How to translate SAT Math word problems

Begin at the final sentence. Underline exactly what must be reported and attach a unit. Then define only the variables you need, each with a meaning and unit. Finally, turn complete relationships into mathematical statements. This order prevents a common failure: calculating a nearby quantity and submitting it because the arithmetic was correct. It also makes equations easier to check, since every symbol has a declared role.

Do not translate one English word at a time. The word ‘more’ can indicate addition, a comparison, or a percent increase depending on the sentence. Translate relationships: ‘A is 7 more than B’ becomes A=B+7, while ‘A is 7 times B’ becomes A=7B. Read the equation back in plain language. If the spoken version reverses the stated relationship, fix it before solving.

Rates compare quantities with different units. Write them as labeled fractions before multiplying. If a pump moves liters per minute and time is measured in hours, convert one unit so they match. Unit cancellation is not decoration; it shows which operation creates the requested label. A value in miles per gallon multiplied by gallons produces miles, while dividing miles by miles per gallon produces gallons.

For combined work or motion, define what stays constant. Two travelers moving toward one another reduce their separation at the sum of their speeds; moving in the same direction uses a difference. An average rate is total change divided by total time, not usually the simple average of two rates. Draw a small table with rate, time, and amount when several stages are involved, then ensure each row uses compatible units.

Every percent statement needs a base. ‘18 percent of 250’ uses 250 as the base; ‘18 percent more than 250’ produces 250(1.18). Percent change divides the change by the original value. When the original is unknown, name it before writing the multiplier. Reversing a percentage is not done by subtracting the same percent: after a 20 percent increase, the new value is 1.2 times the old, so recovering the old value requires division by 1.2.

Repeated equal-percent change is exponential. A quantity that grows r percent per period follows initial value times (1+r)^number of periods when r is written as a decimal. A quantity that decreases uses 1−r. Identify whether the initial value occurs at period 0 or after the first period. When a model uses a different interval, such as a factor every three years, do not silently treat that factor as annual.

Systems appear when two unknown quantities are linked by two independent conditions. A total-count equation often uses x+y=total. A total-value equation weights each count, such as 4x+7y=revenue. Label the variables so coefficients attach to the correct category. If one equation is merely a rearrangement of the other, it supplies no new information; look for a second condition involving value, distance, mixture, or comparison.

Inequalities model limits and ranges. Words such as at least, no more than, minimum, and maximum determine the boundary direction. After solving, apply contextual constraints: counts may need nonnegative integers, lengths must be positive, and a budget may require rounding down to a whole number of purchases. A graph can show the feasible region, but the final answer must still respect the original units and strict or inclusive boundary.

In a linear model, slope carries output units per input unit and the intercept is the predicted output at input zero. In an exponential model, the leading coefficient is the initial value and the base is a multiplicative factor per stated interval. In a quadratic model, zeros, vertex, and intercepts can each have contextual meaning—or none at all if they fall outside the modeled domain. State the meaning with units rather than calling a number merely ‘the slope’ or ‘the answer.’

Finish with a reasonableness audit. Check sign, scale, units, domain, and target. Negative time or a fractional person is generally not meaningful, while a negative temperature or financial change may be. A model can produce a numerical value beyond the observed data, but the prompt may ask whether that extrapolation is justified. Mathematical output is not automatically contextual truth; the last translation goes from symbols back into the situation.