Line Graphs are represented by specific points joined together by straight lines. The points are plotted on a two-dimensional plane taking one parameter on the horizontal axis and the other on vertical axis.

1. One dependent variable historigram,

2. Multi dependent variable historigram.

- Historigram is a graph of time series where the changes in a given variable is illustrated s a function of time.
- From one period of time to another, the graph shows changes in the value of one or more variables.
- These graphs can be either on a natural scale or on a ratio scale.
- Natural scale is used, if the absolute values of a variable are to be represented. In natural scale, equal distances on an axis represent equal values/amounts.
**One dependent variable historigram**is the simplest type. The values of only one variable is to be plotted along the Y-axis and as the time factor being independent variable, it is taken on X-axis.- In
**more than one dependent variable historigram**, distribution of two variables is represented distinctly. For all values, the vertical scale is common.

**Example 1**:

**Directions for questions 1 to 5**: Given below are line graph examples with questions, which show the annual food grain production from 1992 to 1997. Refer to the graph & answer the questions based on line graph as given below.

**1. What is an approximate percentage decrease in production from 1993 to 1994?**

- A. 87.5%

B. 37.5%

C. 9.09%

D. None of these

**Solution**:

- Here we look up the values of the production for the 2 years first. Locate 1993 on the X axis, which denotes years. Move vertically up along the Y axis direction in 1993, and we get value of production in 1993 as 110. In the same way we get the value of production in 1994 as 100.

In calculating % increases & decreases: In this case 1993, it is very important to remember that the original year is the one that is used as the reference year.

Which is 110 – 100 = 10.

Now we have to express 10 as a percentage of the production in 1993, which is 110.

So the required answer is 100 × 10/110 = 9.09%. Hence answer is option C.

**2. The average production of 1994 and 1996 was approximately equal to production of which year?**

- A. 1996

B. 1992

C.1995

D. 1994

**Solution**:

- The production in 1994 was 100, that in 1996 was 62.5.

The average production of 1994 and 1996 = (100 + 62.5)/2 = 81.25.

This was approximately equal to production in 1992. Hence Answer is option B.

**3. The average production of the given years was more than to the production in how many years?**

- A. 1

B. 2

C. 3

D. 4

**Solution**:

- Average production for 6 years = 80 + 110 + 130 + 100 +120 +65 / 6 = 101

Clearly in 1992, 1994 and 1996, the production was less than average. Hence answer is option C.

**4. If the production in 1992 is estimated to be 60% more than that in 1991, the estimate of 1991 production is?**

- A. 120 units

B. 100 units

C.60 units

D. 50 units

**Solution**:

- The production in 1992 is 80 units. The Required production in 1991 =

80 / 160 x 100 = 50 Hence answer is option D.

**5. The maximum increase in food grain production has been in**

- A. 1997

B. 1996

C.1995

D. 1994

**Solution**:

- Here remember that absolute increase has been asked. The difference in Y coordinates of the adjacent years will give us the increase. Since the distance between consecutive years is the same on the X axis, we can check the slopes of various segments. By visual inspection, maximum increase in food grain production has been in the year 1997 (55 units by calculation).Hence, answer is option A

**Example 2**:

**Directions for question 1-5**: Study the following graph carefully and answer accordingly. Following graph shows the percent profit earned by two companies A and B on their investments. (Revenue = Investment + Profit)

**1. Revenue of company B in 2000 was Rs.1239 lakhs. What was the investment in that year (in Rs lakhs) of company B?**

- A. 700

B. 800

C. 650

D. 193.03

**Solution**:

- Investment of company A in 2000 = 1239 x 100 / 177 = Rs.700 lakh

**2. Investment of company B in 1998 was 20% more than that in the previous year. Profit in 1998 of company B was what per cent of its profit in 1997?**

- A. 10%

B. 102(2/3) %

C. 106(2/3) %

D. None of these

**Solution**:

- Let us assume the investment of Company B in 1997 is Rs. 100. Investment of company B in 1998 was 20% more than that in the previous year.

So the investment of Company B in 1998 is Rs. 120.

Profit in 1997 of company B= 90 % of 100 = Rs. 90

Profit in 1998 of company B= 85% of 120 = Rs. 102.

Required % = 102 / 90 x 100 = 113 (1/3) %

**3. In which of the following years is the ratio of investment and profit maximum for company A?**

- A. 2001

B. 1995

C. 1998

D. 2000

**Solution**:

**4. If the revenue of company B in 1996 was same as the revenue of company B in 1999, what would be the ratio of investment of company B in 1999 to the investment of company B in 1996?**

- A. 9 : 10

B. 10 : 9

C. 13 : 15

D. 15 : 13

**Solution**:

- Required ratio = (100 / 185) x (166.5 / 100) = (333 / 370) = (37*9 / 37*10) = 9:10

**5. In which of the following years was the investment minimum for company B?**

- A. 1995

B. 1999

C. 2000

D. Can’t be determined

**Solution**:

- We don’t have the data for investment. So, we can’t determine the answer.

%profit = \(\frac{Income – Expenditure}{Expenditure}\) x 100

- (A) 29 : 20

(B) 20 : 29

(C) 19 : 20

(D) cannot be determined

(E) None of these.

**2. If the total expenditure of the two companies in 2001 was Rs. 18 lakhs and expenditures of companies A and B in that year were in the ratio of 4 : 5 respectively, then what was the income of the company B in that year (in lakhs)?**

- (A) 8

(B) 10

(C) 10.4

(D) All of these

(E) None of these.

**Solution:**

- 1. Since the data given is inadequate hence it cannot be determined.

So, the option(D) is correct answer.

2. The expenditure of company B in 2001 = \(\frac{5}{9} * 18\) = 10 lakhs

Let the income of company B be Rs.\(x\) lakh

Therefore, 40 =\(\frac{x – 10}{10} * 100\)

40 = 10\(x\) – 100

10\(x\) = 140

\(x\) = 14 lakh.

**2. Study the following graph and answer the questions based on it.**

- (A) 50000

(B) 42000

(C) 33000

(D) 21000

(E) 13000

**2. What is the difference between the total productions of the two companies in the given years?**

- (A) 19000

(B) 22000

(C) 26000

(D) 28000

(E) 29000

**3. The production of company Y in 2000 was approximately what percent of the production of company X in the same year?**

- (A) 173

(B) 164

(C) 132

(D) 97

(E) 61

**Solution:**

- 1. Required difference = 128000 – 107000 = 21000.

So, option (D) is correct one.

2. Total production of company X from 1997 to 2002 = 119000 + 99000 + 141000 + 78000 + 120000 + 159000 = 716000.

Total production of company Y from 1997 to 2002 = 139000 + 120000 + 100000 + 128000 + 107000 + 148000 = 742000.

Difference = 742000 – 716000 = 26000.

So, option (C) is correct answer.

3. Required percentage = \(\frac{128000}{78000} * 100\)% = 164%

So, option (B) is correct answer.

**Model 3:** The following line-graph gives the ratio of the amounts of imports by a company to the amount of exports from that company over a period from 1995 to 2001. The questions given below are based on this graph.

(A) Rs. 370 crores

(B) Rs. 320 crores

(C) Rs. 280 crores

(D) Rs. 275 crores

(E) Rs. 264 crores

2. In how many of the given years were the exports more than the imports?

(A) 1

(B) 2

(C) 3

(D) 4

(E) None of these.

**Solution:**

1. Ratio of imports to exports in the year 1996 = 0.85.

Let the exports in 1996 = Rs. \(x\) crores.

Then,

\(\frac{272}{x}\) = 0.85

⇒ \(x\) = \(\frac{272}{0.85}\) = 320

Therefore, exports in 1996 = Rs. 320 crores.

So, option (B) is correct answer.

2. The exports are more than the imports implies that the ratio of value of imports to exports is less than 1.

Now, this ratio is less than 1 in the years 1995, 1996, 1997, and 2000.

Thus there are four such years.

So, option (D) is correct answer.

**Model 4:** Periodical examinations are held every second month in a school. In a session during April 2001 to March 2002, a student of class IX appeared for each of the periodical exams. The aggregate marks obtained in each periodical exam are represented in the below given line graph. Study it and answer the given questions.

(A) 73.25%

(B) 75.5%

(C) 77%

(D) 78.75%

(E) 79.5%

2.The total number of marks obtained in February 02 is what percent of the total marks obtained in April 01?

(A) 110%

(B) 112.5%

(C) 115%

(D) 116.5%

(E) 117.5%

**Solution:**

1. Required percentage = \(\frac{370 + 385}{500 + 500} * 100\)% = \(\frac{755}{1000} * 100\)% = 75.5%

So, option (B) is correct answer.

2. Required percentage = \(\frac{405}{360} * 100\)% = 112.5%

So, option (B) is correct answer.

**Model 5:** Study the following line graph which gives the number of students who joined and left the school in the beginning of year for six years, from 1996 to 2001.

Initial strength of school in 1995 = 3000

(A) 92.13%

(B) 93.75%

(C) 96.88%

(D) 97.25%

2. What was the number of students studying in the school during 1999 ?

(A) 2950

(B) 3000

(C) 3100

(D) 3150

3. The ratio of the least number of students who joined the school to the maximum number of students who left the school in any of the years during the given period is?

(A) 7:9

(B) 4:5

(C) 3:4

(D) 2:3

4.For which year, the percentage rise/fall in the number of students who left the school compared to the previous year is maximum?

(A) 1997

(B) 1998

(C) 1999

(D) 2000

**Solution:**

1. Required percentage = \(\frac{3000}{3200} * 100\)% = 93.75%

So, option (B) is correct answer.

2. From the given data, the number of students studying in the school during 1999 = 3150.

So, option (D) is correct answer.

3. Required percentage = \(\frac{300}{450} * 100\)% = \(\frac{2}{3}\)

So, option (D) is correct answer.

4. The percentage rise/fall in the number of students who left the school (compared to the previous year) during various years are:

For 1997 = \(\frac{450 – 250}{250} * 100\)% = 80% (rise)

For 1998 = \(\frac{450 – 400}{450} * 100\)% = 11.11% (fall)

For 1999 = \(\frac{400 – 350}{400} * 100\)% = 12.5% (fall)

For 2000 = \(\frac{450 – 350}{350} * 100\)% = 28.57% (rise)

For 2001 = \(\frac{450 – 450}{450} * 100\)% = 0% (rise)

Hence, in the year 1997 maximum rise/fall occured.

So, option (A) is correct answer.