A typical problem gives three numbers from a proportional relationship and asks for the fourth: "5 notebooks cost \$8.75; how much for 7?" You already know $y = kx$ from writing equations. Here are three ways to finish the job.
Idea clave
Use the unit rate (find the amount for 1, then multiply), a scale factor (find what one amount was multiplied by), or cross products ($\frac{a}{b} = \frac{c}{d}$ means $ad = bc$). Each works only when the relationship is proportional.
Por qué funciona
Why cross products work: start with $\frac{a}{b} = \frac{c}{d}$ and multiply both sides by $b$ and by $d$. The $b$ cancels on the left and the $d$ cancels on the right, leaving $ad = cb$. It is not a trick; it is two multiplications done to both sides at once.
Worked examples
Ejemplo resuelto: Unit rate
5 notebooks cost \$8.75. How much do 7 notebooks cost?
- One notebook: $8.75 \div 5 = 1.75$ dollars.
- Seven notebooks: $1.75 \times 7 = 12.25$.
Respuesta: \$12.25
Ejemplo resuelto: Scale factor
Solve $\frac{3}{8} = \frac{x}{40}$.
- The denominator went from 8 to 40: that is $\times 5$.
- Do the same to the numerator: $x = 3 \times 5 = 15$.
Respuesta: $x = 15$
Ejemplo resuelto: Cross products
Solve $\frac{6}{x} = \frac{9}{12}$.
- Cross products are equal: $9x = 6 \times 12 = 72$.
- Divide by 9: $x = 8$. Check: $\frac{6}{8} = \frac{3}{4}$ and $\frac{9}{12} = \frac{3}{4}$ ✓.
Respuesta: $x = 8$
Error común
Two traps. Mismatched units: setting miles over gallons equal to gallons over miles. Write the units next to each number; both ratios must compare the same things in the same order. Not proportional: a plant is 4 cm tall and grows 2 cm each week. After 3 weeks it is 10 cm, and after 6 weeks it is 16 cm. The proportion $\frac{3}{10} = \frac{6}{x}$ would say 20 cm, which is wrong, because the height is not a fixed multiple of the weeks. Ask first: does doubling one amount double the other?