Question

Locating a point at a fractional
distance on a number line

Answer 1

To locate a point at a fractional distance on a number line, you can follow these general steps:

1. **Identify the Endpoints:** Determine the two endpoints between which you want to find the fractional distance. Let’s denote these points as \( A \) (the left endpoint) and \( B \) (the right endpoint). Assume \( A \) is at position \( a \) and \( B \) is at position \( b \) on the number line.

2. **Calculate the Total Distance:** Find the total distance between the two points:
\[
\text{Total Distance} = b - a
\]

3. **Determine the Fractional Distance:** Decide on the fraction \( \frac{m}{n} \) of the total distance \( b - a \) that you want to measure. Here, \( m \) is the numerator and \( n \) is the denominator. The fractional distance can be calculated as:
\[
\text{Fractional Distance} = \frac{m}{n} \times (b - a)
\]

4. **Locate the Fractional Point:** Add the calculated fractional distance to the position of point \( A \) to find the new point \( P \):
\[
P = a + \left(\frac{m}{n} \times (b - a)\right)
\]

5. **Final Position:** The coordinate of point \( P \) is therefore:
\[
P = a + \frac{m}{n}(b - a)
\]

### Example
Let's say you want to find the point that is \( \frac{1}{4} \) of the way from \( A = 2 \) to \( B = 6 \).

1. **Endpoints:** \( A = 2 \) and \( B = 6 \)
2. **Total Distance:** \( 6 - 2 = 4 \)
3. **Fractional Distance:** \( \frac{1}{4} \times 4 = 1 \)
4. **Locate the Point:** \( P = 2 + 1 = 3 \)

So, the point located at \( \frac{1}{4} \) of the way from \( 2 \) to \( 6 \) is at \( 3 \).