Examples (Revised) - Chapter 15 - Introduction To Graphs - Ncert Solutions class 8 - Maths

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Chapter 13 - Introduction To Graphs - NCERT Solutions for Class 8 Maths

Example 1:

(A graph on "performance")
The given graph (Fig 13.3) represents the total runs scored by two batsmen $\mathrm{A}$ and $\mathrm{B}$, during each of the ten different matches in the year 2007. Study the graph and answer the following questions.
(i) What information is given on the two axes?
(ii) Which line shows the runs scored by batsman $\mathrm{A}$ ?
(iii) Were the run scored by them same in any match in 2007 ? If so, in which match?
(iii) Among the two batsmen, who is steadier? How do you judge it?

Solution:
(i) The horizontal axis (or the $x$-axis) indicates the matches played during the year 2007. The vertical axis (or the $y$-axis) shows the total runs scored in each match.
(ii) The dotted line shows the runs scored by Batsman A. (This is already indicated at the top of the graph).

(iii) During the 4th match, both have scored the same number of 60 runs. (This is indicated by the point at which both graphs meet).
(iv) Batsman A has one great "peak" but many deep "valleys". He does not appear to be consistent. $\mathrm{B}$, on the other hand has never scored below a total of 40 runs, even though his highest score is only 100 in comparison to 115 of A. Also A has scored a zero in two matches and in a total of 5 matches he has scored less than 40 runs. Since A has a lot of ups and downs, $\mathrm{B}$ is a more consistent and reliable batsman.

Example 2: The given graph (Fig 13.4) describes the distances of a car from a city $\mathrm{P}$ at different times when it is travelling from City $\mathrm{P}$ to City $\mathrm{Q}$, which are $350 \mathrm{~km}$ apart. Study the graph and answer the following:
(i) What information is given on the two axes?
(ii) From where and when did the car begin its journey?
(iii) How far did the car go in the first hour?
(iv) How far did the car go during (i) the 2nd hour? (ii) the 3rd hour?
(v) Was the speed same during the first three hours? How do you know it?
(vi) Did the car stop for some duration at any place? Justify your answer.
(vii) When did the car reach City $\mathrm{Q}$ ?

Solution:
(i) The horizontal $(x)$ axis shows the time. The vertical $(y)$ axis shows the distance of the car from City $\mathrm{P}$.
(ii) The car started from City $\mathrm{P}$ at 8 a.m.
(iii) The car travelled $50 \mathrm{~km}$ during the first hour. [This can be seen as follows. At 8 a.m. it just started from City P. At 9 a.m. it was at the 50 th $\mathrm{km}$ (seen from graph). Hence during the one-hour time between 8 a.m. and 9 a.m. the car travelled $50 \mathrm{~km}$.
(iv) The distance covered by the car during
(a) the 2 nd hour (i.e., from 9 am to $10 \mathrm{am}$ ) is $100 \mathrm{~km},(150-50)$.
(b) the 3rd hour (i.e., from $10 \mathrm{am}$ to $11 \mathrm{am}$ ) is $50 \mathrm{~km}(200-150)$.
(v) From the answers to questions (iii) and (iv), we find that the speed of the car was not the same all the time. (In fact the graph illustrates how the speed varied).
(vi) We find that the car was $200 \mathrm{~km}$ away from city $\mathrm{P}$ when the time was $11 \mathrm{a} . \mathrm{m}$. and also at 12 noon. This shows that the car did not travel during the interval 11 a.m. to 12 noon. The horizontal line segment representing "travel" during this period is illustrative of this fact.
(vii) The car reached City $\mathrm{Q}$ at 2 p.m.

Example 3:

(Quantity and Cost)
The following table gives the quantity of petrol and its cost.

Plot a graph to show the data.
Solution:

(i) Let us take a suitable scale on both the axes (Fig 13.5).

(ii) Mark number of litres along the horizontal axis.
(iii) Mark cost of petrol along the vertical axis.
(iv) Plot the points: $(10,500),(15,750),(20,1000),(25,1250)$.
(v) Join the points.

We find that the graph is a line. (It is a linear graph). Why does this graph pass through the origin? Think about it.

This graph can help us to estimate a few things. Suppose we want to find the amount needed to buy 12 litres of petrol. Locate 12 on the horizontal axis.
Follow the vertical line through 12 till you meet the graph at $\mathrm{P}$ (say).
From $P$ you take a horizontal line to meet the vertical axis. This meeting point provides the answer.
This is the graph of a situation in which two quantities, are in direct variation. (How ?). In such situations, the graphs will always be linear.

Example 4: (Principal and Simple Interest)
A bank gives $10 \%$ Simple Interest (S.I.) on deposits by senior citizens. Draw a graph to illustrate the relation between the sum deposited and simple interest earned. Find from your graph
(a) the annual interest obtainable for an investment of ₹ 250 .
(b) the investment one has to make to get an annual simple interest of ₹ 70 .

Solution:

Steps to follow:
1. Find the quantities to be plotted as Deposit and SI.
2. Decide the quantities to be taken on $x$-axis and on $y$-axis.
3. Choose a scale.
4. Plot points.
5. Join the points.

We get a table of values.

(i) Scale : 1 unit $=₹ 100$ on horizontal axis; 1 unit $=₹ 10$ on vertical axis.
(ii) Mark Deposits along horizontal axis.
(iii) Mark Simple Interest along vertical axis.
(iv) Plot the points : $(100,10),(200,20),(300,30),(500,50)$ etc.
(v) Join the points. We get a graph that is a line (Fig 13.6).

(a) Corresponding to ₹ 250 on horizontal axis, we get the interest to be ₹ 25 on vertical axis.
(b) Corresponding to ₹ 70 on
the vertical axis, we get the sum to be $₹ 700$ on the horizontal axis

Example 5:

(Time and Distance)
Ajit can ride a scooter constantly at a speed of $30 \mathrm{kms} /$ hour. Draw a time-distance graph for this situation. Use it to find
(i) the time taken by Ajit to ride $75 \mathrm{~km}$. (ii) the distance covered by Ajit in $3 \frac{1}{2}$ hours.

Solution

We get a table of values. 

(i) Scale: (Fig 13.7)
Horizontal: 2 units $=1$ hour
Vertical: 1 unit $=10 \mathrm{~km}$
(ii) Mark time on horizontal axis.
(iii) Mark distance on vertical axis.
(iv) Plot the points: $(1,30),(2,60),(3,90),(4,120)$.

(v) Join the points. We get a linear graph.
(a) Corresponding to $75 \mathrm{~km}$ on the vertical axis, we get the time to be 2.5 hours on the horizontal axis. Thus 2.5 hours are needed to cover $75 \mathrm{~km}$.
(b) Corresponding to $3 \frac{1}{2}$ hours on the horizontal axis, the distance covered is $105 \mathrm{~km}$ on the vertical axis.