Step one in fixing a phrase drawback is all the time to learn the issue. You want to have the ability to translate phrases into mathematical symbols, specializing in key phrases that point out the mathematical procedures required to resolve the issue—each the operation and the order of the expression. In a lot the identical means that you could translate Spanish into English, you possibly can translate English phrases into symbols, the language of arithmetic. Many (if not all) key phrases that point out mathematical operations are acquainted phrases.
To start, you translate English phrases into algebraic expressions. An algebraic expression is a group of numbers, variables, operations, and grouping symbols. You’ll translate an unknown quantity because the variable x or n. The grouping symbols are often a set of parentheses, however they will also be units of brackets or braces.
In translating expressions, you need to be nicely acquainted with primary key phrases that translate into mathematical operations: addition key phrases, subtraction key phrases, multiplication key phrases, and division key phrases, that are coated within the 4 following sections.
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Addition key phrases
Some frequent examples of addition key phrases are as follows:
 SUM OF_____ AND _____
 TOTAL OF _____ AND _____
 _____ PLUS _____
 _____ INCREASED BY _____
 GAIN
 RAISE
 MORE
 INCREASE OF
The primary two key phrases (SUM and TOTAL) are referred to as main key phrases as a result of they lead the expression. The second two key phrases (PLUS and INCREASED BY) are key phrases that point out the precise placement of the plus signal. The final 4 key phrases may be present in phrase issues and will point out addition.
When an expression begins with the main key phrases SUM or TOTAL, the main key phrase defines the corresponding AND. The plus signal then bodily replaces the AND within the expression.
Instance 1: Translate the next: the sum of 5 and a quantity
The next steps assist you to translate this drawback:
1. Underline the phrases earlier than and after AND when it corresponds to the main key phrase SUM OF.

 the sum of 5 and a quantity
2. Circle the main key phrase and point out the corresponding AND that it defines.
3. Translate every underlined expression and exchange AND with a plus signal.
 The expression interprets to five + x.
Instance 2: Translate the next: the overall of a quantity and adverse three
Use the next steps to translate this drawback:
1. The key phrase TOTAL OF is a number one key phrase that defines AND, so underline the phrases earlier than and after AND: “a quantity” and “adverse three.”

 the overall of a quantity and adverse three
2. Circle the main key phrase and point out the corresponding AND that it defines.
3. Translate every underlined expression and exchange AND with a plus signal.

 The expression interprets to x + −3.
Instance 3: Translate the next: the sum of seven and adverse 4
Translate this instance within the following means:
1. The phrase SUM OF is a number one key phrase that defines AND, so underline the phrases earlier than and after AND: “seven” and “adverse 4.”

 the sum of seven and adverse 4
2. Circle the main key phrase and point out the corresponding AND that it defines.
3. Translate every underlined expression and exchange AND with a plus signal.

 The expression interprets to 7 + −4.
Reminder: The AND key phrase interprets to imply “plus” as a result of the main key phrase is SUM OF. With different main key phrases (mentioned within the following sections), AND can imply different issues. Additionally discover that you don’t simplify the expression and get “3” for the reply since you are simply translating phrases into symbols and never performing the maths.
Two different key phrases on the addition key phrase listing, PLUS and INCREASED BY, may be accurately translated by the direct translation technique. Within the direct translation technique, you translate every phrase into its corresponding algebraic image, one by one, in the identical order as written, as proven in Instance 4.
Instance 4: Translate the next: a quantity elevated by twenty‐4
 The expression interprets to x + 24.
Some further key phrases, reminiscent of GAIN, MORE, INCREASE OF, and RAISE, are generally present in story issues, as in Instance 5.
Instance 5: Translate the next story drawback right into a mathematical expression concerning the weight of the linebacker: The defensive linebacker weighed 200 twenty‐two kilos firstly of spring coaching. He had a achieve of seventeen kilos after figuring out with the crew for 4 weeks.
 The expression interprets to 222 + 17.
Observe: Not all numbers talked about in a phrase drawback must be included within the mathematical expression. The quantity “4” is simply attentiongrabbing reality, however it isn’t info you want with a purpose to write an expression concerning the linebacker’s weight.
You might also be questioning why the reply is not 239 kilos. That is as a result of the query asks you to translate the story drawback right into a mathematical expression, to not consider the expression.
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Instance 6: Translate the next phrase drawback right into a mathematical expression concerning the cashier’s present hourly wage: A cashier on the nook grocery was incomes $6.25 an hour. He obtained a elevate of 25 cents an hour.
 The expression interprets to six.25 + 0.25.
Observe: The hourly wage is said in {dollars}, and the elevate is said in cents. Any time you’re including two numbers which have models, ensure each numbers are measured with the identical models; if they don’t seem to be, convert one of many numbers to the identical models as the opposite. Having each numbers measured with the identical models known as homogeneous models. On this instance, you change his elevate, the 25 cents, to $0.25 as a result of his hourly wage is measured in {dollars}, not cents, so the elevate should even be in {dollars}.
Subtraction key phrases
Subtraction key phrases additionally embody main key phrases, key phrases that may be translated one phrase at a time, and key phrases which are present in story issues. Take a look at the next listing of subtraction key phrases:
 DIFFERENCE BETWEEN _____ AND _____
 _____ MINUS _____
 _____ DECREASED BY _____
 LOSS
 LESS
 FEWER
 TAKE AWAY
One subtraction key phrase (DIFFERENCE BETWEEN) is a two‐half expression that begins with a number one key phrase that defines the corresponding AND. You need to use the identical strategies of underlining and circling the key phrases proven within the previous part to translate these expressions.
Instance 7: Translate the next: the distinction between 4 and 6
Right here is the way you translate Instance 7:
1. As a result of the key phrase DIFFERENCE BETWEEN is a number one key phrase that defines the corresponding AND, underline the phrases earlier than and after AND: “4” and “six.”

 the distinction between 4 and 6
2. Circle the main key phrase and point out the corresponding AND that it defines.
3. Translate every underlined expression and exchange AND with a minus signal.

 The expression interprets to 4 – 6.
Observe: AND is just not all the time translated to imply addition. Right here, the DIFFERENCE BETWEEN is the main key phrase that defines the AND to imply subtraction.
Different subtraction key phrases, reminiscent of MINUS and DECREASED BY, use the direct translation technique. Instance 8 is a subtraction phrase drawback that’s translated one key phrase at a time, within the precise order of the expression.
Instance 8: Translate the next: twenty‐4 decreased by a quantity
 The expression interprets to 24 – x.
In a narrative drawback, you could discover the subtraction key phrases LOSS, LESS, FEWER, and TAKE AWAY, as proven in Instance 9.
Instance 9: Translate the next phrase drawback right into a mathematical expression concerning the present worth of supplies on the job web site: A building firm saved $1,253 value of supplies on the job web site. The corporate suffered a lack of $300 resulting from storm injury.
 The expression interprets to 1,253 – 300.
Multiplication key phrases
Some frequent examples of multiplication key phrases are as follows:
 MULTIPLY _____ BY _____
 PRODUCT OF _____ AND _____
 _____ TIMES _____
 DOUBLE _____
 TWICE _____
 TRIPLE _____
 PERCENT OF _____
 FRACTION OF _____
For 2 of the multiplication key phrases, MULTIPLY and PRODUCT OF, a number one key phrase defines the corresponding BY or AND, as proven in Instance 10.
Instance 10: Translate the next: the product of seven and a quantity
Translate this instance within the following means:
1. As a result of PRODUCT OF is a number one key phrase that corresponds to AND, underline the phrases earlier than and after AND: “seven” and “a quantity.”

 the product of seven and a quantity
2. Circle the main key phrase and point out the corresponding AND that it defines.
3. Translate every underlined expression and exchange AND with a occasions signal.

 The expression interprets to 7 × x.
Observe: Remember the fact that AND doesn’t all the time point out addition. The key phrase PRODUCT OF defines the AND on this expression to imply multiplication.
A multiplication expression that’s translated by the direct translation technique is proven in Instance 11.
Instance 11: Translate the next: a quantity occasions fifteen
The expression interprets to x × 15.
Some multiplication key phrases, reminiscent of DOUBLE, TWICE, and TRIPLE, translate right into a quantity and the operation of multiplication, as proven in Examples 12 and 13.
Instance 12: Translate the next: twice a quantity
The expression interprets to 2 × x.
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Instance 13: Translate the next phrase drawback right into a mathematical expression: Jennifer had $15 {dollars} within the financial institution. Over the following two weeks she doubled her cash.
The expression interprets to 2 × 15.
One of many key phrases that signifies multiplication is OF. In phrase issues, nonetheless, you may even see a couple of use of the phrase “of.” The one OF that signifies multiplication is the one which follows the key phrase PERCENT, the % signal, the key phrase FRACTION, or a fraction. See Examples 14 and 15.
Instance 14: Translate the next: twenty 5 % of 4 hundred {dollars}
The expression interprets to 0.25 × 400.
Observe: Keep in mind that a % is modified to a decimal earlier than multiplying.
Instance 15: Translate the next: one‐third of twenty‐seven
The expression interprets to .
Division key phrases
Some frequent examples of division key phrases are as follows:
 QUOTIENT OF _____ AND _____
 DIVIDE _____ BY _____
 _____ DIVIDED BY _____
 DIVIDED EQUALLY
 PER
The key phrases PRODUCT OF and QUOTIENT OF are troublesome for some folks to distinguish. Here’s a trace that can assist you bear in mind which one signifies division and which one signifies multiplication: QUOTIENT is a “more durable” phrase than “PRODUCT,” and division is a “more durable” operation than multiplication.
Keep in mind: Main key phrases outline the corresponding AND or BY to imply divide, often designated with the image ÷.
Instance 16: Translate the next: the quotient of seven and a quantity
1. As a result of the key phrase QUOTIENT OF is a number one key phrase that defines AND, underline the phrases earlier than and after AND: “seven” and “a quantity.”

 the quotient of seven and a quantity
2. Circle the main key phrase and point out the corresponding AND that it defines.
3. Translate every underlined expression and exchange AND with a division signal.

 The expression interprets to 7 ÷ n.
Observe: Right here, the key phrase QUOTIENT OF defines AND to imply division.
Instance 17: Translate the next: divide adverse thirty‐six by 9
1. As a result of the phrase DIVIDE is a number one key phrase that defines the BY, underline the phrases earlier than and after BY: “adverse thirty‐six” and “9.”

 divide adverse thirty‐six by 9
2. Circle the main key phrase and point out the corresponding BY that it defines.
3. Translate every underlined expression and exchange BY with a division signal.

 The expression interprets to .
Observe: The primary quantity goes within the numerator when utilizing a fraction bar to point division. The quantity within the numerator (the −36) goes contained in the “home” when utilizing the lengthy division image.
Some division key phrases may be translated one phrase at a time. As a substitute, you simply comply with the sentence and exchange with algebraic notations as you go alongside.
Instance 18: Translate the next: a quantity divided by 16
The expression interprets to .
Usually, in story issues, the key phrase that signifies division is PER. When a narrative drawback asks for the pace of a automobile in miles per hour, arrange the expression to divide the variety of miles by the variety of hours. You not solely instantly translate “miles” ÷ “hours,” but additionally establish the variety of miles and variety of hours by discovering them elsewhere in the issue. See Instance 19.
Instance 19: Translate the next phrase drawback right into a mathematical expression about pace: It takes three hours to journey 150 miles to grandmother’s home. How do you discover your common pace in miles per hour?
You discover “miles” ÷ “hours” within the query. Within the first a part of the phrase drawback, you discover the variety of miles, 150 miles, and the variety of hours, three hours.
The expression interprets to 150 ÷ 3.
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