Do Not Translate One Word at a Time
Consider the phrase:
Three times the sum of a number and five.
The important structure is not merely “three,” “times,” “sum,” and “five.” The phrase says that the complete sum must be formed first and then multiplied by three:
\[ 3(x+5). \]Without the grouping,
\[ 3x+5 \]would mean “five more than three times a number.”
Reliable translation therefore requires us to identify quantities, relationships, operation order, and grouping—not simply replace vocabulary words with symbols.
\[ \boxed{\text{understand the structure first; write the symbols second}} \]A Reliable Translation Method
- Identify the quantities. Determine what is known, unknown, variable, or fixed.
- Define symbols. State clearly what each symbol represents.
- Find the main relationship. Decide whether the result is an expression, equation, inequality, formula, or other object.
- Locate grouped phrases. Words such as “the sum of,” “the difference between,” and “the quantity” often signal a grouped object.
- Preserve order. Subtraction and division depend on which quantity comes first.
- Include conditions and units. Record restrictions and contextual assumptions.
- Translate back into words. Confirm that the symbolic form reproduces the original meaning.
For the sentence “A number decreased by eight is seventeen,” let \(x\) represent the number. The main relationship is equality:
\[ x-8=17. \]Translation stops once the relationship has been represented. Solving the equation is a separate task.
Addition Phrases
| Verbal Phrase | Symbolic Translation |
|---|---|
| The sum of \(x\) and \(7\) | \(x+7\) |
| \(x\) plus \(7\) | \(x+7\) |
| Seven added to \(x\) | \(x+7\) |
| Seven more than \(x\) | \(x+7\) |
| \(x\) increased by \(7\) | \(x+7\) |
| The total of \(x\), \(y\), and \(z\) | \(x+y+z\) |
Addition is commutative, so reversing two addends does not change the value:
\[ x+7=7+x. \]Even so, matching the order of the sentence can make the translation easier to explain.
Subtraction Phrases and Reversed Order
Subtraction is not commutative:
\[ x-7\ne7-x. \]Some subtraction phrases preserve the spoken order, while others reverse it.
| Verbal Phrase | Symbolic Translation | Order Note |
|---|---|---|
| \(x\) minus \(7\) | \(x-7\) | Preserved |
| The difference between \(x\) and \(7\) | \(x-7\) | Preserved |
| \(x\) decreased by \(7\) | \(x-7\) | Preserved |
| Seven less than \(x\) | \(x-7\) | Reversed from spoken order |
| Seven subtracted from \(x\) | \(x-7\) | Reversed from spoken order |
| \(x\) less than \(7\) | \(7-x\) | “Less than” reverses the quantities |
Ask: What quantity is being reduced, and by what amount?
For “nine less than twice a number,” twice the number is reduced by nine:
\[ 2x-9. \]Multiplication Phrases
| Verbal Phrase | Symbolic Translation |
|---|---|
| The product of \(5\) and \(x\) | \(5x\) |
| Five times \(x\) | \(5x\) |
| Five multiplied by \(x\) | \(5x\) |
| Twice \(x\) | \(2x\) |
| Half of \(x\) | \(\frac12x\) |
| \(p\) percent of \(x\) | \(\frac{p}{100}x\) |
The word of often signals multiplication, especially in fraction and percent phrases. Context still matters; not every use of “of” is an operation.
When multiplication applies to a sum or difference, grouping is required:
\[ 4(x-3). \]Four times the difference between \(x\) and three
Division, Ratios, and Fractions
Division also depends on order.
| Verbal Phrase | Symbolic Translation |
|---|---|
| The quotient of \(x\) and \(5\) | \(\frac{x}{5}\) |
| \(x\) divided by \(5\) | \(\frac{x}{5}\) |
| The ratio of \(x\) to \(5\) | \(\frac{x}{5}\) or \(x:5\) |
| Five divided by \(x\) | \(\frac{5}{x}\) |
| One-third of \(x\) | \(\frac{x}{3}\) |
| The sum of \(x\) and \(4\), divided by \(7\) | \(\frac{x+4}{7}\) |
A fraction bar groups the complete numerator and denominator. The phrase
The quotient of the sum of \(x\) and two and the difference between \(x\) and three
is represented by
\[ \frac{x+2}{x-3}, \qquad x\ne3. \]The restriction arises because division by zero is undefined.
Powers and Radicals
| Verbal Phrase | Symbolic Translation |
|---|---|
| The square of \(x\) | \(x^2\) |
| \(x\) cubed | \(x^3\) |
| \(x\) raised to the fifth power | \(x^5\) |
| The square of the sum of \(x\) and \(3\) | \((x+3)^2\) |
| The sum of the squares of \(x\) and \(3\) | \(x^2+3^2\) |
| The square root of \(x\) | \(\sqrt{x}\) |
| The cube root of the difference \(x-4\) | \(\sqrt[3]{x-4}\) |
“The square of a sum” and “the sum of squares” are not equivalent:
\[ (x+3)^2\ne x^2+3^2 \]in general. Grouping determines the base of exponentiation and the radicand of radicalization.
Equations and Relation Phrases
Words such as the following often signal equality:
- is,
- equals,
- is equal to,
- has the same value as,
- results in,
- is equivalent to, when equivalence is intended.
| Verbal Statement | Symbolic Translation |
|---|---|
| Five more than a number is seventeen | \(x+5=17\) |
| Twice a number equals its square | \(2x=x^2\) |
| The sum of two numbers is forty | \(x+y=40\) |
| Distance equals rate times time | \(d=rt\) |
| The perimeter equals twice the length plus twice the width | \(P=2l+2w\) |
The equal sign should connect complete expressions. Determine where the verbal statement divides into its two sides.
Inequality Phrases
| Verbal Phrase | Relation |
|---|---|
| Is less than | \(<\) |
| Is greater than | \(>\) |
| Is at most | \(\le\) |
| Is no more than | \(\le\) |
| Does not exceed | \(\le\) |
| Is at least | \(\ge\) |
| Is no less than | \(\ge\) |
| Is not equal to | \(\ne\) |
“At least” and “at most” include their boundary values.
A person must be at least eighteen years old.
If \(a\) represents age in years, then
\[ a\ge18. \]The package may weigh no more than fifty pounds.
If \(w\) is the weight in pounds, then
\[ w\le50. \]Grouping Multi-Step Phrases
Longer verbal phrases should be divided into meaningful layers.
| Phrase | Translation |
|---|---|
| Five more than twice a number | \(2x+5\) |
| Twice the sum of a number and five | \(2(x+5)\) |
| Three less than the square of a number | \(x^2-3\) |
| The square of three less than a number | \((x-3)^2\) |
| Half the difference between \(x\) and \(7\) | \(\frac{x-7}{2}\) |
| The difference between half of \(x\) and \(7\) | \(\frac{x}{2}-7\) |
Useful grouping cues include:
- the sum of,
- the difference between,
- the product of,
- the quotient of,
- the square of,
- the quantity,
- all of,
- the total.
These phrases often name a complete object that should be formed before the next operation is applied.
Percent, Rate, and Proportion Language
Percent of a Quantity
\[ 0.20x. \]Twenty percent of \(x\)
Percent Increase
\[ P+\frac{r}{100}P =P\left(1+\frac{r}{100}\right). \]A value \(P\) increased by \(r\) percent
Constant Rate
\[ N=qt. \]A machine produces \(q\) units per hour for \(t\) hours.
Fixed Amount Plus a Rate
\[ C=F+rh. \]A service charges \(F\) dollars plus \(r\) dollars per hour for \(h\) hours.
Units help confirm the translation:
\[ \left(\frac{\text{dollars}}{\text{hour}}\right)(\text{hours})=\text{dollars}. \]Context, Units, and Assumptions
Natural language may leave details implicit. A complete algebraic translation records the details that affect the model.
Suppose a theater sells adult tickets for \(\$15\) and student tickets for \(\$9\). If \(a\) and \(s\) represent the numbers sold, total revenue is
\[ R=15a+9s. \]The context adds restrictions:
\[ a,s\in\{0,1,2,3,\ldots\}. \]Ticket counts cannot be negative or fractional.
Important translation details may include:
- units and dimensions,
- permitted domains,
- whether values must be whole numbers,
- whether a rate is constant,
- whether endpoints are included,
- whether measurements are exact or approximate.
Worked Example: A Number Relationship
Translate:
Seven more than three times a number is two less than the square of the number.
Define the Quantity
Let \(x\) represent the number.
Translate the Left Side
“Three times a number” is \(3x\). “Seven more than” gives
\[ 3x+7. \]Translate the Relation
The word “is” signals equality.
Translate the Right Side
“The square of the number” is \(x^2\). “Two less than” gives
\[ x^2-2. \]Construct the Equation
\[ \boxed{3x+7=x^2-2}. \]Translate Back
The equation reads: seven more than three times \(x\) equals two less than the square of \(x\). The original meaning has been preserved.
Worked Example: A Practical Cost Model
A repair company charges a \(\$45\) visit fee plus \(\$70\) for each hour of labor. Translate the total cost for \(h\) hours and state reasonable restrictions.
Define the Quantities
- Let \(h\) be the number of labor hours.
- Let \(C\) be the total cost in dollars.
Translate the Variable Charge
\[ 70h. \]Add the Fixed Fee
\[ C=45+70h. \]State a Domain
At minimum, labor time cannot be negative:
\[ h\ge0. \]If the company bills in quarter-hour increments, a more specific domain could be
\[ h\in\left\{0,\frac14,\frac12,\frac34,1,\ldots\right\}. \]The billing policy determines the appropriate restriction.
Reverse-Translation Check
After writing a symbolic form, read it back into natural language without looking at the original phrase.
Suppose we translated a phrase as
\[ 5(x-2)^2+3. \]Read from the inside outward:
- the difference between \(x\) and two,
- the square of that difference,
- five times that square,
- three more than the resulting product.
If this reconstructed sentence matches the intended relationship, the grouping and operation order are probably correct.
A numerical spot check can provide additional evidence. Choose a simple value and compare the verbal procedure with the symbolic evaluation.
Common Translation Errors
Translating Word by Word
Individual keywords do not reveal the full structure. Identify grouped phrases and the main relationship.
Reversing Subtraction Incorrectly
“Five less than \(x\)” means \(x-5\), not \(5-x\).
Reversing Division
“The quotient of \(x\) and five” means \(\frac{x}{5}\), not \(\frac{5}{x}\).
Losing Grouping
“Twice the sum of \(x\) and four” is \(2(x+4)\), not \(2x+4\).
Confusing “At Least” and “At Most”
At least \(5\) means \(x\ge5\); at most \(5\) means \(x\le5\).
Solving When Only Translation Was Requested
Constructing the symbolic relationship and solving it are separate stages.
Ignoring Units and Context
A correct-looking expression may still be an incomplete model if domains, units, or restrictions are missing.
Forcing Precision onto Ambiguous Language
If a phrase genuinely supports several interpretations, request clarification or state the chosen assumption rather than silently guessing.
Practice
- Translate “eight more than a number.”
- Translate “eight less than a number.”
- Translate “a number subtracted from eight.”
- Translate “four times the sum of a number and three.”
- Translate “the sum of four times a number and three.”
- Translate “the quotient of the square of \(x\) and the sum \(x+1\),” stating any restriction.
- Translate “five more than twice a number is at most seventeen.”
- Translate “the square of the difference between \(x\) and six.”
- A tank begins with \(40\) liters and fills at \(6\) liters per minute. Write a model for the volume after \(t\) minutes.
- A price \(P\) is reduced by \(15\%\). Write the new price in two equivalent forms.
Selected Answers
- \(x+8\)
- \(x-8\)
- \(8-x\)
- \(4(x+3)\)
- \(4x+3\)
- \(\frac{x^2}{x+1}\), with \(x\ne-1\).
- \(2x+5\le17\)
- \((x-6)^2\)
- \(V=40+6t\), with \(t\ge0\) while the model remains applicable.
- \(P-0.15P\) or \(0.85P\).
Summary
- Translate mathematical structure rather than isolated vocabulary words.
- Define the quantities and identify the main relationship before writing symbols.
- Addition and multiplication are commutative, but subtraction and division require careful order.
- Grouping preserves phrases such as sums, differences, products, quotients, powers, and radicals.
- Equality and inequality phrases determine the relation connecting complete expressions.
- “At least” includes a lower boundary; “at most” includes an upper boundary.
- Rates, percentages, fixed charges, and variable charges must be connected with compatible units.
- Domains, restrictions, assumptions, and measurement precision belong to the translation when they affect meaning.
- Reverse translation is a practical check that the symbolic form preserves the original statement.