Questions: On a number line, point D is at -3, and point E is at 6. The point F lies on DE. The ratio of DF to FE is 2:3. Where does point F lie on the number line? Point F is at on the number line.

On a number line, point D is at -3, and point E is at 6. The point F lies on DE. The ratio of DF to FE is 2:3. Where does point F lie on the number line?

Point F is at on the number line.

Transcript text: On a number line, point D is at -3 , and point E is at 6 . The point F lies on $\overline{\mathrm{DE}}$. The ratio of $D F$ to $F E$ is $2: 3$. Where does point F lie on the number line? Point $F$ is at $\square$ on the number line.

Solution

Solution Steps

To find the position of point F on the number line, we need to divide the segment DE into parts that reflect the given ratio of DF to FE, which is 2:3. This means the entire segment DE is divided into 5 equal parts, with DF consisting of 2 parts and FE consisting of 3 parts. We can calculate the length of each part and then determine the position of F by adding the length of DF to the position of D.

Step 1: Determine the Total Length of Segment DE

The points D and E are located at $-3$ and $6$ on the number line, respectively. The total length of segment DE is calculated as: \[ \text{Length of DE} = E - D = 6 - (-3) = 9 \]

Step 2: Divide the Segment DE According to the Given Ratio

The ratio of $DF$ to $FE$ is $2:3$. This means the segment DE is divided into $2 + 3 = 5$ equal parts. Each part has a length of: \[ \text{Part length} = \frac{\text{Length of DE}}{\text{Total parts}} = \frac{9}{5} = 1.8 \]

Step 3: Calculate the Position of Point F

Point F is located such that $DF$ consists of 2 parts. Therefore, the position of F is: \[ F = D + 2 \times \text{Part length} = -3 + 2 \times 1.8 = -3 + 3.6 = 0.6 \]

Final Answer

The position of point F on the number line is $\boxed{0.6}$.

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