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On a number line, point F is at 4, and point G is at -2. Point H lies between point F and point G. If the ratio of FH to HG is 3:9, where does point H lie on the number line?
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.
