Pre-Algebra · Class Pack · Class 11 · Grades 5–6

Pre-Algebra Class 11: Variables and Algebraic Expressions — Translating Words — 8 SOMATH Posters, Theory & 40 Practice Questions

A complete, illustrated grade 5–6 walk through variables vs constants, translating addition, subtraction, multiplication, and division words, multi-step phrases, and using parentheses to keep groups together. Eight SOMATH posters, full theory under each one, and 40 practice questions with click-to-reveal step-by-step solutions — plus a bonus set of 20 real-life word problems with hidden step-by-step answers in increasing difficulty. Class 11 of the SOMATH Pre-Algebra rolling syllabus — Upper West Side, NYC.

· By SOMATH · Class 11 of the Pre-Algebra rolling syllabus · (646) 668-6151 · 226 W 79th St, New York, NY 10024

Short answer

To translate words into an algebraic expression, use a variable (a letter like x, n, or y) for the unknown number and a constant for any specific number. Then match key words to operations: sum, plus, more than, increased by, total mean +; minus, difference, decreased by, less than, fewer than, subtracted from mean −; product, times, twice, triple, of mean ×; quotient, divided by, ratio, per, one-half of mean ÷. Watch the order-flip trap: less than and subtracted from put the second number first (so “5 less than n” is n − 5, not 5 − n). Use parentheses when words like the sum of, the difference of, the quantity group two things together (so “twice the sum of x and 4” is 2(x + 4), not 2x + 4).

SOMATH Pre-Algebra · Grades 5–6 · Class 11 of 48

This is one class from our Pre-Algebra course

SOMATH's Pre-Algebra course meets in person on the Upper West Side (226 W 79th St). It's a 48-class rolling syllabus for grades 5–6 covering integers, fractions, decimals, ratios, percent, and the transition into algebraic thinking. Small in-person groups of up to 6 students. Every family starts with a free in-person evaluation.

See the full Pre-Algebra course →   Book a free evaluation  ·  (646) 668-6151

What's inside this class

  1. 1. Variables and constants
  2. 2. Translating addition words
  3. 3. Translating subtraction words
  4. 4. Translating multiplication words
  5. 5. Translating division words
  6. 6. Translating multi-step phrases
  7. 7. Using parentheses
  8. 8. Translating words into algebraic expressions
  9. Bonus: 20 word problems (Easy → Hard → Challenge)

1. Variables and constants

Poster 1 of 8 — Pre-Algebra Class 11

SOMATH poster 1: variables and constants. A variable represents an unknown or changing number. A constant is a fixed number that does not change. Example: in x + 5, the x is a variable (can represent any number) and the 5 is a constant (always has the same value).

A variable stands for an unknown number. A constant is a fixed number.

Every algebraic expression is built from two kinds of pieces:

  • A variable is a letter (usually x, y, n, a, m, or similar) that stands for an unknown or changing number. Its value can change — that's why we use a letter instead of a specific number.
  • A constant is a specific, fixed number whose value never changes: 5, 12, 100, 3.14, 1/2.

In the expression x + 5:

  • The x is a variable — it could be 1, 100, or any other number.
  • The 5 is a constant — it is always exactly 5.

Two more names you'll hear in the same breath:

  • Coefficient — the number multiplied by a variable, like the 3 in 3x. If a variable has no visible coefficient (like plain x), the coefficient is 1.
  • Term — a single piece of an expression separated by + or −. In 3x + 5 there are two terms: 3x and 5.
Big idea: Variables can change. Constants stay the same. Algebra is the language we use to write true statements about numbers we don't yet know.

5 practice questions with hidden solutions

Q1.1 In the expression 7y − 4, list every variable and every constant.
Variable: y. Constants: 7 and 4. (The 7 is also called the coefficient of y.)
Q1.2 True or false: in the expression n + 12, the letter n always equals the same number.
False. A variable can take different values in different problems. Only constants always have the same value.
Q1.3 Which pieces of 2a + 9 − b are variables and which are constants?
Variables: a and b. Constants: 2 and 9. (The 2 is the coefficient of a; b has an invisible coefficient of 1.)
Q1.4 Write an algebraic expression that uses one variable (call it x) and one constant (7), joined by an addition sign.
x + 7 (or 7 + x — both are correct because addition doesn't care about order).
Q1.5 A pencil costs 25¢ and Maya buys x pencils. Write an expression for the total cost in cents. Identify the variable and the constant.
Total cost = 25x cents. Variable: x (number of pencils, which can change). Constant: 25 (price per pencil, fixed).

2. Translating addition words

Poster 2 of 8 — Pre-Algebra Class 11

SOMATH poster 2: words that mean addition — sum, plus, increased by, more than, added to, the total of. Examples: 7 more than x becomes x + 7. The sum of a and 5 becomes a + 5. x increased by 12 becomes x + 12. A number plus 3 becomes n + 3. The total of y and 8 becomes y + 8.

Words that mean + (addition)

When you see any of these words in a phrase, use a + sign in your expression:

Word / phraseExample phraseExpression
sum ofthe sum of a and 5a + 5
plusa number plus 3n + 3
more than7 more than xx + 7
increased byx increased by 12x + 12
added to4 added to yy + 4
the total ofthe total of y and 8y + 8

Good news about addition: the order doesn't matter. x + 7 and 7 + x represent the same number, so both are correct translations of “7 more than x.” That's the commutative property of addition. Order only matters for subtraction and division (coming up).

Convention: Even though 7 + x and x + 7 are equal, in algebra we usually write the variable first when we start with addition: x + 7. It reads more naturally.

5 practice questions with hidden solutions

Q2.1 Translate: “the sum of x and 9.”
x + 9. The word “sum” = +.
Q2.2 Translate: “15 more than a number.” Use n for the unknown.
n + 15. Addition doesn't flip the order — it doesn't matter that the 15 came first in the sentence.
Q2.3 Translate: “y increased by 6.”
y + 6.
Q2.4 Translate: “the total of a number and twenty.” Use x.
x + 20.
Q2.5 Two numbers add to give x and 11. Write two equivalent expressions and explain why they are the same.
x + 11 and 11 + x. Both equal the same value because of the commutative property of addition — changing the order of the two things you're adding doesn't change the sum.

3. Translating subtraction words

Poster 3 of 8 — Pre-Algebra Class 11

SOMATH poster 3: words that mean subtraction — difference, minus, decreased by, less than, fewer than, subtracted from. Examples: x minus 4 becomes x − 4. The difference of a and 3 becomes a − 3. y decreased by 6 becomes y − 6. 5 less than n becomes n − 5. 2 subtracted from z becomes z − 2. Warning: 'less than' and 'subtracted from' flip the order.

Words that mean − (subtraction)

These words all mean subtract — but some of them flip the order:

Word / phraseExample phraseExpressionOrder
minusx minus 4x − 4Normal
difference ofthe difference of a and 3a − 3Normal
decreased byy decreased by 6y − 6Normal
less than5 less than nn − 5Flipped!
fewer than3 fewer than xx − 3Flipped!
subtracted from2 subtracted from zz − 2Flipped!

The order-flip trap. The phrases less than, fewer than, and subtracted from put the numbers in the opposite order from how they appear in the sentence. Read carefully:

  • “5 less than n” means start with n, and take 5 away. So it's n − 5, NOT 5 − n.
  • “2 subtracted from z” means start with z, subtract 2. So it's z − 2, NOT 2 − z.

Trick: the word than or from tells you what to start with. Whatever comes right after than or from goes first in the expression.

Why the order matters: Subtraction is not commutative. 7 − 3 = 4, but 3 − 7 = −4. So “5 less than n” and “n less than 5” are completely different expressions.

5 practice questions with hidden solutions

Q3.1 Translate: “x minus 8.”
x − 8. Normal order — “minus” does not flip.
Q3.2 Translate: “9 less than a number.” Use n.
n − 9. “Less than” flips the order — start with n, then subtract 9. NOT 9 − n.
Q3.3 Translate: “the difference of y and 12.”
y − 12. “Difference of” keeps normal order.
Q3.4 Translate: “7 subtracted from z.”
z − 7. “Subtracted from” flips the order — the word from tells you to start with what comes right after it (that's z). NOT 7 − z.
Q3.5 A student writes “3 less than x” as 3 − x. What is the correct expression, and what mistake did the student make?
Correct: x − 3. The student wrote the numbers in the same order they appeared in the sentence — but “less than” flips the order. The word after “than” (x) is what you start with.

4. Translating multiplication words

Poster 4 of 8 — Pre-Algebra Class 11

SOMATH poster 4: words that mean multiplication — product, times, multiplied by, twice, double, triple, four times, of. Examples: three times a number becomes 3x. The product of n and 6 becomes 6n. Twice the value of y becomes 2y. Five times a number plus 4 becomes 5x + 4. The product of (a + 3) and 2 becomes 2(a + 3).

Words that mean × (multiplication)

Word / phraseExample phraseExpression
product ofthe product of n and 66n
timesthree times a number3x
multiplied bya multiplied by 55a
twice / doubletwice the value of y2y
tripletriple a number3x
four times, five times…four times b4b
of (with a fraction)one-half of a number(1/2)x or x/2

Algebra shorthand. Between a number and a variable, we drop the multiplication sign completely. We write 3x, not 3 × x or 3 · x. The number goes in front of the variable (called the coefficient).

Between two numbers, we cannot drop the sign (3 · 5 is not the same as 35). Between two variables, we can drop it (xy means x · y).

Watch: “the product of n and 6” is written 6n — the number goes first as the coefficient, even though it appeared second in the sentence. That's just a convention: x3 looks like a typo, so we always write 3x.

Multiplication is commutative: 3 · x = x · 3, so “3 times x” and “x times 3” are the same expression. Order doesn't change the answer — unlike subtraction.

5 practice questions with hidden solutions

Q4.1 Translate: “the product of 7 and a number.” Use x.
7x.
Q4.2 Translate: “twice a number decreased by 3.” Use n.
2n − 3. Two steps: “twice a number” = 2n, then “decreased by 3” = − 3.
Q4.3 Translate: “triple y plus 8.”
3y + 8. “Triple” = × 3.
Q4.4 Translate: “four times z.”
4z. Number in front, no times sign.
Q4.5 A student writes “the product of x and 5” as x5. Is this correct? If not, what is the standard way to write it?
Technically it means the same thing (5 · x), but x5 is not the standard convention — it reads like a typo and can be confused with an exponent. Write it as 5x. Coefficient (the number) always goes in front of the variable.

5. Translating division words

Poster 5 of 8 — Pre-Algebra Class 11

SOMATH poster 5: words that mean division — quotient, divided by, ratio, per, one-half of, one-third of, out of, per each. Examples: the quotient of x and 4 becomes x/4. A number divided by 7 becomes n/7. One-half of a number becomes (1/2)x. The ratio of a to 5 becomes a/5. x divided by (y + 3) becomes x/(y+3).

Words that mean ÷ (division)

Word / phraseExample phraseExpression
quotient ofthe quotient of x and 4x/4
divided bya number divided by 7n/7
ratio of … to …the ratio of a to 5a/5
per, per eachmiles per hourm/h
one-half of, one-third ofone-half of a number(1/2)x or x/2
out of3 out of n3/n

Algebra shorthand. Instead of the ÷ sign, we write division as a fraction. So “x divided by 4” is written x/4 — the first thing goes on top, the second goes on the bottom.

Order matters. Division is not commutative: 12/3 = 4 but 3/12 = 1/4. So “the quotient of x and 4” is x/4, not 4/x. Whatever is named first goes on top.

Grouping. When a phrase inside the division has its own operation, wrap it in parentheses so the whole thing goes together: “x divided by (y + 3)” is x / (y + 3), meaning add y and 3 first, then divide x by that result.

5 practice questions with hidden solutions

Q5.1 Translate: “the quotient of y and 8.”
y/8. First thing named goes on top.
Q5.2 Translate: “one-third of a number.” Use x.
(1/3)x or equivalently x/3. The word “of” with a fraction means multiply.
Q5.3 Translate: “the ratio of a to b.”
a/b. Whatever is named first goes on top.
Q5.4 Translate: “5 divided by the sum of x and 2.”
5 / (x + 2). The 5 goes on top; the sum x + 2 goes on the bottom — wrap it in parentheses because it is one grouped quantity.
Q5.5 A car travels m miles in h hours. Write an expression for its speed (miles per hour).
m / h. The word “per” means divide — miles go on top, hours on the bottom.

6. Translating multi-step phrases

Poster 6 of 8 — Pre-Algebra Class 11

SOMATH poster 6: multi-step phrases use more than one operation. Words to notice: plus, minus, times, divided by, more than, less than, sum of, product of. Examples: Three times a number plus 5 becomes 3x + 5. 7 less than twice y becomes 2y − 7. The sum of a number and 4, then divided by 2 becomes (x + 4)/2. 5 more than n divided by 3 becomes n/3 + 5. The product of 2 and a number, minus 1 becomes 2x − 1.

Multi-step phrases — go one step at a time

Some phrases use more than one operation. The trick is to slow down and read one step at a time. Follow the order the words appear, but pause on the two things that flip:

  1. “less than” and “subtracted from” still flip the order.
  2. “the sum of … and …”, “the difference of … and …”, “the product of … and …”, “the quantity …” group two things — do that operation first, in parentheses if needed.

Examples from the poster:

  • “Three times a number plus 5”: first “three times a number” = 3x, then “plus 5” = + 5. Answer: 3x + 5.
  • “7 less than twice y”: first “twice y” = 2y, then “7 less than” flips the order — subtract 7 from 2y. Answer: 2y − 7 (NOT 7 − 2y).
  • “The sum of a number and 4, then divided by 2”: “the sum of” groups x + 4 in parentheses, then divide by 2. Answer: (x + 4)/2.
  • “5 more than n divided by 3”: careful — this means “n divided by 3” first (n/3), then add 5. Answer: n/3 + 5. (If the sentence said “5 more than n, divided by 3” with a comma, it would be different.)
  • “The product of 2 and a number, minus 1”: “product of 2 and x” = 2x, then “minus 1” = − 1. Answer: 2x − 1.
Study move: circle each key word (times, plus, sum of, less than, divided by, product of). Write the operation above it. Then translate one chunk at a time, left to right, honoring the two flip cases.

5 practice questions with hidden solutions

Q6.1 Translate: “four times a number, plus 9.” Use x.
4x + 9.
Q6.2 Translate: “3 less than five times n.”
5n − 3. First “five times n” = 5n. Then “3 less than” flips the order: subtract 3 from 5n. NOT 3 − 5n.
Q6.3 Translate: “the quotient of a number and 4, plus 6.” Use x.
x/4 + 6.
Q6.4 Translate: “half of a number, decreased by 5.” Use n.
n/2 − 5 (or (1/2)n − 5).
Q6.5 Translate: “the difference of twice a number and 7.” Use x.
2x − 7. “The difference of” keeps normal order — first thing named (2x) comes first, then subtract the second thing (7).

7. Using parentheses

Poster 7 of 8 — Pre-Algebra Class 11

SOMATH poster 7: parentheses show that numbers or variables belong together as one group. Words to notice: sum of, difference of, the quantity, the total of, the whole expression, group together. Examples: Twice the sum of x and 4 becomes 2(x + 4). Three times the difference of n and 2 becomes 3(n − 2). The quantity y + 5, divided by 2 becomes (y+5)/2. Four times the sum of a and 1 becomes 4(a + 1). The total of m and 6, then multiplied by 3 becomes 3(m + 6).

Parentheses = “keep these together, do them first”

Parentheses are the algebra way to say “this whole thing is one group.” They keep parts of an expression together and tell you what to do first under the order of operations.

Words that signal parentheses:

  • the sum of … and …
  • the difference of … and …
  • the product of … and …
  • the quantity …
  • the total of …
  • the whole expression, group together

Examples from the poster:

PhraseExpression
Twice the sum of x and 42(x + 4)
Three times the difference of n and 23(n − 2)
The quantity y + 5, divided by 2(y + 5) / 2
Four times the sum of a and 14(a + 1)
The total of m and 6, then multiplied by 33(m + 6)

Common mistake: writing “twice the sum of x and 4” as 2x + 4. That is WRONG. 2x + 4 doubles only the x. What the phrase actually asks for is to add first (x + 4), then double the whole thing — so 2(x + 4). Try x = 3: 2(3 + 4) = 14, but 2(3) + 4 = 10. Different answers.

Rule of thumb: if the phrase pairs a grouping word (sum of, difference of, quantity, total of) with any other operation (like “twice”, “three times”, “divided by”), you almost certainly need parentheses.

5 practice questions with hidden solutions

Q7.1 Translate: “three times the sum of x and 5.”
3(x + 5). Add first, then multiply.
Q7.2 Translate: “the quantity n minus 2, then divided by 4.”
(n − 2) / 4.
Q7.3 Translate: “twice the difference of a number and 7.” Use y.
2(y − 7).
Q7.4 Are “twice the sum of x and 4” and “twice x, plus 4” the same expression? Show why.
No. “Twice the sum of x and 4” = 2(x + 4). “Twice x, plus 4” = 2x + 4. Try x = 5: the first gives 2(9) = 18, the second gives 10 + 4 = 14. Different values, so different expressions.
Q7.5 Translate: “the total of m and 8, then multiplied by 5.”
5(m + 8).

8. Translating words into algebraic expressions

Poster 8 of 8 — Pre-Algebra Class 11

SOMATH poster 8: Translating words into algebraic expressions — putting it all together. Use everything you have learned about addition, subtraction, multiplication, division, and parentheses to translate complete verbal statements into algebraic expressions. Read the words carefully, identify the operations, and write the expression. Examples: Five less than twice a number becomes 2x − 5. Three times the sum of a number and 7 becomes 3(x + 7). Half the difference of a number and 4 becomes (1/2)(x − 4). Four more than the quotient of a number and 3 becomes x/3 + 4. The product of 2 and the quantity x + 5 becomes 2(x + 5). The same words can be used in different ways — practice helps you get it right.

Putting it all together

Now you have every tool you need. The 4-step method works for any word phrase, no matter how long:

  1. Read the phrase. Look for the unknown — assign it a variable (usually x, n, or whatever makes sense).
  2. Identify the operations. Underline every key word: sum, plus, more than, minus, less than, times, divided by, product of, quotient of, twice, half of, the quantity…
  3. Watch for the two order-flippers — “less than” and “subtracted from” — and for grouping words that need parentheses.
  4. Write the expression one step at a time, left to right, honoring the flips and the parentheses.

Examples from the poster:

PhraseExpression
Five less than twice a number2x − 5
Three times the sum of a number and 73(x + 7)
Half the difference of a number and 4(1/2)(x − 4)
Four more than the quotient of a number and 3x/3 + 4
The product of 2 and the quantity x + 52(x + 5)

Sanity-check trick. When you're done, plug in a small number for the variable and check that the expression matches what the words describe. If “5 less than x” is supposed to be x − 5, try x = 10: x − 5 = 5. That matches “5 less than 10”. If you had written 5 − x, plugging in 10 gives −5 — a red flag.

Class 12 (next class) puts these expressions to work: evaluating expressions and combining like terms. Class 13 introduces equations — setting two expressions equal to each other and solving for the variable.

5 practice questions with hidden solutions

Q8.1 Translate: “seven more than three times a number.” Use x.
3x + 7.
Q8.2 Translate: “the product of 4 and the sum of x and 2.”
4(x + 2). “The sum of” groups x and 2 together in parentheses, then multiply by 4.
Q8.3 Marta has x dollars. She spends $12 and then triples what is left. Write an expression for how much she has now.
Left after spending: x − 12. Then triple: 3(x − 12). Parentheses matter — we triple the leftover, not just x.
Q8.4 Translate: “six less than half of a number.” Use n.
n/2 − 6 (or (1/2)n − 6). “Half of a number” = n/2. “Six less than” flips the order: subtract 6 from n/2.
Q8.5 A rectangle has length x and width that is 3 less than the length. Write an expression for the perimeter of the rectangle.
Width = x − 3. Perimeter = 2(length) + 2(width) = 2x + 2(x − 3) = 2x + 2(x − 3). That simplifies to 4x − 6, but simplification is Class 12 — for now, the correct translation is 2x + 2(x − 3).

20 word problems — translate then solve

Bonus set of 20 word problems — increasing difficulty (Easy → Medium → Hard → Challenge). Read the words, name the unknown, write the expression, then answer the question. W1–W10 build up from single-operation phrases to multi-step scenarios; W11–W20 are Hard and Challenge tier and demand careful parenthesization, order-of-flip attention, and multi-step reasoning.

How to attack a word problem

  1. Name the unknown. Pick a variable and write exactly what it means (“let x = number of pencils”).
  2. Translate the sentence into an expression. Use everything from Class 11: key words for operations, the order-flip on “less than”/“subtracted from”, parentheses for grouping.
  3. Set up an equation only if the problem tells you what the expression equals. Otherwise, just evaluate.
  4. Solve and check. Plug the answer back in and re-read the sentence to be sure it fits.

20 word problems with hidden step-by-step solutions

W1. Easy. Lucas has x baseball cards. His sister gives him 8 more. Write an expression for the total number of cards Lucas has now, then find the total if x = 15.
Expression: x + 8. “8 more” = + 8. Evaluate at x = 15: 15 + 8 = 23 cards.
W2. Easy. A pack of markers costs $6. Mia buys n packs. Write an expression for the total cost in dollars, then find the cost when n = 4.
Expression: 6n. Cost per pack × number of packs. Evaluate at n = 4: 6(4) = $24.
W3. Easy–Medium. A book has 200 pages. Ana has already read p pages. Write an expression for the number of pages she has left, then find how many are left when p = 78.
Expression: 200 − p. Total minus pages already read. Evaluate at p = 78: 200 − 78 = 122 pages.
W4. Medium. Sam is y years old. His mom is 3 years less than four times Sam’s age. Write an expression for the mom’s age, then find her age when y = 11.
Translate: “four times Sam’s age” = 4y. “3 years less than” flips the order: subtract 3 from 4y, so expression: 4y − 3 (NOT 3 − 4y). Evaluate at y = 11: 4(11) − 3 = 44 − 3 = 41 years old.
W5. Medium. A pizza is cut into s equal slices. Six friends share the pizza equally. Write an expression for the number of slices each friend gets, then find the answer when s = 24.
Expression: s / 6. “Shared equally” among 6 = divide by 6. Evaluate at s = 24: 24 / 6 = 4 slices each.
W6. Medium. A movie theater charges $12 per ticket plus a flat $3 booking fee for the whole order. Write an expression for the total cost of t tickets, then find the total for 5 tickets.
Expression: 12t + 3. $12 per ticket × t tickets, then add the one-time $3 fee. Evaluate at t = 5: 12(5) + 3 = 60 + 3 = $63. Notice we did NOT write 12(t + 3) — the $3 fee is only charged once, not per ticket.
W7. Medium–Hard. Maya’s uncle earns d dollars. He puts half of what he earns into savings, then spends $50 of what’s left on groceries. Write an expression for how much money he has left after groceries, then evaluate when d = 800.
Translate step by step: half of what he earns = d/2, so half is left over: d/2. Then he spends $50 of that leftover, so expression: d/2 − 50. Evaluate at d = 800: 800/2 − 50 = 400 − 50 = $350.
W8. Hard. A rectangle has length x centimeters and width that is 4 cm less than the length. Write an expression for the perimeter of the rectangle, then find the perimeter when x = 10.
Width: ‘4 less than the length’ flips the order — x − 4. Perimeter = 2(length) + 2(width) = 2x + 2(x − 4), so expression: 2x + 2(x − 4) (which simplifies to 4x − 8, but simplification is Class 12). Evaluate at x = 10: 2(10) + 2(10 − 4) = 20 + 2(6) = 20 + 12 = 32 cm. Sanity check with 4x − 8: 4(10) − 8 = 32. ✓
W9. Hard. A summer camp charges $75 per child. A family of c children gets $10 off the whole total (not per child). Write an expression for what the family pays. Then find the price for a family of 3 children, and separately for a family of 5 children.
Full price: 75c. $10 off the total: subtract 10 from the whole amount, so expression: 75c − 10. This is different from 75(c − 10), which would mean “$10 off each child” — not what the sentence says. For c = 3: 75(3) − 10 = 225 − 10 = $215. For c = 5: 75(5) − 10 = 375 − 10 = $365.
W10. Challenge. A phone plan costs $40 per month plus $0.15 per minute of international calls. In one month, Elena talks m minutes internationally. (a) Write an expression for her total bill for the month. (b) Write an expression for the average cost per minute of international calls (total bill divided by minutes). (c) Evaluate the total bill for m = 200 minutes, and find the average cost per minute.
(a) Total bill: flat $40 plus $0.15 for each minute = 40 + 0.15m dollars.
(b) Average cost per minute: total bill divided by minutes — the whole bill is one quantity, so wrap it in parentheses: (40 + 0.15m) / m dollars per minute.
(c) At m = 200: Total bill = 40 + 0.15(200) = 40 + 30 = $70. Average per minute = 70 / 200 = $0.35 per minute. Notice the average is more than $0.15 — that’s because the $40 flat fee gets spread over the 200 minutes. As m gets bigger, the average per minute drops toward $0.15.
W11. Hard. A rectangle’s length is x centimeters. Its width is 5 centimeters more than half the length. (a) Write an expression for the width. (b) Write an expression for the perimeter. (c) Evaluate the perimeter at x = 12.
(a) Width: “half the length” is x/2. “5 more than half the length” adds 5 to that half: width = x/2 + 5.
(b) Perimeter: P = 2(length) + 2(width) = 2x + 2(x/2 + 5). Distribute 2 across the parentheses: 2x + x + 10 = 3x + 10.
(c) At x = 12: 3(12) + 10 = 36 + 10 = 46 cm. Sanity-check: length 12, width 12/2 + 5 = 11, perimeter 2(12) + 2(11) = 24 + 22 = 46. ✓
W12. Hard. A store marks up every item by 20% above cost, then charges 8% sales tax on the marked-up price. Let c be the cost. (a) Write an expression for the price after markup. (b) Write an expression for the final price the customer pays (markup + tax). (c) Evaluate at c = $50.
(a) After markup: 20% above cost means add 0.20c to c, or multiply by 1.20: 1.20c (equivalent to c + 0.20c).
(b) Final price: tax is 8% of the marked-up price, so multiply the whole marked-up quantity by 1.08. Wrap the marked-up expression in parentheses: 1.08(1.20c) = 1.296c.
(c) At c = 50: 1.296(50) = $64.80. Common mistake: adding 20% and 8% to get 28% and computing 1.28c = $64.00 — wrong, because the 8% tax is applied to the marked-up price, not the original cost.
W13. Hard. Two consecutive whole numbers add up to some total. Let n be the smaller number. (a) Write an expression for the next consecutive whole number. (b) Write an expression for their sum. (c) Write an expression for the sum of their squares. (d) Evaluate parts (b) and (c) at n = 7.
(a) Next consecutive number: n + 1.
(b) Sum: n + (n + 1) = 2n + 1. This is always odd — a nice number-theory nugget.
(c) Sum of their squares: square each first, then add: n2 + (n + 1)2. Careful: this is not the same as (n + n + 1)2.
(d) At n = 7: Sum = 2(7) + 1 = 15. Sum of squares = 72 + 82 = 49 + 64 = 113.
W14. Hard. A gym charges a $60 one-time signup fee plus $35 per month. Let m be the number of months of membership. (a) Write an expression for the total cost after m months. (b) Write an expression for the average cost per month (total divided by months). (c) Evaluate the average at m = 12 and m = 24. What do you notice as m grows?
(a) Total cost: flat $60 plus $35 for each month = 60 + 35m dollars.
(b) Average per month: divide the whole total by m, so parenthesize: (60 + 35m) / m dollars per month. Do not write 60 + 35m/m = 60 + 35 — that would only be dividing the last term.
(c) At m = 12: (60 + 420)/12 = 480/12 = $40/month. At m = 24: (60 + 840)/24 = 900/24 = $37.50/month. As m grows, the $60 signup fee gets spread thinner and the average drifts toward the pure monthly rate of $35.
W15. Hard. Ana is a years old. Ben is 2 years older than Ana. Carla’s age is 3 less than twice Ben’s age. (a) Write an expression for Ben’s age. (b) Write an expression for Carla’s age, fully simplified. (c) Write an expression for the sum of all three ages. (d) Evaluate at a = 9.
(a) Ben: a + 2.
(b) Carla: “twice Ben” is 2(a + 2) — you must parenthesize before doubling. “3 less than” flips the order and subtracts 3: 2(a + 2) − 3 = 2a + 4 − 3 = 2a + 1.
(c) Sum: a + (a + 2) + (2a + 1) = 4a + 3.
(d) At a = 9: Ana 9, Ben 11, Carla 2(9) + 1 = 19. Sum = 4(9) + 3 = 39. Check: 9 + 11 + 19 = 39. ✓
W16. Challenge. A bank account earns simple interest of 4% per year on the original deposit. You deposit P dollars and leave it for t years. (a) Write an expression for the interest earned. (b) Write an expression for the total balance after t years. (c) Evaluate for P = $500 and t = 3.
(a) Interest: 4% of P, once per year, for t years: 0.04Pt (which is P · 0.04 · t).
(b) Total balance: original deposit plus the interest = P + 0.04Pt, which factors as P(1 + 0.04t). Both forms are correct.
(c) At P = 500, t = 3: Interest = 0.04(500)(3) = $60. Total = 500 + 60 = $560, or 500(1 + 0.12) = 500(1.12) = 560. ✓ Common mistake: writing 0.04(P + t) — the 4% is a rate on the money, not on some sum of dollars and years.
W17. Challenge. A movie theater sells adult tickets for $14 and child tickets for $9. A family buys a adult tickets and c child tickets, then uses a coupon for $5 off the entire order (not per ticket). (a) Write an expression for the total the family pays. (b) If they also want popcorn at $6 per person for everyone in the family, write a new expression for the grand total. (c) Evaluate at a = 2 and c = 3.
(a) After coupon: tickets cost 14a + 9c. “$5 off the entire order” subtracts 5 from the whole ticket total, so no per-ticket discount: 14a + 9c − 5. This is not 14(a − 5) or 9(c − 5).
(b) With popcorn: everyone in the family = a + c people, at $6 each: 6(a + c). Add to the after-coupon total: 14a + 9c − 5 + 6(a + c) = 14a + 9c − 5 + 6a + 6c = 20a + 15c − 5.
(c) At a = 2, c = 3: 20(2) + 15(3) − 5 = 40 + 45 − 5 = $80. Check the long way: tickets 14(2) + 9(3) = 28 + 27 = 55, minus $5 coupon = 50, plus popcorn 6(5) = 30, grand total = 80. ✓
W18. Challenge. A car travels d miles from home to a campground, then returns home along the same route. The trip out is at an average of 50 mph; the trip back is at 60 mph. (a) Write an expression for the time (in hours) for the trip out. (b) Write an expression for the time for the trip back. (c) Write an expression for the total time for the round trip. (d) Evaluate at d = 150.
(a) Time out: distance divided by speed, so d / 50 hours.
(b) Time back: d / 60 hours.
(c) Total time: d/50 + d/60. To simplify with a common denominator of 300: 6d/300 + 5d/300 = 11d / 300 hours.
(d) At d = 150: 150/50 + 150/60 = 3 + 2.5 = 5.5 hours. Or 11(150)/300 = 1650/300 = 5.5. ✓ Common mistake: averaging the speeds (50 + 60)/2 = 55 mph and using 2d/55 — wrong, because the car spends more time at the slower speed, so the true average is less than 55.
W19. Challenge. A rectangular garden has length L feet and width W feet. A path of uniform width 2 feet is added around the outside of the garden. (a) Write an expression for the outer length (garden + path). (b) Write an expression for the outer width. (c) Write an expression for the area of just the path (outer area minus garden area). (d) Evaluate at L = 10 and W = 6.
(a) Outer length: 2 feet of path on each end, so L + 4.
(b) Outer width: same reasoning: W + 4.
(c) Path area: outer area − garden area = (L + 4)(W + 4) − LW. Expand (L + 4)(W + 4) = LW + 4L + 4W + 16. Subtracting LW leaves 4L + 4W + 16, which factors as 4(L + W + 4).
(d) At L = 10, W = 6: 4(10) + 4(6) + 16 = 40 + 24 + 16 = 80 square feet. Sanity-check: outer area 14 × 10 = 140, garden area 10 × 6 = 60, path = 140 − 60 = 80. ✓
W20. Challenge. A student scored x, y, and z on three tests. Each test is out of 100 points. The teacher drops the lowest score and averages the remaining two, but only if that average is at least 85; otherwise she averages all three. (a) Assuming z is the lowest score, write an expression for the “drop-the-lowest” average. (b) Write an expression for the plain three-test average. (c) At x = 92, y = 88, z = 60, which rule applies and what is the final grade? (d) At x = 80, y = 78, z = 40, which rule applies and what is the final grade?
(a) Drop-the-lowest average (with z lowest): the top two scores are x and y. Their average is the sum divided by 2, so wrap the sum in parentheses: (x + y) / 2.
(b) Plain three-test average: (x + y + z) / 3. Again, parenthesize the whole numerator — x + y + z/3 would only divide the last score by 3.
(c) x = 92, y = 88, z = 60: drop-lowest average = (92 + 88)/2 = 180/2 = 90. Since 90 ≥ 85, the drop-the-lowest rule applies. Final grade = 90.
(d) x = 80, y = 78, z = 40: drop-lowest average = (80 + 78)/2 = 158/2 = 79. Since 79 < 85, we fall back to the plain three-test average: (80 + 78 + 40)/3 = 198/3 = 66. Final grade = 66. This is a real algebra trap: whether you can even use the friendlier rule depends on the value of the expression you just wrote, not just on the scores themselves.

Ready to see how SOMATH teaches this in class?

Pre-Algebra meets in person on the Upper West Side. Small groups of up to 6 students. Real teachers, not tutors or apps. Every family starts with a free in-person evaluation so we can place your child at the right point in the 48-class syllabus.

SOMATH · 226 W 79th St, 1st Floor, New York, NY 10024 · (646) 668-6151 · See class schedule · Pre-Algebra course page

Frequently asked questions

What is the difference between a variable and a constant?
A variable is a letter (like x, y, n, a) that stands for an unknown or changing number. A constant is a specific number whose value is fixed and never changes. In x + 5, x is the variable and 5 is the constant.
How do you translate "less than" and "subtracted from" in algebra?
Both flip the order. "5 less than n" is n − 5, NOT 5 − n. "2 subtracted from z" is z − 2, NOT 2 − z. The word "than" or "from" tells you what to start with.
What words mean addition in algebra?
Sum, plus, added to, more than, increased by, and the total of all mean +. Example: "the sum of a and 5" = a + 5.
What words mean subtraction in algebra?
Minus, difference, decreased by, less than, fewer than, subtracted from. "Less than" and "subtracted from" flip the order — the others do not.
What words mean multiplication in algebra?
Product, times, multiplied by, twice, double, triple, four times, and "of" (with a fraction) all mean ×. We drop the multiplication sign between a number and a variable and write the number first, so 3 · x is written 3x.
What words mean division in algebra?
Quotient, divided by, ratio, per, per each, out of, one-half of, one-third of. Division is written as a fraction — the first thing named goes on top.
When do you use parentheses in an algebraic expression?
When words group two things together — "the sum of", "the difference of", "the quantity", "the total of" — pair those with another operation like "twice" or "divided by" and you almost always need parentheses. "Twice the sum of x and 4" is 2(x + 4), not 2x + 4.
How do you translate multi-step word phrases into algebra?
Read one operation at a time. Circle each key word, decide the operation, write it left to right — but pause on "less than", "subtracted from", and grouping words. Example: "7 less than twice y" has two steps — "twice y" = 2y, then "7 less than" flips: 2y − 7.

Related SOMATH class packs and posts