To determine where point H lies on the number line given that point F is at 4 and point G is at -2 with the ratio of FH to HG as 3:9, follow these steps:
1. Understand the Positions and Ratio:
- Point F is at position 4.
- Point G is at position -2.
- The ratio FH:HG is 3:9. This can be simplified to 1:3 (dividing both terms by 3).
2. Calculate the Total Distance:
- The distance between F and G is calculated as:
[tex]\[
\text{Total distance} = F - G
\][/tex]
Here:
[tex]\[
\text{Total distance} = 4 - (-2) = 4 + 2 = 6
\][/tex]
3. Simplify the Ratio:
- The simplified ratio FH:HG is 1:3. This means that the segment from F to H is one-quarter of the total distance from F to G, and the segment from H to G is three-quarters of the total distance.
4. Determine the Distance FH:
- Calculate FH using the simplified ratio:
[tex]\[
\text{distance } FH = \text{Total distance} \times \frac{1}{1+3} = 6 \times \frac{1}{4} = 1.5
\][/tex]
5. Find the Position of H:
- From the position of F, move 1.5 units towards G:
[tex]\[
H = F - \text{distance } FH
\][/tex]
Therefore:
[tex]\[
H = 4 - 1.5 = 2.5
\][/tex]
Therefore, point H is at position 2.5 on the number line.
