A proportion is an equation stating that two ratios are equal. The arithmetic is usually short; the main reasoning task is pairing corresponding quantities in the same numerator-denominator order.
Structure and vocabulary
The values \(a\) and \(d\) are the extremes; \(b\) and \(c\) are the means. Denominators \(b\) and \(d\) must be nonzero.
Why cross products are equal
Start with \(a/b=c/d\). Multiplying both sides by the common denominator \(bd\) clears the fractions: \(bd(a/b)=bd(c/d)\). Canceling \(b\) on the left and \(d\) on the right produces \(ad=bc\). This is ordinary multiplication of equal quantities, not a separate trick.
This conclusion assumes \(b\ne0\) and \(d\ne0\).
Verifying a proposed proportion
Cross-product check
- Preserve order
Write the two ratios with corresponding quantities aligned.
- Compute both products
Find the product of the extremes and the product of the means.
- Compare
Equal cross products confirm a proportion; unequal products reject it.
Decimal verification
Do \(1.8/4.5\) and \(2.4/6\) form a proportion?
- Write the equality
Test \(1.8/4.5=2.4/6\).
- Cross products
\(1.8(6)=10.8\) and \(4.5(2.4)=10.8\).
- Compare
The products are equal.
Solving proportions
A missing numerator
Solve \(5/9=x/27\).
- Cross products
\(5)(27)=9x\).
- Simplify
\(135=9x\).
- Solve
\(x=15\).
- Check
\(5/9=15/27\) after simplifying both ratios.
An expression in a denominator
Solve \(4/(x+2)=6/9\).
- Restriction
Require \(x+2\ne0\), so \(x\ne-2\).
- Cross products
\(4)(9)=6(x+2)\).
- Solve
\(36=6x+12\), so \(24=6x\) and \(x=4\).
- Check the denominator
\(4+2=6\ne0\), so the solution is allowed.
Maps, models, and scale factors
A scale proportion must pair model measurement with model measurement and actual measurement with actual measurement. Linear scale factors apply once to lengths. For areas, the linear scale factor is squared; a map rectangle scaled by \(k\) in each dimension has area scaled by \(k^2\).
| Ratio position | First object | Second object |
|---|---|---|
| Numerator | model length | model length |
| Denominator | actual length | actual length |
Model-to-actual scale
A display model uses \(3\) centimeters for \(8\) meters. How many actual meters correspond to \(16.5\) model centimeters?
- Align
\(3\text{ cm}/8\text{ m}=16.5\text{ cm}/x\text{ m}\).
- Cross products
\(3x=8(16.5)=132\).
- Solve
\(x=44\).
Common proportion mistakes
- Mismatching corresponding quantities between the two ratios.
- Reversing only one ratio instead of reversing both.
- Multiplying adjacent terms rather than the actual cross products.
- Using addition because both quantities increased, even though proportional scaling is multiplicative.
- Dropping units and failing to notice an inverted setup.
- Rounding an intermediate scale factor before the final calculation.
- Accepting a solution that makes an original denominator equal to zero.
Check your understanding
Which equality correctly compares \(4\) notebooks costing \(\$7\) with \(x\) notebooks costing \(\$21\)?
- \(4/7=x/21\)
- \(4/7=21/x\)
- \(7/4=x/21\)
- \(4+x=7+21\)
Show answer and explanation
Answer: \(4/7=x/21\)
Notebook counts occupy both numerators and dollar costs occupy both denominators.
What to remember
- A proportion is an equality between two ratios with matching correspondence.
- Cross products follow from clearing nonzero denominators in an ordinary equation.
- Equal cross products verify a proportion; unequal products disprove it.
- Record denominator restrictions before solving expression-based proportions.
- For scale problems, distinguish a linear scale factor from an area scale factor.
Put these notes into practice
Apply the ideas with SAT-style questions, then reinforce key details with flashcards.