Question 15 Marks
The points $P, Q, R, S, T, U, A$ and $B$ on the number line are such that $T R=R S=S U$ and $A P=P Q=Q B$. Name the rational numbers represented by $P, Q, R$ and $S$.
Answer
View full question & answer→Given points $P, Q, R, S, T, U, A$ and $B$ are on the number line such that $T R=R S=S U$ and $A P=P Q=Q B$.
It means that distance between $A B$ is divided into 3 equal parts and similarly $U T$ on the left of zero is divided into 3 equal parts.
Now, point $P$ is on the right of zero on the number line and between 2 and 3 .
So, the rational number represented by $P=2+\frac{1}{3}$
$\left[\right.$ since, $A B$ is divided into 3 equal parts and each part shows $\left.\frac{1}{3}\right]$
$=\frac{2 \times 3}{1 \times 3}+\frac{1}{3}=\frac{6+1}{3}=\frac{7}{3}$
Similarly, $Q$ is on the right of zero and between 2 and 3.
So, the rational number represented by $Q=2+\frac{1}{3}+\frac{1}{3}$
[since, $Q$ shows 2 equal parts out of 3 between 2 and 3]
$=\frac{2 \times 3}{3}+\frac{1}{3}+\frac{1}{3}=\frac{6+1+1}{3}=\frac{8}{3}$
Also, $R$ and $S$ lie on the left of zero and between -1 and -2 , then each part shows $\frac{-1}{3}$.
So, the rational number represented by
$R=-1+\left(\frac{-1}{3}\right)=\frac{-1 \times 3}{3}-\frac{1}{3}=\frac{-3-1}{3}=\frac{-4}{3}$
and the rational number represented by
$S=-1+\left(\frac{-1}{3}\right)+\left(\frac{-1}{3}\right)$
[since, $S$ shows 2 equal parts out of 3 parts between -1 and -2 ]
$=\frac{-1 \times 3}{3}-\frac{1}{3}-\frac{1}{3}=\frac{-3-1-1}{3}=\frac{-5}{3}$
It means that distance between $A B$ is divided into 3 equal parts and similarly $U T$ on the left of zero is divided into 3 equal parts.
Now, point $P$ is on the right of zero on the number line and between 2 and 3 .
So, the rational number represented by $P=2+\frac{1}{3}$
$\left[\right.$ since, $A B$ is divided into 3 equal parts and each part shows $\left.\frac{1}{3}\right]$
$=\frac{2 \times 3}{1 \times 3}+\frac{1}{3}=\frac{6+1}{3}=\frac{7}{3}$
Similarly, $Q$ is on the right of zero and between 2 and 3.
So, the rational number represented by $Q=2+\frac{1}{3}+\frac{1}{3}$
[since, $Q$ shows 2 equal parts out of 3 between 2 and 3]
$=\frac{2 \times 3}{3}+\frac{1}{3}+\frac{1}{3}=\frac{6+1+1}{3}=\frac{8}{3}$
Also, $R$ and $S$ lie on the left of zero and between -1 and -2 , then each part shows $\frac{-1}{3}$.
So, the rational number represented by
$R=-1+\left(\frac{-1}{3}\right)=\frac{-1 \times 3}{3}-\frac{1}{3}=\frac{-3-1}{3}=\frac{-4}{3}$
and the rational number represented by
$S=-1+\left(\frac{-1}{3}\right)+\left(\frac{-1}{3}\right)$
[since, $S$ shows 2 equal parts out of 3 parts between -1 and -2 ]
$=\frac{-1 \times 3}{3}-\frac{1}{3}-\frac{1}{3}=\frac{-3-1-1}{3}=\frac{-5}{3}$