Lesson 1.4 · Unit 1 · A-SSE.1
Turning a sentence into algebra, including the two phrases that reverse
Most of the difficulty in word problems is not the algebra; it is the translation. English has several ways to say each operation, and two common phrases reverse the order of what follows them. Getting those two right removes a large share of the errors students make on modeling problems.
- Define the variable first, in a full sentence with units. Write "let \( n \) be the number of tickets sold", not just "let \( n = \) tickets". A vague definition produces an equation nobody can check.
- Addition: sum, plus, increased by, more than, total, added to.
- Subtraction: difference, minus, decreased by, less, fewer than, take away.
- Multiplication: product, times, of, twice, doubled, each, per.
- Division: quotient, divided by, per, ratio, split equally, out of.
- "Less than" and "subtracted from" reverse the order. "Five less than \( x \)" is \( x - 5 \), not \( 5 - x \). "Five subtracted from \( x \)" is also \( x - 5 \). Read them as instructions about what to take away from what.
- "Of" almost always means multiply, especially with fractions and percentages. Twenty percent of \( x \) is \( 0.20x \).
- Brackets show what a phrase applies to. "Twice the sum of \( x \) and 3" is \( 2(x + 3) \), while "twice \( x \), plus 3" is \( 2x + 3 \). The word "sum" marks a grouped quantity.
Where students lose marks: writing "5 less than \( x \)" as \( 5 - x \). The phrase describes starting at \( x \) and taking 5 away, so it is \( x - 5 \). The same reversal happens with "subtracted from" and with "fewer than", and these three phrases account for a large fraction of setup errors.
The problem. A theater charges $9 per adult ticket and $6 per child ticket, plus a flat $75 fee to run the show. On one night the number of child tickets sold was 12 fewer than twice the number of adult tickets. Write an expression for the theater's profit in terms of the number of adult tickets sold.
Step one: define the variable precisely. Let \( a \) be the number of adult tickets sold on that night. It must be a whole number and cannot be negative, which is worth noting even before any algebra.
Step two: translate the relationship between the two ticket counts. "Twelve fewer than twice the number of adult tickets." Work the phrase from the inside out. "Twice the number of adult tickets" is \( 2a \). "Twelve fewer than" that quantity reverses the order, so it is \( 2a - 12 \), not \( 12 - 2a \).
Step three: check that translation against a number. If 20 adult tickets were sold, twice that is 40, and twelve fewer is 28. Substituting into \( 2a - 12 \) gives \( 2(20) - 12 = 28 \), which matches. Substituting into \( 12 - 2a \) gives \( -28 \), a negative number of children, which is impossible. Testing one value catches the reversal immediately.
Step four: write the revenue from adults. At $9 each, \( a \) adult tickets bring in \( 9a \) dollars.
Step five: write the revenue from children. At $6 each and \( 2a - 12 \) tickets, the revenue is \( 6(2a - 12) \) dollars. The brackets are essential: the $6 applies to the whole count, not just to the \( 2a \).
Step six: assemble total revenue and subtract the cost. Profit is revenue minus cost: \[ P = 9a + 6(2a - 12) - 75 \]
Step seven: simplify. Distribute the 6 across both terms inside the bracket, watching the sign on the second: \[ P = 9a + 12a - 72 - 75 = 21a - 147 \]
Step eight: check the simplified form against the unsimplified one. Take \( a = 20 \). The original gives \( 9(20) + 6(28) - 75 = 180 + 168 - 75 = 273 \). The simplified form gives \( 21(20) - 147 = 420 - 147 = 273 \). They agree, so the simplification is sound. The expression also says something readable: each additional adult ticket adds $21 of profit, because it brings $9 directly and two more child tickets worth $12, and the show starts $147 in the hole.
Problems 1 to 5 are routine. Problems 6 to 10 are multi-step or ask for reasoning.
- Write an expression for "seven more than a number \( n \)".
Show the full solution
\( n + 7 \)
- Write an expression for "four less than a number \( n \)".
Show the full solution
"Less than" reverses the order. \( n - 4 \)
- Write an expression for "the product of 5 and a number \( x \)".
Show the full solution
\( 5x \)
- Write an expression for "twice the sum of \( x \) and 8".
Show the full solution
"Sum" marks a grouped quantity. \( 2(x + 8) \)
- Write an expression for "30 percent of a number \( p \)".
Show the full solution
\( 0.30p \)
- Write expressions for "three less than twice a number" and "twice, three less than a
number", and explain why they differ.
Show the full solution
The first is \( 2n - 3 \): double the number, then take 3 away. The second is \( 2(n - 3) \): take 3 away first, then double, so the 3 is doubled as well. Expanding the second gives \( 2n - 6 \), which is 3 smaller than the first for every value of \( n \). The phrase order tells you which operation happens inside the bracket. \( 2n - 3 \) and \( 2(n - 3) = 2n - 6 \)
- A plumber charges a $65 callout fee plus $48 per hour. Write an expression for the cost
of a job lasting \( h \) hours, and state what each part represents.
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The callout fee is charged once regardless of duration, so it is a constant. The hourly charge accumulates, so it is multiplied by the number of hours. \( C = 48h + 65 \), where \( 48h \) is the labor cost in dollars and 65 is the fixed callout fee in dollars. \( C = 48h + 65 \)
- A rectangle's length is 4 cm more than three times its width. Write expressions for the
length and for the perimeter in terms of the width \( w \).
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"Three times its width" is \( 3w \), and "4 more than" that is \( 3w + 4 \), so the length is \( 3w + 4 \) cm. Perimeter is twice the length plus twice the width: \[ P = 2(3w + 4) + 2w = 6w + 8 + 2w = 8w + 8 \] Length \( 3w + 4 \) cm; perimeter \( 8w + 8 \) cm
- A student writes "8 less than a number" as \( 8 - n \). Show with a specific value why
this is wrong.
Show the full solution
Take the number to be 20. Eight less than 20 is 12. The student's expression gives \( 8 - 20 = -12 \), which is not 12. The correct expression \( n - 8 \) gives \( 20 - 8 = 12 \). The phrase names the amount removed and the thing it is removed from, in that order, so the subtraction is written in the opposite order to the words. \( n - 8 \); at \( n = 20 \) the student's version gives \( -12 \) instead of 12
- A phone plan charges $25 per month plus $0.10 for each text over 200. Write an expression
for the monthly cost when \( t \) texts are sent, assuming \( t \gt 200 \), and explain why
the expression is not simply \( 25 + 0.10t \).
Show the full solution
Only the texts beyond 200 are charged, so the billable count is \( t - 200 \), not \( t \). The cost is therefore \[ C = 25 + 0.10(t - 200) = 25 + 0.10t - 20 = 0.10t + 5 \] Writing \( 25 + 0.10t \) would charge for all texts including the 200 already covered by the monthly fee, overcharging by \( 0.10 \times 200 = \$20 \) every month. Checking at \( t = 200 \) confirms the correct version: it gives \( 0.10(200) + 5 = \$25 \), exactly the base fee, as it should. Note the expression is only valid for \( t \gt 200 \); below that the cost is a flat $25. \( C = 25 + 0.10(t - 200) = 0.10t + 5 \) for \( t \gt 200 \)