In Grade 3 you saw with pictures that 1/2 and 2/4 name the same amount. Now you will see why, and get a rule that works for any fraction.
Idea clave
Multiply the numerator and the denominator by the same number, and you get an equivalent fraction: $\dfrac{2}{3} = \dfrac{2 \times 4}{3 \times 4} = \dfrac{8}{12}$. The pieces get smaller, there are more of them, and the amount stays the same.
Por qué funciona
Start with $\dfrac{2}{3}$ of a bar: 2 of 3 equal pieces are shaded. Now cut every piece into 4 smaller pieces. The bar has $3 \times 4 = 12$ pieces, and the shaded part has $2 \times 4 = 8$ of them. Nothing was added or taken away; you only cut. So $\dfrac{2}{3} = \dfrac{8}{12}$. Multiplying the bottom by 4 makes the pieces 4 times smaller; multiplying the top by 4 counts 4 times as many of them. The two changes cancel out.
Ejemplo resuelto: Find a missing numerator
Fill in: $\dfrac{3}{5} = \dfrac{?}{20}$
- The denominator went from 5 to 20: it was multiplied by 4.
- Multiply the numerator by 4 too: $3 \times 4 = 12$.
Respuesta: $\dfrac{3}{5} = \dfrac{12}{20}$
Ejemplo resuelto: Test a claim
Is $\dfrac{2}{3}$ equal to $\dfrac{4}{5}$?
- $\dfrac{4}{5}$ comes from adding 2 to the top and to the bottom, not multiplying.
- Compare with the same denominator: $\dfrac{2}{3} = \dfrac{10}{15}$ and $\dfrac{4}{5} = \dfrac{12}{15}$.
Respuesta: No: $\dfrac{2}{3} < \dfrac{4}{5}$.
Error común
Adding the same number to the top and bottom, thinking $\dfrac{2}{3} = \dfrac{4}{5}$. Adding does not cut pieces evenly, so the amount changes. Only multiplying (or dividing) both by the same number keeps it the same.