2, which is different from taking half of $k$ squared. Choice C, $\frac{k^2 - 4}{2}$ , translates to "half of the quantity $k$ squared decreased by 4." Here, you first subtract 4 from $k^2$ , then take half of that entire result — the order of operations is wrong. Choice D, $\frac{k - 4}{2}$ , gives us "half of $k$ decreased by 4." This completely misses the squaring of $k$ and instead subtracts 4 from $k$ itself before halving. Remember: When translating word problems, identify the main operation first, then work outward. Here, "half" and "decreased by" are applied to "the square of $k$ " as the core expression.
In a game, players earn $3$ points for each correct answer and lose $1$ point for each incorrect answer. If a player answers $c$ questions correctly out of $n$ total questions, which expression represents the player's final score?
Answer: $4c - n$ If the player answers $c$ questions correctly out of $n$ , then $n - c$ answers are incorrect. Points earned: $3c$ . Points lost: $1 \times (n - c) = n - c$ . Final score: $3c - (n - c) = 3c - n + c = 4c - n$ . The expression $3c - n$ subtracts a point for every question, including the ones answered correctly. The expression $2c - n$ subtracts the correct answers instead of adding them back when distributing the negative. The expression $3c - (c - n)$ reverses the count of incorrect answers, which gives $2c + n$ and would make the score grow with every wrong answer.
Translate "the average of $r$ and 10" into an algebraic expression.
Answer: $\dfrac{r+10}{2}$ When you encounter a problem asking you to translate a word phrase into an algebraic expression, focus on identifying the key mathematical operation and what numbers or variables are involved. The phrase "the average of $r$ and 10" tells you to find the mean of two values: the variable $r$ and the number 10. To find the average of any set of numbers, you add them all up and divide by how many numbers you have. Since you have two numbers here, you add $r + 10$ and divide by 2, giving you $\dfrac{r+10}{2}$ . Choice A, $\dfrac{r+10}{2}$ , correctly represents this average formula. Choice B, $\dfrac{r-10}{2}$ , uses subtraction instead of addition, which would give you half the difference between $r$ and 10, not their average. Choice C, $\dfrac{r+10}{10}$ , adds the numbers correctly but divides by 10 instead of 2 — this mistake comes from focusing on the number 10 in the problem rather than counting how many values you're averaging. Choice D, $r+5$ , might seem tempting because it's equivalent to $\dfrac{r+10}{2}$ when simplified in certain contexts, but it doesn't show the proper average formula structure. Remember that "average of" always means add the values and divide by the count. When you see translation problems, identify the operation first (average, sum, difference, product), then carefully set up the expression using the correct numbers and variables in the right order.
A rectangular pool is surrounded by a walkway of uniform width $w$ feet. If the pool itself measures $L$ feet by $W$ feet, which expression represents the area of the walkway only?
Answer: $(L + 2w)(W + 2w) - LW$ The walkway area is the area of the large outer rectangle minus the area of the pool. The walkway adds $w$ feet on every side, so each dimension grows by $2w$ . The outer rectangle is $(L + 2w)$ by $(W + 2w)$ . The pool is $L$ by $W$ . So the walkway area is $(L + 2w)(W + 2w) - LW$ . Expanding gives $LW + 2wL + 2wW + 4w^2 - LW = 4w^2 + 2w(L + W)$ , the same area written another way. Using $(L + w)(W + w) - LW$ adds the width on only one side of each dimension. The expression $4w^2 + w(L + W)$ counts only one strip along each length and each width instead of two. The expression $2w(L + W)$ covers the four side strips but leaves out the four corner squares, each $w^2$ . The area of the walkway is $(L + 2w)(W + 2w) - LW$ .
Translate the phrase "the sum of $k$ and its reciprocal" into an expression.
Answer: $k+\dfrac{1}{k}$ When you encounter a phrase like "the sum of $k$ and its reciprocal," you need to break it down into its mathematical components. The key is understanding what "reciprocal" means and how "sum" translates to mathematical operations. The reciprocal of a number is $\frac{1}{\text{that number}}$ . So the reciprocal of $k$ is $\frac{1}{k}$ . The word "sum" indicates addition, so you're looking for $k + \frac{1}{k}$ , which is choice A. Let's examine why the other options are incorrect. Choice B gives $k-\frac{1}{k}$ , which represents "the difference between $k$ and its reciprocal," not the sum. The minus sign changes the entire meaning of the expression. Choice C, $\frac{k}{k+1}$ , doesn't involve the reciprocal of $k$ at all - instead, it's $k$ divided by $(k+1)$ . Choice D, $\frac{1}{k+k}$ or $\frac{1}{2k}$ , represents the reciprocal of twice $k$ , which is completely different from what the phrase describes. Remember that translation problems require you to identify each component separately: first find what each part means ( $k$ and "its reciprocal" which is $\frac{1}{k}$ ), then apply the operation ("sum" means addition). Always double-check by reading your final expression back in words to ensure it matches the original phrase.
Translate "the square of the quantity $y$ minus 9" into an expression.
Answer: $(y-9)^{2}$ When translating word problems into mathematical expressions, you need to carefully parse the language to identify the order of operations and what gets grouped together. The phrase "the square of the quantity $y$ minus 9" has a crucial structural element: "the quantity $y$ minus 9." The word "quantity" signals that everything following it should be grouped together before any other operations are applied. So first you form the quantity $(y-9)$ , then you square that entire quantity, giving you $(y-9)^2$ . Choice A, $(y-9)^2$ , correctly captures this structure by grouping $y-9$ in parentheses first, then applying the square to the entire quantity. Choice B, $y^2-9$ , represents " $y$ squared minus 9," which would be written as "the square of $y$ , minus 9" in words. This misses the grouping indicated by "quantity." Choice C, $y^2+9$ , represents " $y$ squared plus 9," which doesn't match our expression at all since we need subtraction, not addition. Choice D, $(9-y)^2$ , represents "the square of the quantity 9 minus $y$ ," which reverses the order of the subtraction. While this expression has the same value as choice A when squared (since $(-1)^2 = 1$ ), it doesn't match the specific wording given. Strategy tip: When you see "the [operation] of the quantity [expression]," always use parentheses to group the entire expression first, then apply the operation to that grouped quantity.
Which expression matches "four fewer than the square of $x$ "?
Answer: $x^{2}-4$ When you encounter word problems asking you to translate English phrases into mathematical expressions, you need to carefully parse each part of the phrase and convert it step by step. Let's break down "four fewer than the square of $x$ ": - "The square of $x$ " means $x^2$ - "Four fewer than" means we subtract 4 from whatever comes after this phrase - So we take $x^2$ and subtract 4 from it: $x^2 - 4$ This confirms that choice A is correct. Now let's see why the other options are wrong: Choice B, $(x-4)^2$ , represents "the square of four less than $x$ " or "the square of the quantity $x$ minus 4." This changes the order of operations completely—we're squaring the result of $x-4$ , not subtracting from $x^2$ . Choice C, $4-x^2$ , means "four minus the square of $x$ " or "four decreased by $x^2$ ." This reverses the subtraction, giving us the opposite of what we want. Choice D, $x^2+4$ , represents "four more than the square of $x$ " or "the square of $x$ plus four." This uses addition instead of subtraction. The key strategy here is to identify the main quantity first ( $x^2$ ), then apply the operation ("four fewer than" = subtract 4). Watch out for phrases like "fewer than," "less than," or "decreased by"—these signal subtraction where the main quantity comes first, followed by minus the specified amount.

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