Card 1 of 20
EasyStudy card 1
Prompt
What is a proportion?
Corresponding quantities must occupy matching positions.
Answer
An equation stating that two ratios are equal.
Example
\(3/5=12/20\) is a proportion.
MathChapter 6: Ratios, Rates, and Proportions
Flashcards
Proportions Flashcards
Twenty focused cards on proportion structure, cross products, correspondence, denominator restrictions, scales, and common mistakes.
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Card 2 of 20
EasyStudy card 2
Prompt
In \(a/b=c/d\), which terms are the extremes?
They occupy the outer positions.
Answer
\(a\) and \(d\).
Example
In \(3/7=9/21\), the extremes are \(3\) and \(21\).
Card 3 of 20
EasyStudy card 3
Prompt
In \(a/b=c/d\), which terms are the means?
They occupy the inner positions.
Answer
\(b\) and \(c\).
Example
In \(3/7=9/21\), the means are \(7\) and \(9\).
Card 4 of 20
EasyStudy card 4
Prompt
What cross-product equality follows from \(a/b=c/d\)?
Multiply diagonally opposite terms.
Answer
\(ad=bc\), provided \(b\ne0\) and \(d\ne0\).
Example
From \(3/5=12/20\), \(3(20)=5(12)\).
Card 5 of 20
MediumStudy card 5
Prompt
Why do cross products work?
It is ordinary multiplication of an equation.
Answer
Multiplying both sides of \(a/b=c/d\) by the common denominator \(bd\) clears the fractions and leaves \(ad=bc\).
Example
\(bd(a/b)=bd(c/d)\) simplifies to \(ad=bc\).
Card 6 of 20
MediumStudy card 6
Prompt
How do you verify whether two ratios form a proportion?
Equal products confirm equality of the ratios.
Answer
Compute both cross products and compare them.
Example
\(4/9\) and \(12/27\) work because \(4(27)=9(12)=108\).
Card 7 of 20
MediumStudy card 7
Prompt
What should be checked before solving a proportion with a variable denominator?
Restrictions come from the unsimplified equation.
Answer
Identify values that make any original denominator zero and exclude them.
Example
In \(5/(x-2)=3/7\), require \(x\ne2\).
Card 8 of 20
MediumStudy card 8
Prompt
How do you solve \(a/b=x/d\) for \(x\)?
Follow the correct cross-product diagonal.
Answer
Use \(bx=ad\), so \(x=ad/b\).
Example
\(4/7=x/35\) gives \(x=20\).
Card 9 of 20
MediumStudy card 9
Prompt
How do you solve \(a/b=c/x\) for \(x\)?
The variable is part of a denominator but its cross product is linear.
Answer
Use \(ax=bc\), so \(x=bc/a\).
Example
\(6/15=10/x\) gives \(x=25\).
Card 10 of 20
MediumStudy card 10
Prompt
What makes a proportion's correspondence valid?
Label the positions before inserting values.
Answer
Each numerator describes the same kind of quantity, and each denominator describes another matching kind.
Example
Items over dollars must equal items over dollars.
Card 11 of 20
MediumStudy card 11
Prompt
Can both ratios in a proportion be reversed?
Reverse both, not only one.
Answer
Yes. If \(a/b=c/d\), then \(b/a=d/c\) when all denominators are nonzero.
Example
\(2/3=8/12\) implies \(3/2=12/8\).
Card 12 of 20
MediumStudy card 12
Prompt
Why is reversing only one ratio usually wrong?
Both sides must describe the same ordered comparison.
Answer
It breaks the correspondence and replaces one ratio with its reciprocal while leaving the other unchanged.
Example
\(2/5\) does not equal \(15/6\), though it equals \(6/15\).
Card 13 of 20
MediumStudy card 13
Prompt
How should decimals be handled in a proportion?
Early decimal division can hide equality or introduce error.
Answer
Use exact cross products and postpone rounding until the final answer.
Example
Compare \(1.2(10)\) with \(3(4)\), not rounded decimal approximations.
Card 14 of 20
MediumStudy card 14
Prompt
What ratio setup works for a scale drawing?
Keep like quantities in matching positions.
Answer
Model over actual equals model over actual, or the reciprocal on both sides.
Example
\(2\text{ cm}/5\text{ m}=8\text{ cm}/x\text{ m}\).
Card 15 of 20
MediumStudy card 15
Prompt
How does a linear scale factor affect area?
Area has two dimensions.
Answer
Area is multiplied by the square of the linear scale factor.
Example
A length factor of \(3\) produces an area factor of \(3^2=9\).
Card 16 of 20
MediumStudy card 16
Prompt
What is a common wrong-cross-product pattern?
Trace the two diagonals.
Answer
Multiplying terms on the same side or row instead of diagonally opposite terms.
Example
For \(3/5=x/20\), use \(3(20)=5x\), not \(3x=5(20)\).
Card 17 of 20
MediumStudy card 17
Prompt
How can a solved proportion be checked?
Check arithmetic and domain together.
Answer
Substitute the value, confirm both ratios are equal, and verify no denominator is zero.
Example
For \(5/8=x/24\), \(x=15\) gives \(5/8=15/24\).
Card 18 of 20
HardStudy card 18
Prompt
What indicates that a real-world relationship is proportional?
A proportional graph passes through the origin.
Answer
Every corresponding pair has the same multiplicative ratio and the relationship has no fixed additive offset.
Example
A constant price per item with no fixed fee is proportional.
Card 19 of 20
HardStudy card 19
Prompt
Why does equal additive change not prove proportionality?
Test ratios, not differences.
Answer
Proportional relationships preserve a constant factor, not merely a constant difference.
Example
Pairs \((2,5)\) and \((4,7)\) share difference \(3\) but not one ratio.
Card 20 of 20
HardStudy card 20
Prompt
What final audit should a scale proportion pass?
A correct cross product cannot repair mismatched quantities.
Answer
Matching model/actual order, compatible units, correct linear or area factor, and a contextually reasonable result.
Example
If model length increases, corresponding actual length should scale in the same direction.
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