Card 1 of 20
EasyStudy card 1
Prompt
What is a conversion factor?
The units differ, but the measurements represent the same amount.
Answer
A fraction made from equivalent measurements, so its value is \(1\).
Example
\(60\text{ min}/1\text{ h}=1\).
MathChapter 6: Ratios, Rates, and Proportions
Flashcards
Ratios, Rates, and Proportions Word Problems Flashcards
Twenty focused cards on conversion factors, cancellation, scales, recipes, production, group ratios, dimensions, and proportional validity.
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Card 2 of 20
EasyStudy card 2
Prompt
How should a conversion factor be oriented?
Place the unwanted unit opposite its occurrence in the given quantity.
Answer
So the unwanted unit cancels and the desired unit remains.
Example
Feet to inches uses \(12\text{ in}/1\text{ ft}\).
Card 3 of 20
EasyStudy card 3
Prompt
Why is unit cancellation mathematically valid?
Treat units like algebraic factors.
Answer
Units multiply and divide with their quantities, and a conversion factor equals \(1\).
Example
\(18\text{ ft})(12\text{ in}/1\text{ ft})=216\text{ in}\).
Card 4 of 20
EasyStudy card 4
Prompt
What indicates an incorrectly oriented conversion factor?
Audit units before numbers.
Answer
The unwanted unit appears twice or remains instead of canceling.
Example
\(18\text{ ft})(1\text{ ft}/12\text{ in})\) leaves square feet per inch, not inches.
Card 5 of 20
MediumStudy card 5
Prompt
How should multi-step conversions be organized?
Plan the unit path first.
Answer
As one chain in which each factor cancels the current unwanted unit and introduces the next needed unit.
Example
Miles per hour can become kilometers per minute using a distance factor and a time factor.
Card 6 of 20
MediumStudy card 6
Prompt
Why should intermediate conversion results not be rounded early?
Keep exact fractions or full calculator precision.
Answer
Rounding error can accumulate through later multiplication or division.
Example
Retain \(5/12\) instead of replacing it too early with \(0.42\).
Card 7 of 20
MediumStudy card 7
Prompt
How does a linear conversion affect area units?
Area has two dimensions.
Answer
Square the linear conversion factor.
Example
\(1\text{ yd}^2=(3\text{ ft})^2=9\text{ ft}^2\).
Card 8 of 20
MediumStudy card 8
Prompt
How does a linear conversion affect volume units?
Volume has three dimensions.
Answer
Cube the linear conversion factor.
Example
\(1\text{ m}^3=(100\text{ cm})^3=1{,}000{,}000\text{ cm}^3\).
Card 9 of 20
MediumStudy card 9
Prompt
What correspondence must a scale model preserve?
Keep the same ratio orientation.
Answer
Model measurement must align with model measurement and actual measurement with actual measurement.
Example
\(2\text{ cm}/5\text{ m}=8\text{ cm}/x\text{ m}\).
Card 10 of 20
MediumStudy card 10
Prompt
How is a recipe scaled correctly?
Do not add one fixed amount to every ingredient.
Answer
Multiply every ingredient amount by the same serving or batch scale factor.
Example
Doubling servings doubles each ingredient.
Card 11 of 20
MediumStudy card 11
Prompt
When is production proportional to machine count?
Check the stated assumptions.
Answer
When machine efficiency, operating time, and conditions remain unchanged.
Example
Doubling identical machines for the same time doubles output.
Card 12 of 20
MediumStudy card 12
Prompt
What is a machine-hour?
It combines two proportional inputs.
Answer
One machine operating for one hour; total machine-hours equal machines multiplied by hours.
Example
\(6\) machines for \(4\) hours provide \(24\) machine-hours.
Card 13 of 20
MediumStudy card 13
Prompt
When is the common-multiplier approach efficient for a group ratio?
Represent all groups using one \(x\).
Answer
When a total, a changed category, or several category amounts are involved.
Example
For \(3:5\), use \(3x\) and \(5x\).
Card 14 of 20
MediumStudy card 14
Prompt
When is a proportion efficient for a group ratio?
Align category over category.
Answer
When one category amount is known and one corresponding category amount is requested.
Example
\(3/5=18/x\) directly finds the second group.
Card 15 of 20
MediumStudy card 15
Prompt
What form identifies a directly proportional relationship?
Its graph passes through the origin.
Answer
\(y=kx\), with constant ratio \(y/x=k\) and no additive offset.
Example
Cost \(C=4n\) is proportional to item count \(n\).
Card 16 of 20
MediumStudy card 16
Prompt
Why is a fixed-fee relationship not proportional?
Linear does not always mean proportional.
Answer
The output-to-input ratio changes because the relationship does not pass through the origin.
Example
\(C=3m+12\) includes a fixed \(12\)-dollar fee.
Card 17 of 20
MediumStudy card 17
Prompt
What is additive reasoning, and why can it fail in ratio problems?
Ratios preserve multiplicative structure.
Answer
It adds the same amount to quantities instead of multiplying them by one common factor, so the ratio may change.
Example
Adding \(2\) to both parts of \(2:5\) gives \(4:7\), not an equivalent ratio.
Card 18 of 20
HardStudy card 18
Prompt
How can a changing group-ratio problem be modeled?
Change only the affected category.
Answer
Write original categories as scaled ratio parts, apply the stated addition or removal, then form the new ratio equation.
Example
From \(3x:5x\), adding \(8\) to the first group gives \((3x+8)/(5x)\).
Card 19 of 20
HardStudy card 19
Prompt
What is the safest way to solve a conversion-heavy SAT problem?
Let units expose inverted factors.
Answer
Write one factor chain, cancel units visibly, keep full precision, and compare the remaining unit with the question.
Example
A final distance answer should not retain hours, minutes, or miles if kilometers were requested.
Card 20 of 20
HardStudy card 20
Prompt
What final test determines whether proportional reasoning is appropriate?
Do not assume every word problem is proportional.
Answer
Check for one constant multiplicative factor under unchanged conditions and verify there is no fixed additive component.
Example
Output per worker may stop being constant if crowding changes efficiency.
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