Question

b.
the figure shows two triangles. the left triangle is divided into 6 small triangles, 4 of which are shaded. the right triangle is divided into 3 small triangles. the equation is \\(\frac{4}{6} = \frac{\quad}{3}\\)

Explanation:

Step1: Simplify the fraction $\frac{4}{6}$

To simplify a fraction, we divide both the numerator and the denominator by their greatest common divisor (GCD). The GCD of 4 and 6 is 2. So, we divide the numerator and denominator of $\frac{4}{6}$ by 2: $\frac{4\div2}{6\div2}=\frac{2}{3}$? Wait, no, wait. Wait, the right side denominator is 3. Wait, maybe I made a mistake. Wait, let's check again. Wait, the left triangle is divided into 6 equal parts, and 4 are shaded. The right triangle is divided into 3 equal parts. Let's see the equivalent fraction. Wait, $\frac{4}{6}$ can be simplified by dividing numerator and denominator by 2: $\frac{4\div2}{6\div2}=\frac{2}{3}$? But the right side is $\frac{?}{3}$. Wait, no, maybe the problem is to find the numerator for $\frac{?}{3}$ such that it's equivalent to $\frac{4}{6}$. Let's use the concept of equivalent fractions: $\frac{4}{6}=\frac{x}{3}$. Cross - multiply: $6x = 4\times3=12$. Then $x=\frac{12}{6}=2$. Wait, but let's check the shading. The left triangle has 6 parts, 4 shaded. The right has 3 parts. If we simplify $\frac{4}{6}$, divide numerator and denominator by 2, we get $\frac{2}{3}$. Wait, maybe the original problem is to fill in the blank. Wait, the equation is $\frac{4}{6}=\frac{\square}{3}$. To find the numerator, we can see that 6 divided by 2 is 3, so we divide the numerator 4 by 2 as well. $4\div2 = 2$. So $\frac{4}{6}=\frac{2}{3}$. Wait, but let's check the triangles. The left triangle: 6 small triangles, 4 shaded. The right triangle: 3 larger triangles. If we group the 6 small triangles into 3 groups of 2, then 4 shaded small triangles would be 2 groups of 2, so 2 shaded larger triangles. So the numerator should be 2.

Step2: Verify the equivalent fraction

We know that two fractions $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent if $a\times d=b\times c$. For $\frac{4}{6}$ and $\frac{2}{3}$, $4\times3 = 12$ and $6\times2 = 12$. So they are equivalent.

Answer:

The missing numerator is 2, so $\frac{4}{6}=\frac{2}{3}$