Ratios and rates both compare quantities by division. A ratio usually compares quantities in the same or related categories, while a rate compares measurements carrying different units. SAT questions often test the order of a comparison as carefully as its arithmetic.
Ratio language and notation
- Ratio
- A comparison of two quantities by division, written in a specified order.
All three forms compare \(a\) with \(b\). The first quantity is the numerator when fraction notation is used.
Equivalent ratios and common scale factors
If two quantities are in the ratio \(a:b\), they can be represented as \(ax\) and \(bx\) for a common positive scale factor \(x\). Multiplying or dividing both parts by the same nonzero value preserves the comparison.
Recover quantities from a ratio and a total
A workshop has mentors and students in the ratio \(3:8\). There are \(55\) people altogether. How many are mentors?
- Represent both groups
Mentors \(=3x\) and students \(=8x\).
- Use the total
\(3x+8x=55\), so \(11x=55\).
- Find the scale factor
\(x=5\).
- Answer the requested category
Mentors \(=3(5)=15\).
Rates and unit rates
- Rate
- A ratio comparing measurements with different units.
- Unit rate
- A rate rewritten per \(1\) unit of the denominator quantity.
| Quantity | Formula | Example unit |
|---|---|---|
| Unit price | \(\text{package price}/\text{number of units}\) | dollars per item |
| Gas mileage | \(\text{miles traveled}/\text{gallons used}\) | miles per gallon |
| Speed | \(\text{distance}/\text{time}\) | kilometers per hour |
| Density | \(\text{mass}/\text{volume}\) | grams per cubic centimeter |
| Reading rate | \(\text{pages}/\text{time}\) | pages per minute |
Compare package prices
A box of \(18\) markers costs \(\$12.60\). A box of \(25\) markers costs \(\$16.25\). Which has the lower unit price?
- First unit price
\(12.60/18=0.70\), or \(\$0.70\) per marker.
- Second unit price
\(16.25/25=0.65\), or \(\$0.65\) per marker.
- Compare like units
Both answers are dollars per marker, and \(0.65<0.70\).
Numerator, denominator, and unit order
A rate and its reciprocal answer different questions
Miles per gallon
\(\text{miles}/\text{gallon}\) measures distance obtained from one gallon. Larger values generally mean better fuel efficiency.
Gallons per mile
\(\text{gallons}/\text{mile}\) measures fuel used for one mile. It is the reciprocal and answers a different question.
Distributing a total through a multi-part ratio
Ratio-parts method
- Represent categories
For ratio \(a:b:c\), write the quantities as \(ax\), \(bx\), and \(cx\).
- Add the parts
The total is \((a+b+c)x\).
- Solve for the scale
Set the ratio-parts total equal to the known total and find \(x\).
- Find the requested amount
Multiply \(x\) by the requested category's ratio part.
- Check
The category amounts should add to the original total and simplify to the stated ratio.
Triangle-angle distribution
The angles of a triangle are in the ratio \(2:3:4\). Find the largest angle.
- Represent
The angles are \(2x\), \(3x\), and \(4x\).
- Use the angle sum
\(2x+3x+4x=180\), so \(9x=180\).
- Solve
\(x=20\).
- Find the largest
\(4x=4(20)=80\).
Dimensions from a ratio
A rectangular display has length-to-width ratio \(5:3\) and perimeter \(128\) centimeters. Find its width.
- Represent
Let length \(=5x\) and width \(=3x\).
- Use perimeter
\(2(5x)+2(3x)=128\), so \(16x=128\).
- Solve and answer
\(x=8\), so width \(=3(8)=24\) centimeters.
Common ratio and rate mistakes
- Reversing the requested comparison, such as reporting yellow-to-green when green-to-yellow was requested.
- Ignoring units or combining measurements before converting them to compatible units.
- Calling a rate a unit rate before its denominator equals \(1\).
- Treating \(a:b\) as \(a+b\) instead of a comparison by division.
- Using the known total as one ratio part rather than as the sum of all scaled parts.
- Adding the ratio parts incorrectly before solving for the common scale factor.
- Computing gallons per mile when the question asks for miles per gallon.
Check your understanding
A printer produces \(168\) pages in \(7\) minutes. Which expression gives pages per minute?
- \(168/7\)
- \(7/168\)
- \(168+7\)
- \(168(7)\)
Show answer and explanation
Answer: \(168/7\)
The requested numerator is pages and the requested denominator is minutes, giving \(24\) pages per minute.
What to remember
- A ratio's order must match the wording of the requested comparison.
- Equivalent ratios multiply or divide every part by one common scale factor.
- A unit rate has denominator \(1\) and must include its compound units.
- For a known total, add the ratio parts before solving for the common multiplier.
- Check both numerical value and unit orientation before choosing an SAT answer.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.