SHSAT Math Quiz: Words To Expressions

20 questions · exam conditions

0:00

Words To ExpressionsQuestion 1 of 20

Which expression matches "triple a number $n$ decreased by four times the same number"?

3n-4n

4n-3n

3(n-4)

n(3-4)

← Back to quizzes

SHSAT Math Quiz

SHSAT Math Quiz: Words To Expressions

Practice Words To Expressions in SHSAT 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 Words To Expressions, giving you a quick way to practice the rules, question types, and explanations that matter most for SHSAT 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.

All questions

Question 1

Which expression matches "triple a number $n$ decreased by four times the same number"?

$3n-4n$ (correct answer)
$4n-3n$
$3(n-4)$
$n(3-4)$

Explanation: When you encounter word problems asking you to translate phrases into algebraic expressions, work methodically through each part of the description to build your expression step by step. Let's break down "triple a number

n

decreased by four times the same number." The phrase "triple a number

n

" means

3n

. The word "decreased by" signals subtraction, so we subtract what comes next. "Four times the same number" means

4n

(since we're still talking about the number

n

). Putting it together:

3n - 4n

. Choice A (

3n - 4n

) correctly captures this translation. This expression represents three times

n

minus four times

n

, which matches our phrase exactly. Choice B (

4n - 3n

) reverses the order, giving us "four times a number decreased by triple the same number" - the opposite of what we want. Choice C (

3(n-4)

) represents "triple the quantity of a number decreased by four," which would mean

3n - 12

when distributed. This misinterprets where the grouping occurs. Choice D (

n(3-4)

) represents "a number times the quantity three decreased by four," which simplifies to

-n

. This completely misunderstands the phrase structure. Strategy tip: When translating word problems, identify the operations in order as you read left to right, and pay special attention to keywords like "decreased by," "less than," or "reduced by" that indicate subtraction. Always double-check by reading your final expression back in words to see if it matches the original phrase.

Question 2

What algebraic expression represents "the number of minutes $m$ , reduced by 12, divided by 4"?

$\dfrac{m-12}{4}$ (correct answer)
$4(m-12)$
$\dfrac{m}{4}-12$
$\dfrac{12-m}{4}$

Explanation: 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.

Question 3

Translate the phrase "the sum of $k$ and its reciprocal" into an expression.

$k+\dfrac{1}{k}$ (correct answer)
$k-\dfrac{1}{k}$
$\dfrac{k}{k+1}$
$\dfrac{1}{k+k}$

Explanation: 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

\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}

\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.

Question 4

What expression represents the perimeter of a square with side length $x$ centimeters?

$4x$ (correct answer)
$x^{2}$
$2x$
$x+4$

Explanation: 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.

Question 5

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?

$2w(L + W)$
$(L + w)(W + w) - LW$
$(L + 2w)(W + 2w) - LW$ (correct answer)
$4w^2 + 2w(L + W)$

Explanation: When you encounter problems involving shapes within shapes, visualize the total area and subtract what you don't want. This question asks for just the walkway area, not the entire space. The correct approach is to find the total area (pool plus walkway) and subtract the pool's area. The walkway extends

w

feet on all sides of the pool, so the total dimensions become

(L + 2w)

(W + 2w)

. The walkway area equals

(L + 2w)(W + 2w) - LW

, which is answer choice C. Let's examine why the other options fail. Choice A gives

2w(L + W)

, which only accounts for walkway strips along two sides of the pool, missing the corner areas entirely. Choice B uses

(L + w)(W + w) - LW

, incorrectly assuming the walkway extends

w

feet in total rather than

w

feet on each side. This would mean the walkway only goes halfway around the pool. Choice D,

4w^2 + 2w(L + W)

, is actually equivalent to choice C when expanded, but it's presented in a form that obscures the logical approach and could confuse you during the test. The key insight is recognizing that "uniform width

w

" means the walkway is

w

feet wide all around the perimeter. This adds

2w

to each pool dimension (

w

on each side). Always draw a quick sketch for these problems—it helps you visualize whether you're adding

w

2w

to each dimension.

Question 6

Which expression matches "two-thirds of the difference between $p$ and 8"?

$\dfrac{2}{3}(p-8)$ (correct answer)
$\dfrac{2}{3}(8-p)$
$\dfrac{p-8}{2/3}$
$\dfrac{p}{3}-8$

Explanation: When you encounter word problems asking you to translate English phrases into algebraic expressions, work through the phrase step by step, identifying each mathematical operation and the order in which they should be performed. Let's break down "two-thirds of the difference between

p

and 8." Start from the inside out: "the difference between

p

and 8" means

p - 8

(when we say "difference between A and B," we typically mean A minus B in the order given). Next, "two-thirds of" this difference means we multiply the entire difference by

\frac{2}{3}

. This gives us

\frac{2}{3}(p-8)

, which is choice A. Choice B,

\frac{2}{3}(8-p)

, reverses the subtraction order. While "difference" could technically work either way, the conventional interpretation of "difference between

p

and 8" is

p - 8

, not

8 - p

. Choice C,

\frac{p-8}{2/3}

, divides the difference by two-thirds instead of multiplying by it. Remember that "of" in mathematics typically indicates multiplication, not division. Choice D,

\frac{p}{3}-8

, incorrectly applies the fraction only to

p

and subtracts 8 separately. This completely misinterprets the phrase structure—the "two-thirds" should apply to the entire difference, not just to

p

. Strategy tip: When translating word problems, identify grouping words like "of," "the difference between," or "the sum of" first. These tell you what operations to perform and in what order. Always work from the innermost operation outward.

Question 7

Which algebraic expression represents "7 more than twice a number $n$ "?

$2n+7$ (correct answer)
$7n+2$
$2(n-7)$
$7-2n$

Explanation: 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.

Question 8

What is the expression for "half the square of $k$ decreased by 4"?

$\dfrac{k^{2}}{2}-4$ (correct answer)
$\left(\dfrac{k}{2}\right)^{2}-4$
$\dfrac{k^{2}-4}{2}$
$\dfrac{k-4}{2}$

Explanation: 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.

Question 9

Which algebraic expression represents "five less than the product of 4 and $m$ "?

$4m-5$ (correct answer)
$5-4m$
$4(m-5)$
$5(4-m)$

Explanation: 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.

Question 10

The number of bacteria in a culture triples every hour. If there are initially $n$ bacteria, which expression represents the number of bacteria after the culture has tripled $k$ times and then $50$ more bacteria are added?

$3^k \cdot n + 50$ (correct answer)
$3^k \cdot (n + 50)$
$3kn + 50$
$n \cdot 3k + 50$

Explanation: After tripling k times, the bacteria count is

n \cdot 3^k

. Then 50 bacteria are added, giving

3^k \cdot n + 50

. Choice B incorrectly adds the 50 bacteria before the tripling occurs. Choice C uses multiplication instead of exponentiation for the tripling factor. Choice D confuses

3^k

with $3k$.

Question 11

A number is decreased by $20\%$ , then the result is increased by $25\%$ . If the original number is $x$ , which expression represents the final result?

$1.05x$
$x$ (correct answer)
$0.95x$
$1.25x$

Explanation: After decreasing by 20%:

x - 0.20x = 0.80x

. Then increasing by 25%:

0.80x + 0.25(0.80x) = 0.80x(1 + 0.25) = 0.80x \cdot 1.25 = 1.00x = x

. The final result equals the original number. Choice A gives

1.05x

. Choice C gives

0.95x

. Choice D gives

1.25x

, representing only the 25% increase.

Question 12

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?

$3c - n$
$3c - (n - c)$
$2c - n$
$4c - n$ (correct answer)

Explanation: The player gets

3c

points for correct answers and loses

1

point for each of the

(n - c)

incorrect answers. Score =

3c - 1(n - c) = 3c - n + c = 4c - n

. Choice A doesn't account for losing only 1 point per wrong answer. Choice B doesn't simplify the expression. Choice C incorrectly calculates the point values.

Question 13

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?

$\frac{9d}{5}$ (correct answer)
$\frac{9d}{5} + 32$
$\frac{9d + 32}{5}$
$\frac{9(d + 32)}{5}$

Explanation: 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.

Question 14

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?

$6w + 6$ (correct answer)
$6w + 6 + 2x$
$6w + 6 - 2x$
$6w + 6 + 4x$

Explanation: 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

Question 15

A store offers a discount where customers pay only $75\%$ of the marked price, but then a $2 processing fee is added to every purchase. If the marked price of an item is $ $p$ $dollars, which expression represents the total amount a customer pays for$ $n$ $ such items?

$0.75np + 2$
$0.75n(p + 2)$
$0.75np + 2n$ (correct answer)
$n(0.75p + 2)$

Explanation: Each item costs

0.75p

after discount, plus a $2 processing fee per item. For $n

items: total cost =

n(0.75p + 2) = 0.75np + 2n$. Choice A applies only one processing fee for all items instead of per item. Choice B incorrectly applies the discount to the processing fee. Choice D is mathematically equivalent to C but in factored form rather than expanded form.

Question 16

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?

$\frac{d}{3} - 5$
$\frac{d}{6} - 5$
$d - \frac{d}{3} - \frac{d}{3} - 5$
$\frac{d - 15}{6}$ (correct answer)

Explanation: After lunch, Marcus has

\frac{2d}{3}

left. He spends

\frac{1}{2} \cdot \frac{2d}{3} + 5 = \frac{d}{3} + 5

on the book. Money left:

\frac{2d}{3} - (\frac{d}{3} + 5) = \frac{d}{3} - 5 = \frac{d - 15}{6}

. Choice A stops after the book purchase calculation. Choice B incorrectly calculates the remaining amount. Choice C sets up the wrong subtraction.

Question 17

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?

$40 + 0.05t$
$40 + 0.05(t - 500)$ (correct answer)
$40 + 25 + 0.05(t - 500)$
$40 + 0.05(500 - t)$

Explanation: 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)

Question 18

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?

$6 + 2w$ (correct answer)
$4w$
$2 + 6w$
$12 + 2(w - 3)$

Explanation: First find the rate per pound: the cost increases by $6 for 3 additional pounds, so the rate is $2 per pound. Using the 3-pound package: $12 = flat fee + 2(3), so the flat fee is $6. The expression is

6 + 2w

. Choice B omits the flat fee. Choice C reverses the coefficients. Choice D is an equivalent form but uses a more complex reference point.

Question 19

If $n$ represents an integer, which expression represents the next consecutive integer?

$n+1$ (correct answer)
$n-1$
$2n$
$n+2$

Explanation: When you encounter questions about consecutive integers, you're working with numbers that follow each other in order on the number line, like 5, 6, 7 or -3, -2, -1. If

n

represents any integer, then the "next" consecutive integer is simply one more than

n

. Think about specific examples: if

n = 5

, the next consecutive integer is 6, which equals

5 + 1

. If

n = -10

, the next consecutive integer is -9, which equals

-10 + 1

. This pattern holds for any integer value of

n

. Choice A (

n + 1

) correctly represents this relationship by adding 1 to the original integer. Choice B (

n - 1

) gives you the previous consecutive integer, not the next one. If

n = 5

, then

n - 1 = 4

, which comes before 5, not after. Choice C (

2n

) doubles the original integer rather than finding the next one. When

n = 5

, we get

2(5) = 10

, which skips several integers and isn't consecutive to 5. Choice D (

n + 2

) gives you the integer that's two positions ahead. If

n = 5

, then

n + 2 = 7

, but this skips over 6, so it's not the immediate next consecutive integer. Remember: consecutive integers are always exactly 1 apart. Whether you're looking for the next consecutive integer (add 1) or the previous one (subtract 1), you're always moving by just one step on the number line.

Question 20

Translate "the square of the quantity $y$ minus 9" into an expression.

$(y-9)^{2}$ (correct answer)
$y^{2}-9$
$y^{2}+9$
$(9-y)^{2}$

Explanation: 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.