7th grade math (The question is in the description)

Question

The distance between two points, a and b, on a number line is expressed as

|a – b|, where a and b are rational numbers. Think about locating a point halfway between a and b on a number line. How can you find the coordinate for such a point? Develop a formula in terms of a and b that you can use to find the coordinate. Your formula must work for positive and negative values of a and b. Does absolute value play a role in your formula? Why or why not? Under what circumstances might your formula be used in the real world? Explain.

Discuss how you can find the coordinate of a point located other fractional distances from a to b (or from b to a). For example, locate a point three-fourths of the distance from a to b. What kind of calculation is involved?

Thank you for helping!!!

Answer 1

a and b are two points on a real number line

the distance between them is |a-b| as distances are usually considered positive, unless you want to designate both magnitude and direction, with + in one direction and - the opposite direction

which makes the absolute value of a-b

mean a-b if a>b

or b-a if b>a

a point half way between a and b is the arithmetic average, the mean, (a+b)/2 = (b+a)/2

which works for both positive and negative values of a and b

for example if a=4 and b=-2, midpoint = 1

(4-2)/2 = 1 = (-2+4)/2 which also = |4-2|/2

but if a=-4 and b =2

(-4+2)/2 =-1 = (2-4).2 which does not equal |-4+2|/2

so the absolute value sign is not only not necessary, but gives the wrong answer for the midpoint half the time

(a-b)/2 = (1/2)(a-b)

for other fractional in between points

just replace 1/2 by the fraction

such as

(3/4)(a-b) for 3/4

which = (3a-3b)/4

for short distances, this works well for 2 points on earth,

but for large distances, it gives a midpoint that is below the earth's surface