SAT · Problem-Solving and Data Analysis · TOPIC LESSON
Keep quantities proportional and let units guide every conversion.
BY THE END, YOU CAN
BEFORE YOU START
Original educational lesson. Work through the examples before revealing the practice answers. This lesson supports SAT Math and Reading and Writing skill study; it does not reproduce official questions or guarantee a score.
LEARN THROUGH ORIGINAL PRACTICE
The teaching passages, worked examples and exercises in this lesson are original. They are not official College Board questions or past-exam reproductions. Blueprint and observed-practice analytics describe separate factual metadata; those source questions are not displayed here.
01 / UNDERSTAND & APPLY
The ratio a:b means a/b when b ≠ 0; it does not by itself state the total. If red:blue = 3:5, write red = 3k and blue = 5k, so total = 8k. Red is 3/8 of the total, not 3/5. Keep the order attached to the quantity names.
Equivalent ratios multiply both quantities by the same factor. Adding the same number to both does not generally preserve a ratio.
02 / UNDERSTAND & APPLY
Add the ratio parts, divide the total by that sum to find the size of one part, then multiply by each component. Whole-number objects require whole-number counts; a model with fractional counts may be incompatible with the context.
ORIGINAL WORKED EXAMPLE
A mosaic has 72 tiles with green:white = 5:7. How many green tiles are there?
Answer 30 green tiles
03 / UNDERSTAND & APPLY
A direct proportional relationship is y = kx, with a constant ratio y/x = k for nonzero x. A fixed starting fee makes y = kx + b nonproportional when b ≠ 0 even though it remains linear. Use a proportion only when the underlying rate or scale is constant.
ORIGINAL WORKED EXAMPLE
A recipe uses 450 g of flour for 12 rolls. At the same recipe ratio, how much flour is needed for 20 rolls?
Answer 750 g
04 / UNDERSTAND & APPLY
Speed is distance/time, density is mass/volume, and a unit price is cost/quantity. State which quantity is “per” which. If a vehicle travels equal distances at two different speeds, averaging the speeds directly is wrong because it spends different amounts of time at those speeds. Total average speed is always total distance divided by total time.
ORIGINAL WORKED EXAMPLE
A cyclist travels 18 km in 1 hour and then 12 km in 0.5 hour. What is the average speed for the whole trip?
Answer 20 km/h
05 / UNDERSTAND & APPLY
A conversion factor represents equal quantities in different units, such as 100 cm/1 m. Arrange it so the unwanted unit cancels. In a multistep conversion, keep units through the full product and round only at the end if needed.
When the unit is in a denominator, its conversion direction reverses. Converting “per minute” to “per hour” multiplies the numerical rate by 60, whereas converting a duration in minutes to hours divides by 60.
ORIGINAL WORKED EXAMPLE
Convert 7.5 meters per second to kilometers per hour.
Answer 27 km/h
06 / UNDERSTAND & APPLY
A scale drawing gives corresponding lengths, not areas directly. If a drawing uses 1 cm for 3 m, multiply drawing lengths by 3 m/cm. For same-unit similar shapes with length ratio k, perimeter scales by k, area by k² and volume by k³. A two-dimensional drawing can encode a three-dimensional object only when the needed dimensions are given.
Full-size scientific diagram. Swipe or scroll horizontally; keyboard users can focus the diagram and use arrow keys.
ORIGINAL WORKED EXAMPLE
A garden is shown as 6 cm by 4 cm at scale 1 cm : 3 m. Find its real area.
Answer 216 m²
07 / UNDERSTAND & APPLY
Because 1 m = 100 cm, 1 m² = (100 cm)² = 10,000 cm² and 1 m³ = 1,000,000 cm³. Applying a length conversion only once to an area or volume is a dimensional error. Density conversions combine a mass conversion with a cubed length conversion.
ORIGINAL WORKED EXAMPLE
Convert 0.08 m³ to cm³.
Answer 80,000 cm³
08 / UNDERSTAND & APPLY
Some derived units are products rather than quotients. Energy in kilowatt-hours equals power in kilowatts multiplied by operating time in hours. A kilowatt measures a rate of energy use; a kilowatt-hour measures an amount of energy. Do not treat kW and kWh as interchangeable.
If an appliance changes power level, add the energy used during each interval rather than multiplying one rate by the entire time.
ORIGINAL WORKED EXAMPLE
A heater runs at 1.8 kW for 2 hours and then at 0.6 kW for 3 hours. How much electrical energy does it use?
Answer 5.4 kWh
09 / UNDERSTAND & APPLY
A quantity expressed in a smaller unit has a larger numerical count: 2 m is 200 cm. The physical length stays the same. A rate can change in the opposite numerical direction if the smaller unit is in its denominator, so use cancellation instead of that heuristic alone.
Calculator arithmetic helps with several factors, but only a correctly arranged model gives meaningful units. Compare magnitude with a rough estimate; a walking speed of thousands of kilometers per hour signals a conversion error.
10 / UNDERSTAND & APPLY
Keep exact values until the final step.
ORIGINAL PRACTICE · 1
27
There are 5 parts, each 9; blue uses 3 parts.
ORIGINAL PRACTICE · 2
$3.50
Divide cost by count: 17.50/5 = 3.50 dollars per notebook.
ORIGINAL PRACTICE · 3
No
A direct proportion has form y = kx and goes through the origin; this rule has a fixed offset 9.
ORIGINAL PRACTICE · 4
0.9 m
Multiply by 1 m/100 cm, giving 90/100 m.
ORIGINAL PRACTICE · 5
150 minutes
Multiply by 60 minutes per hour.
ORIGINAL PRACTICE · 6
30,000 cm²
The length factor 100 must be squared: 3 × 100².
ORIGINAL PRACTICE · 7
6 g/cm³
Density is mass divided by volume: 420/70.
ORIGINAL PRACTICE · 8
1:64
For similar three-dimensional objects, cube the length ratio: 1³:4³.
ORIGINAL PRACTICE · 9
45 liters
Two minutes is 120 seconds, three times the measured interval, so volume triples.
KEEP THE BIG IDEAS
READ WITH CONTEXT
The teaching notes, passages, worked explanations and practice are original. Official references identify the assessment framework and skill scope; they do not endorse this lesson or license official question reproduction.
Official skill taxonomy reference. All teaching prose, examples, diagrams and practice in this lesson are original; this is not a past-question collection.
SAT analytics distinguish official blueprint weights from independently classified linear-practice question occurrences. These are not historical administered-exam frequencies.