Simple Addition Phrase

"A number increased by 7 equals 15." Write as an equation.

Identify the unknown: "A number" is unknown = Let $x$ = the number

Find the operation word: "increased by" means addition = $x + 7$

Find the result: "equals 15" means $= 15$ = $x + 7 = 15$

Answer: $x + 7 = 15$ (solving: $x = 8$)

Multiplication Phrase

"Twice a number decreased by 4 is 12." Write as an equation.

Identify the unknown: "a number" is unknown = Let $x$ = the number

Translate "twice a number": "Twice" means multiply by 2 = $2x$

Translate "decreased by 4": "Decreased by" means subtract = $2x - 4$

Write the full equation: "is 12" means equals 12 = $2x - 4 = 12$

Answer: $2x - 4 = 12$ (solving: $x = 8$)

Two-Part Word Problem

"The sum of a number and 5, multiplied by 3, gives 24."

Identify the unknown: "a number" is unknown = Let $x$ = the number

Translate "sum of a number and 5": Sum means addition = $x + 5$

Translate "multiplied by 3": Use parentheses to group, then multiply = $3(x + 5)$

Complete the equation: "gives 24" means equals 24 = $3(x + 5) = 24$

Answer: $3(x + 5) = 24$ (solving: $x = 3$)

Age Problem

"Maria is 4 years older than twice her brother's age. Maria is 22." Find her brother's age.

Identify the unknown: Brother's age is unknown = Let $x$ = brother's age

Translate "twice her brother's age": Twice means multiply by 2 = $2x$

Translate "4 years older than": Older than means add = $2x + 4$

Maria is 22: This expression equals 22 = $2x + 4 = 22$

Solve for $x$: $2x = 18$, then $x = 9$ = Brother is 9 years old

Answer: $2x + 4 = 22$; Maria's brother is 9 years old.

Mistake: Reversing subtraction: Writing "5 less than $x$" as $5 - x$ instead of $x - 5$

Why: "Less than" means you subtract from the number. "5 less than $x$" = "$x$ minus 5"

Correct: "5 less than $x$" translates to $x - 5$, not $5 - x$

Mistake: Forgetting parentheses: Writing "3 times the sum of $x$ and 2" as $3x + 2$

Why: "The sum of $x$ and 2" is a group that gets multiplied. Without parentheses, only $x$ gets multiplied.

Correct: "3 times the sum of $x$ and 2" is $3(x + 2)$, not $3x + 2$

Mistake: Confusing "more than" with "is more than": "$x$ is 5 more than $y$" vs. "$x$ is more than 5"

Why: "5 more than $y$" means $y + 5$. "Is more than" is a comparison (inequality).

Correct: "$x$ is 5 more than $y$" means $x = y + 5$

Shopping and Discounts

Stores describe prices in words that you can translate to find deals.

"The sale price is 20 dollars less than the original price" becomes $s = p - 20$, where $s$ is sale price and $p$ is original price.

Salary and Earnings

Job pay is often described in words that become equations.

"You earn 15 dollars per hour plus a 50 dollar bonus" becomes $E = 15h + 50$, where $E$ is total earnings and $h$ is hours worked.

Sports Statistics

Athletes' stats are compared using phrases that translate to equations.

"She scored 8 more points than her previous game" becomes $s = p + 8$, where $s$ is this game's score and $p$ is the previous score.

Q: How do I know which variable to use?

A: Any letter works! Using $x$ is common, but you can use letters that make sense, like $a$ for age, $t$ for time, or $p$ for price.

Q: What's the difference between 'is' and 'is more than'?

A: 'Is' usually means equals (=). 'Is more than' or 'is greater than' means a comparison (>). Context helps: 'The price is 50' means $p = 50$. 'The price is more than 50' means $p > 50$.

Q: How do I handle 'of' in phrases like '1/2 of a number'?

A: 'Of' typically means multiply. So '1/2 of a number' is $\frac{1}{2} \times x$ or $\frac{x}{2}$. Similarly, '25% of a number' is $0.25x$.