Linear Equations in One Variable Pattern - Translate Context

Translating a real-world scenario into a linear equation

This pattern tests your ability to convert a word problem into an equation — or to pick the correct equation from four options. You'll see contexts like budgets, inventory, rates, fees, and discounts. The core skill is identifying which quantity is unknown, what operations connect the pieces, and writing the equation that ties them together. On easier questions you compute a straightforward answer; on harder ones, the SAT scrambles the roles of the numbers in the wrong answer choices to trip you up.

 

Part-Part-Whole

The simplest version: a total is split into two categories, and you need one of the parts.

A bookshelf holds $32$ books in total. The books are either fiction or nonfiction. If $19$ of the books are fiction, how many nonfiction books are on the bookshelf?

The total equals the sum of the parts: $19 + n = 32$, so $n = 32 - 19 =$ $13$.

When a discount or deduction is involved, the same structure appears with multiplication:

A soccer coach ordered $15$ team jerseys, all at the same unit price. The total cost was reduced by a $150 sponsorship contribution. The final amount the team had to pay was $450. What was the original price of a single jersey?

Let $j$ be the price per jersey. Total before discount: $15j$. After the sponsorship: $15j - 150 = 450$. Add $150$ to both sides: $15j = 600$. Divide by $15$: $j = 40$.

 

Start + Rate $\times$ Time

Many problems describe a quantity that begins at some value and increases at a constant rate. The equation is: $\text{start} + \text{rate} \times \text{time} = \text{total}$.

A baker starts the day with $48$ cookies already prepared for an order. The baker then makes additional cookies at a constant rate of $72$ cookies per hour. If the total number of cookies needed for the order is $480$, how many hours will the baker need to work?

Let $h$ be the number of hours. The equation is $48 + 72h = 480$. Subtract $48$: $72h = 432$. Divide by $72$: $h = 6$.

 

Start $-$ Rate $\times$ Time

When a quantity is decreasing, the equation flips to subtraction:

A large water tank holds $1{,}200$ gallons. Water is drained at a constant rate of $15$ gallons per minute. At this rate, how many minutes will it take for the volume to reach $450$ gallons?

Let $m$ be the number of minutes. Set up $1{,}200 - 15m = 450$. Subtract $1{,}200$ from both sides: $-15m = -750$. Divide by $-15$: $m = 50$.

 

Fixed Fee + Rate $\times$ Quantity

A one-time fee plus a per-unit charge is one of the most common setups. The SAT often asks you to identify the correct equation rather than solve it.

A car rental company charges a one-time fee of $45 plus $0.20 per mile driven. If a customer's total bill was $95 and they drove $m$ miles, which equation represents this situation?

The total cost is the fixed fee plus the variable cost: $0.20m + 45 = 95$.

Trap: A wrong choice like $45m + 0.20 = 95$ swaps which number multiplies the variable. The per-unit rate multiplies the variable; the fixed fee is added separately.

 

Translating Phrases: "More Than" and "Less Than"

The SAT uses phrases like "5 less than 6 times a number" or "4 more than twice." The order in English is the reverse of the arithmetic:

• "5 less than $6y$" $\;\rightarrow\;$ $6y - 5$ (not $5 - 6y$)
• "4 more than $2b$" $\;\rightarrow\;$ $2b + 4$

Maria's current age is $4$ years more than twice her brother's age, $b$. If Maria is $18$ years old, which equation represents this relationship?

"Twice her brother's age" $= 2b$. "4 years more than" that $= 2b + 4$. Set equal to Maria's age: $2b + 4 = 18$.

 

Compound Rates: X per Y Units

When the rate isn't given per single unit — for example, "loses 5 liters every 2 minutes" — find how many complete cycles occurred, then convert to the unit the question asks about.

A large water tank initially holds $500$ liters of water. A leaky faucet causes the water level to drop, losing $5$ liters every $2$ minutes. If the tank currently contains $350$ liters, for how many minutes has the faucet been leaking?

Total water lost: $500 - 350 = 150$ liters. Each cycle removes $5$ liters, so the number of cycles is $\dfrac{150}{5} = 30$. Each cycle takes $2$ minutes: $30 \times 2 =$ $60$ minutes.

 

Flat Fee for an Initial Period, Then a Per-Unit Rate

Some pricing structures charge a lump sum for the first few units, then a per-unit rate after that. The variable cost applies only to the extra units beyond the initial period.

The cost to rent a car is $250 for the first $3$ days and an additional $65 per day for each day after the first $3$. If the total cost for $d$ days (where $d > 3$) was $445, which equation represents this situation?

The first $3$ days cost a flat $250. The remaining days number $d - 3$, each at $65. Total: $250 + 65(d - 3) = 445$.

Trap: An option like $250 + 65d = 445$ charges $65 for all days instead of just the extra ones. The expression $(d - 3)$ is the key detail.

 

Revenue $-$ Cost $=$ Profit

When a problem gives a selling price, a cost, and a profit, the relationship is: price $\times$ quantity $-$ cost $=$ profit.

A bakery sells custom cakes for $35 each. On Saturday, the bakery's profit was $655 after accounting for $150 in ingredient costs. How many custom cakes were sold?

Let $c$ be the number of cakes. Revenue: $35c$. Profit $=$ revenue $-$ cost: $35c - 150 = 655$. Add $150$: $35c = 805$. Divide by $35$: $c = 23$.

 

Three Categories with a Multiplier (and Distractor Information)

The hardest version: three groups where one is expressed as a multiple of another, plus a fixed third group. The SAT throws in irrelevant details — sizes, prices, or weights — to distract you. The question asks about counts, not total value.

A farm has chickens, goats, and pigs. The number of chickens is $6$ times the number $g$ of goats. The number of pigs is $12$. If there is a total of $236$ animals on the farm, which equation represents this situation?

Goats: $g$. Chickens: $6g$. Pigs: $12$. Total animals: $g + 6g + 12 = 236$, which simplifies to $7g + 12 = 236$.

Trap: The choice $6g + 12 = 236$ forgets to count the goats themselves. The goats contribute "$g$" and the chickens contribute "$6g$" — you need both, giving $7g$.

 

The TRANSLATE Method

1. Read the whole problem. Identify what's unknown — that's your variable — and what you're asked to find (a value or an equation).
2. Label every quantity. Write each piece in terms of your variable and the given constants.
3. Identify the relationship. Is it a total? A difference? Revenue minus cost? Write the equation connecting the pieces.
4. Check units and roles. The per-unit rate multiplies the variable; the fixed amount is added or subtracted separately. Don't swap them.
5. If choosing an equation, plug in a simple number for the variable and check that the equation makes sense before committing.

 

Watch Out For

• "Less than" reversal. "5 less than $6y$" is $6y - 5$, not $5 - 6y$. The quantity after "less than" comes first in the expression.
• Forgetting to count all groups. When one group is "6 times" another, the total includes both groups: $g + 6g = 7g$, not just $6g$.
• Distractor numbers. Hard questions mention prices, sizes, or weights for each category, but the question asks about the total number of items. Don't multiply by the per-item attribute.
• Compound rate errors. When a rate is "5 liters every 2 minutes," the rate per minute is $\dfrac{5}{2}$, not $\dfrac{2}{5}$. Work in complete cycles to avoid fraction mistakes.