Card 1 of 20
EasyStudy card 1
Prompt
What is a ratio?
The category named first occupies the first ratio position.
Answer
A comparison of two quantities by division in a specified order.
Example
\(12\) blue to \(8\) gold is \(12:8=3:2\).
MathChapter 6: Ratios, Rates, and Proportions
Flashcards
Ratios and Rates Flashcards
Twenty focused cards on ratio order, equivalent ratios, unit rates, formulas, multi-part totals, units, and common traps.
Current card1 / 20
Session progress0 of 20 studied
Card 1 of 20. Answer hidden.
Card 2 of 20
EasyStudy card 2
Prompt
What three notations can express the ratio of \(a\) to \(b\)?
Each notation preserves the same order.
Answer
\(a\) to \(b\), \(a:b\), and \(a/b\).
Example
\(3\) to \(5\), \(3:5\), and \(3/5\) are equivalent.
Card 3 of 20
EasyStudy card 3
Prompt
How are quantities in ratio \(a:b\) represented with a common scale factor?
Both categories must scale together.
Answer
As \(ax\) and \(bx\) for the same positive scale factor \(x\).
Example
A \(2:7\) ratio can produce \(6\) and \(21\) when \(x=3\).
Card 4 of 20
EasyStudy card 4
Prompt
What is a rate?
Look for a compound unit joined by ‘per.’
Answer
A ratio comparing measurements with different units.
Example
\(72\) kilometers per hour is a rate.
Card 5 of 20
EasyStudy card 5
Prompt
What is a unit rate?
Normalize the denominator quantity.
Answer
A rate whose denominator is \(1\) unit.
Example
\(24\) pages per \(1\) minute is a unit rate.
Card 6 of 20
MediumStudy card 6
Prompt
How is unit price calculated?
The requested unit is dollars per item.
Answer
Divide package price by the number of units in the package.
Example
\(\$14/20=\$0.70\) per item.
Card 7 of 20
MediumStudy card 7
Prompt
How is gas mileage in miles per gallon calculated?
Miles belong above gallons.
Answer
Divide miles traveled by gallons of fuel used.
Example
\(360/12=30\) miles per gallon.
Card 8 of 20
MediumStudy card 8
Prompt
How is speed calculated from distance and time?
Use compatible distance and time units.
Answer
\(\text{speed}=\text{distance}/\text{time}\).
Example
\(150\) kilometers in \(3\) hours is \(50\) kilometers per hour.
Card 9 of 20
MediumStudy card 9
Prompt
How is density calculated?
Mass belongs in the numerator.
Answer
\(\text{density}=\text{mass}/\text{volume}\).
Example
\(240\) grams over \(20\) cubic centimeters is \(12\) grams per cubic centimeter.
Card 10 of 20
MediumStudy card 10
Prompt
Why does numerator-denominator order matter in a rate?
Miles per gallon is not gallons per mile.
Answer
Reversing the order creates the reciprocal rate and answers a different question.
Example
\(30\) miles per gallon has reciprocal \(1/30\) gallon per mile.
Card 11 of 20
MediumStudy card 11
Prompt
How do you distribute a known total through ratio \(a:b:c\)?
Add every ratio part before finding \(x\).
Answer
Represent the categories as \(ax\), \(bx\), and \(cx\), then solve \((a+b+c)x=\text{total}\).
Example
For \(2:3:5\) totaling \(100\), one part is \(10\).
Card 12 of 20
MediumStudy card 12
Prompt
What mistake occurs when a total is treated as one ratio part?
Use the sum of the ratio coefficients.
Answer
Every category becomes too large because the total represents the sum of all scaled parts.
Example
For \(3:4\) totaling \(56\), solve \(7x=56\), not \(3x=56\).
Card 13 of 20
MediumStudy card 13
Prompt
How do you simplify a ratio?
Do not divide only one position.
Answer
Divide every part by the same greatest common factor.
Example
\(24:36\) simplifies to \(2:3\).
Card 14 of 20
MediumStudy card 14
Prompt
How can you check two ratios for equivalence without decimals?
One multiplier must transform the entire ratio.
Answer
Compare cross products or verify that both parts use the same scale factor.
Example
\(4:7\) and \(20:35\) both use factor \(5\).
Card 15 of 20
MediumStudy card 15
Prompt
In a triangle-angle ratio problem, what total should the scaled angle expressions equal?
Use the angle sum of a triangle.
Answer
\(180^\circ\).
Example
For \(2:3:4\), solve \(2x+3x+4x=180\).
Card 16 of 20
MediumStudy card 16
Prompt
How should mixed measurement units be handled before calculating a rate?
A compound unit cannot mix unmatched time scales silently.
Answer
Convert them to compatible units, then divide in the requested order.
Example
Convert \(1.5\) hours to \(90\) minutes before finding pages per minute.
Card 17 of 20
MediumStudy card 17
Prompt
What is the best first step when a question asks for ‘items per dollar’?
The wording specifies the fraction's order.
Answer
Write items above dollars before inserting the values.
Example
\(12\) items for \(\$3\) gives \(12/3=4\) items per dollar.
Card 18 of 20
HardStudy card 18
Prompt
How can a relation such as \(5p=3q\) be converted to ratio \(p:q\)?
Coefficients move to the opposite ratio positions.
Answer
Isolate \(p/q\): \(p/q=3/5\), so \(p:q=3:5\).
Example
Dividing \(5p=3q\) by \(5q\) gives \(p/q=3/5\).
Card 19 of 20
HardStudy card 19
Prompt
How do you solve a ratio problem when the difference between categories is known?
The coefficient difference represents the stated difference.
Answer
Subtract the scaled expressions, solve for the common multiplier, then find the requested amount.
Example
For \(7x-3x=20\), \(x=5\).
Card 20 of 20
HardStudy card 20
Prompt
What final checks should a ratio or rate answer pass?
Audit meaning as well as arithmetic.
Answer
Correct order, simplified value, compatible units, and agreement with the original total or measurements.
Example
A mileage answer should be labeled miles per gallon and reproduce the total miles when multiplied by gallons.
Deck complete
You rated every card. Revisit difficult cards or shuffle for another pass.
Keyboard: Space reveals, left/right arrows navigate, 1–4 rate, and S shuffles.