Cross-multiply proportions and compare fractions rapidly.
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When two fractions are equal, each numerator times the other denominator gives the same product: if \(\frac{a}{b} = \frac{c}{d}\), then \(ad = bc\).
Simplify before you cross-multiply to keep the numbers small.
Common slips: multiplying straight across (top times top), distributing to only the first term in the parentheses, and setting up the fractions with different quantities on top (keep the same quantity on top on both sides).
Cross-multiply, each numerator times the OTHER denominator: \(7(x - 1) = 3(x + 11)\). Distribute: \(7x - 7 = 3x + 33\). Collect terms: \(4x = 40\), so \(x = 10\). (Multiplying only the \(x\)-terms, \(7x - 1 = 3x + 11\), gives \(3\). Multiplying each numerator by its own denominator, \(3(x - 1) = 7(x + 11)\), gives \(-20\). Distributing \(3(x + 11)\) as \(3x - 33\) gives \(-13/2\).)
Cross-multiply: \(12 \times 15 = 20 \times x\), which gives \(x = 9\). (Cross-multiplying the wrong diagonal pair gives 25. Stopping at the cross product without dividing gives 180. Keeping the same difference instead of scaling, \(12 + (15 - 20)\), gives 7.)
\(k = 72 \div 20 = 3.6\), so at \(x = 27\), \(y = 3.6 \times 27 = 97.2\). (Using \(20 \div 72\) as the rate gives 7.5. Adding the change in \(x\), \(72 + 7\), gives 79. Multiplying \(72 \times 27\) without dividing by 20 gives 1944.)