Question

The points P, Q, R, S, T, U, A and B on the number line are such that TR=RS=SU and AP=PQ=QB. Name rational numbers represented by P, Q, R, S .

Answer 1

Hint: We are given a number line with some letters and numbers placed on it. To find the number in between numbers divide the unit in parts provided with numbers of parts in the denominator of that number.

Complete step-by-step answer:
First observe the number line.
A, P, Q, B are on the right hand of origin.
U, R, S, T are on the left of the origin.
The segment or unit AB is divided in three equal parts.
Same for segment UT.
Numbers represented by P and Q:
That segment AB is of 1 unit. It is divided into three equal parts. So the value of each unit will be .
Value of P = \[2 + \dfrac{1}{3} = \dfrac{7}{3}\]
Value of Q = \[\dfrac{7}{3} + \dfrac{1}{3} = \dfrac{8}{3}\]
In case of this we have taken 2 complete units.
Now let’s verify for AP=PQ=QB
AP = \[\dfrac{7}{3} - 2 = \dfrac{{7 - 6}}{3} = \dfrac{1}{3}\]
PQ = \[\dfrac{8}{3} - \dfrac{7}{3} = \dfrac{1}{3}\]
QB = \[3 - \dfrac{8}{3} = \dfrac{{9 - 8}}{3} = \dfrac{1}{3}\]
Hence the values are correct.
Numbers represented by S and R:
That segment TU is of 1 unit. It is divided into three equal parts. So the value of each unit will be \[\dfrac{1}{3}\].
Value of R = \[ - 1 + \left( {\dfrac{{ - 1}}{3}} \right) = \dfrac{{ - 3 - 1}}{3} = \dfrac{{ - 4}}{3}\]
Value of S = \[\dfrac{{ - 4}}{3} + \left( {\dfrac{{ - 1}}{3}} \right) = \dfrac{{ - 4 - 1}}{3} = \dfrac{{ - 5}}{3}\]
In this case we have taken 1 complete unit.
Now let’s verify for TR=RS=SU
TR = \[\dfrac{{ - 4}}{3} - \left( { - 1} \right) = \dfrac{{ - 4 + 3}}{3} = \dfrac{{ - 1}}{3}\]
RS = \[\dfrac{{ - 5}}{3} - \left( {\dfrac{{ - 4}}{3}} \right) = \dfrac{{ - 5 + 4}}{3} = \dfrac{{ - 1}}{3}\]
SU = \[ - 2 - \left( {\dfrac{{ - 5}}{3}} \right) = \dfrac{{ - 6 + 5}}{3} = \dfrac{{ - 1}}{3}\]
Hence the values are correct.
So ,
P = \[\dfrac{7}{3}\] , Q = \[\dfrac{8}{3}\] , R = \[\dfrac{{ - 4}}{3}\], S = \[\dfrac{{ - 5}}{3}\]


Note: Dividing whole units by number of subdivisions between two units is your hint to find the value of each subunit that helps in finding remaining numbers.