Label the quantities before calculating. A ratio describes parts; a rate links different units; a conversion must cancel the units you start with.
Ratio and rate questions reward keeping quantities visible. Write units next to numbers, define what each ratio part represents, and decide whether the relationship is proportional before applying a shortcut.
Part-to-part is different from part-to-whole
These are original worked examples, not released SAT questions. Suppose a garden's tomatoes and peppers are in the ratio 5:3. Write the counts as 5k and 3k, so the total is 8k. Tomatoes are 5/8 of the total, rather than 5/3 of it.
If 24 tomatoes are added and the ratio becomes 7:3, peppers are unchanged. Thus (5k + 24)/(3k) = 7/3, giving 5k + 24 = 7k and k = 12. Originally there were 60 tomatoes and 36 peppers. After adding 24, 84:36 simplifies to 7:3. Keeping the unchanged quantity fixed is the key.
Use unit cancellation for rates
Convert 72 kilometres per hour to metres per second:
72 km/h × (1,000 m/1 km) × (1 h/3,600 s) = 20 m/s.
The kilometre and hour units cancel. Multiplying by 3,600 instead of dividing would leave the units wrong. For a speed-time-distance problem, keep time in a compatible unit: at 20 m/s, a constant-speed trip lasting 45 seconds covers 900 metres.
A rate can be written in either direction
A vehicle uses 6 litres per 100 kilometres. At that constant rate, a 250-kilometre trip uses 250 km × (6 L/100 km) = 15 L. The reciprocal rate is 100/6 kilometres per litre. Choose the direction that cancels the given quantity; “per” does not always mean dividing the two numbers as printed.
Direct and inverse proportional relationships
If identical parts cost $4 each without a fixed charge, total cost C = 4n is directly proportional to the count n. Doubling n doubles C. With a $12 delivery charge, C = 4n + 12 is linear but is not directly proportional: the fixed charge prevents the same scaling.
If 3 identical pumps fill a tank in 8 hours, and each pump works at the same constant rate, the work takes 24 pump-hours. Six such pumps take 24/6 = 4 hours. This inverse relationship depends on the stated assumption that the pumps' rates combine without interference.
Read graph scales and dimensional changes
An axis labelled “distance, hundreds of kilometres” gives 250 kilometres at a plotted value of 2.5. The displayed coordinate is not automatically the quantity itself. Read the label before putting the number into a rate equation.
A scale factor of 3 multiplies lengths by 3, areas by 9, and volumes by 27 for similar figures. Decide what kind of quantity is being scaled rather than applying the length factor to every measurement.
Check the setup before cross-multiplying
For two ratios, confirm corresponding quantities and compatible units. For a rate, check the requested unit. For a total, check whether a ratio is part-to-part or part-to-whole. A correctly solved equation can still answer the wrong question.
Record these setup errors separately from arithmetic mistakes. For changing percentages, continue to the percentage multiplier method.
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