# SAT Math Word Problems: Equations, Rates and Units | topin

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## Translate the relationship before selecting a method

College Board’s [Algebra content page](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/algebra) includes creating and solving linear equations and systems. Its [data-analysis content page](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/problem-solving) includes ratios, rates and units. The examples below are original practice illustrations, not official questions. Their purpose is to make the interpretation visible rather than simply show a numerical answer.

Our four-line setup is unknown, units, relationship and target. For example: x is adult tickets; units are tickets and pounds; revenue equals price times quantity; target is the number of student tickets. That last line prevents you from reporting x if the question asks another quantity. Write only enough notation to keep the structure clear. A long rewritten version of the entire story is rarely necessary.

Test the equation with a simple value before solving when the wording is unfamiliar. If a cost contains a fixed charge, x = 0 should leave that fixed charge. If a quantity is a rate per hour, multiplying by hours should produce the total quantity. These checks do not prove every model, but they can expose a backwards relationship before you spend time solving a perfectly correct equation for the wrong story.

## Original example: build a two-price revenue equation

A club sells 40 tickets. Adult tickets cost £12 and student tickets £7\. Total revenue is £380\. How many student tickets were sold? Let a be adult tickets, so student tickets are 40 − a. The revenue equation is 12a + 7(40 − a) = 380\. Expanding gives 12a + 280 − 7a = 380, hence 5a = 100 and a = 20.

The requested student count is 40 − 20 = 20\. Check both totals: 20 + 20 = 40 tickets, and 20 × 12 + 20 × 7 = 240 + 140 = £380\. Two checks are useful because satisfying the headcount alone does not establish the revenue. The units make the equation sensible: pounds per ticket multiplied by tickets produces pounds.

A tempting incorrect equation is 12a + 7a = 380\. It assigns the same count to both categories without justification and ignores the total. Another is 12a + 7(40) = 380, which treats every ticket as student plus some extra adult revenue. Repair the model by defining complementary counts before adding prices. If asked for the adult fraction instead, the final quantity would be 20/40 = 1/2, not the count.

## Original example: interpret fixed charges and unit rates

A bicycle rental costs a fixed £9 plus £4 per hour. A customer pays £29\. How long was the rental? Let h represent hours. The total model is C = 9 + 4h, giving 29 = 9 + 4h and h = 5\. The coefficient has units pounds per hour; the intercept has units pounds. Interchanging them gives a different service, not merely a different algebraic form.

Check the fixed charge by evaluating h = 0: the model gives £9\. Check one additional hour: the cost increases by £4\. If a question asks what the number 4 represents, answer the change in total cost for each extra hour, rather than the initial charge or the number of customers. The meaning comes from the units and relationship, not the location of the digit alone.

Now suppose £29 covers 300 minutes. The same five hours become 300 minutes because 5 × 60 = 300\. A student who enters 5 when the question asks minutes has solved the model but missed the target. Write the final unit before calculation or underline it on scratch paper. The [Math pacing guide](/articles/sat-math-module-pacing) explains how to preserve this check without redoing the whole solution.

## Original example: cancel units explicitly

A pump moves 18 litres per minute. How many cubic metres does it move in 25 minutes, given 1000 litres = 1 cubic metre? Write 18 litres/minute × 25 minutes × 1 cubic metre/1000 litres. Minutes and litres cancel, leaving cubic metres. Numerically, 18 × 25 / 1000 = 0.45\. The conversion factor equals one physical quantity expressed in two units.

If you multiply by 1000 instead of dividing, the numerical answer becomes 450,000 while the intended magnitude should be less than one cubic metre. The unit cancellation shows why the factor is backwards. A conversion is not selected by the vague instruction “bigger unit means multiply”; it is chosen so the unwanted unit cancels and the desired unit remains.

For squared or cubed units, apply the conversion to the dimension. If 1 metre = 100 centimetres, then 1 square metre = 10,000 square centimetres, not 100\. A rectangle 2 metres by 3 metres has area 6 square metres; as 200 by 300 centimetres it has 60,000 square centimetres. These original values verify the squared factor directly and show why a linear conversion cannot be copied to area.

## Original example: compare parts of a total

A 10-litre mixture contains 30% juice. It is combined with 5 litres containing 60% juice. What percentage of the combined mixture is juice? The juice amounts are 0.30 × 10 = 3 litres and 0.60 × 5 = 3 litres. The combined mixture is 15 litres with 6 litres of juice, so the concentration is 6/15 = 0.40, or 40%.

The average of 30% and 60% is 45%, but that treats the two mixtures as equally large. They are not. The larger lower-concentration mixture pulls the combined percentage closer to 30%. Use total component divided by total mixture, and keep litres in both parts. The final ratio has no litre unit because identical units cancel. This is a weighted calculation, not an unweighted average of percentages.

If the question changes to equal volumes of the two mixtures, 45% becomes valid because the weights are equal. This variation is a useful review technique: identify which assumption makes a tempting answer correct. You then understand the structural difference instead of memorising that “averaging percentages is wrong”. Use the [reverse-percentage guide](/articles/sat-reverse-percentages) for cases where the base amount itself is unknown.

## Use quantity and unit checks to diagnose common traps

__Original modelling checks; these are mathematical methods, not score rules__
| Situation                       | Model or check                  | Typical wrong turn                  |
| ------------------------------- | ------------------------------- | ----------------------------------- |
| Two categories with fixed total | a + s = total                   | Give both categories the same count |
| Fixed fee plus use              | C = fixed + rate × use          | Multiply the fixed fee by use       |
| Distance and time               | Distance = speed × time         | Mix hours and minutes               |
| Mixture percentage              | Component total / mixture total | Average unequal-size percentages    |
| Area conversion                 | Square the length factor        | Use a linear factor on square units |
| Requested expression            | Evaluate it after solving       | Report the variable only            |

When reviewing a mistake, identify the first incorrect relationship rather than the final wrong number. If you used minutes with a kilometres-per-hour rate, the error begins before multiplication. If you answered adults when students were asked, the equation may be entirely correct. Those mistakes need different repairs, and putting both under “careless” makes the next practice session less useful.

Our suggested correction sentence states an action: “Convert all time to hours before using the rate” or “Write the requested category next to the variable definition.” Test the action on a fresh variation. If you can solve only the original numbers after reading the explanation, the modelling repair has not yet transferred. Return to the relationship and units before increasing speed.

## Practise models in short sets, then return to the timer

Create a short set with one ticket problem, one fixed-charge model, one conversion and one mixture. Work untimed first and explain each equation in a sentence. Then change one number or requested quantity and solve again. The examples on this page are instructional, so remembering their answers is not strong evidence of readiness. Fresh conditions require you to select the model independently.

[topin’s free SAT mock](/sat/practice-test), marked on the official scale, can test whether the method survives mixed timed work. topin has its own graphing calculator; practise official Bluebook tools separately. A graph or calculation is useful after the relationship is correct. Do not spend a timed module repeatedly entering different expressions until one produces a number resembling a choice.

Use the [error-log framework](/articles/sat-error-log-score-plateau) to retain the exact modelling cause and the successful repair. Your goal is fewer wrong equations and incorrect target quantities, not longer scratch work. A good model is concise enough to use under time and explicit enough to check. Finish every solution by asking whether the units, magnitude and requested quantity all describe the same answer.

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## FAQs

How do I start an SAT word problem?

Define the unknown and its unit, write the relationship and identify the final requested quantity before calculating.

Can the graphing calculator solve a word problem?

It can help solve a correctly modelled equation. You still need to interpret the story and choose the quantity being asked for.

Why is averaging mixture percentages sometimes wrong?

Unequal mixture sizes need weighting by their amounts. Divide the combined component amount by the total mixture.

How should I choose a conversion factor?

Write it so the unwanted unit cancels and the desired unit remains. Square or cube the factor for area or volume.

How do I review a modelling error?

Locate the first wrong relationship, write a specific corrective action and test it on a fresh variation.

## Sources (checked 5 October 2026)

* [College Board: Algebra content (checked 5 October 2026)](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/algebra)
* [College Board: Problem-Solving and Data Analysis content (checked 5 October 2026)](https://satsuite.collegeboard.org/sat/whats-on-the-test/math/types/problem-solving)
* [College Board: using the Student Question Bank (checked 5 October 2026)](https://satsuite.collegeboard.org/practice/student-question-bank)

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