Line Graphs

Read values, rises, falls and trends from a line graph, estimate values between the readings where they make sense, compare two lines, draw a line graph with a suitable scale, and predict with care.

Before you start

In Primary school you read and drew line graphs: points plotted for a set of readings and joined by straight lines.
To read any scale, first find what one step stands for, as you do for a bar graph. On a scale marked 0, 50, 100, 150, …, each step is 50, and a point halfway between 250 and 300 stands for 275.
A percentage increase is worked out of the starting value: from 200 to 250 is an increase of 50, and $\frac{50}{200} \times 100\% = 25\%$ .

The Parts of a Line Graph

A line graph shows how a quantity changes over time. The times go along the horizontal axis, in order and in equal steps. The values go up the vertical axis, against a scale. Each reading is plotted as a point, and the points are joined in order by straight lines, so that the rises and falls can be seen at a glance.

A drinks stall at a hawker centre kept a record of its sales each day for one week. The table shows the record, and the line graph below it is drawn from the table.

Daily sales of a drinks stall
Day	Mon	Tue	Wed	Thu	Fri	Sat	Sun
Sales ($)	200	250	275	250	350	400	350

What a line graph has

A title that says what the data are.

A horizontal axis for the time (days, months, years, hours), marked in equal steps and in order, with a label.

A vertical axis with a label (and the unit of the values) and a scale in equal steps. It starts at 0, or above 0 with a zigzag break marked on the axis.

A point for each reading, the points joined in order by straight lines drawn with a ruler.

Reading Values and Changes

To read a point, go from the point straight across to the vertical axis and read the scale there. A point halfway between two gridlines stands for the number halfway between them: on the sales graph each step is 50, so the Wednesday point, halfway between 250 and 300, stands for $275.

A line graph shows changes as well as values. The change from one reading to the next is the later value minus the earlier one: a rise if the line goes up, a fall if it goes down. When the time steps are equal, the steeper a piece of the line, the greater the change.

Rises and falls

Change from one reading to the next $=$ later value $-$ earlier value.

The greatest rise is on the piece of the line that goes up most steeply; the greatest fall on the piece that goes down most steeply. Work out each change to be sure.

Example 1: Reading the sales graph

Use the line graph of the daily sales of the drinks stall above.

(a) What were the sales on Wednesday?

(b) On which day were the sales highest, and on which day lowest?

(d) Between which two days was the greatest fall in sales? How much was it?

(e) On how many days were the sales more than $260?

Solution

Each step of the scale is 50. Reading every point: Mon 200, Tue 250, Wed 275 (halfway between 250 and 300), Thu 250, Fri 350, Sat 400, Sun 350.

(a) The Wednesday point is halfway between the gridlines for 250 and 300, so it stands for 275.

(b) The highest point is Saturday, at 400. The lowest point is Monday, at 200.

Mon to Tue: $250 - 200 = 50$ ; Tue to Wed: $275 - 250 = 25$ ; Wed to Thu: $250 - 275 = -25$ ;

Thu to Fri: $350 - 250 = 100$ ; Fri to Sat: $400 - 350 = 50$ ; Sat to Sun: $350 - 400 = -50$ .

The greatest rise is 100, from Thursday to Friday. It is also the steepest piece of the line going up.

(d) The two falls are 25 (Wed to Thu) and 50 (Sat to Sun). The greater is 50, from Saturday to Sunday.

(e) The points above 260 are Wednesday (275), Friday (350), Saturday (400) and Sunday (350): 4 days.

(a) $275 (b) Highest on Saturday ($400), lowest on Monday ($200). (c) From Thursday to Friday, a rise of $100. (d) From Saturday to Sunday, a fall of $50. (e) On 4 days.

Describing a Trend

The trend of a line graph is the general way the values change over the whole time: going up, going down, staying the same, or going up and down. A trend is described in words, with the numbers that support it.

On the sales graph, the sales generally rose from $200 on Monday to $400 on Saturday, with one small dip on Thursday, and then fell to $350 on Sunday.

Describing a trend

Say the general direction: increasing (rising), decreasing (falling), staying the same (levelling off), or rising and falling.

Say where the direction changes, and whether the change is fast (a steep line) or slow (a gentle line).

Support each statement with values read from the graph.

Example 2: The height of a plant

Hui Min measured the height of a plant at the end of each week for 8 weeks.

(a) Describe the trend in the height of the plant.

(b) Between which weeks did the plant grow fastest?

Solution

Each step of the scale is 10. The heights are 5, 10, 20, 35, 50, 60, 65 and 65 cm (5, 35 and 65 are halfway between gridlines).

The growth each week: $10 - 5 = 5$ , $20 - 10 = 10$ , $35 - 20 = 15$ , $50 - 35 = 15$ , $60 - 50 = 10$ , $65 - 60 = 5$ , $65 - 65 = 0$ .

(a) The plant grew every week until week 7: slowly at first (5 cm, then 10 cm), then fastest in the middle weeks (15 cm a week), then more slowly again (10 cm, then 5 cm). From week 7 to week 8 the line is flat: the height stayed at 65 cm.

(b) The steepest pieces of the line are from week 3 to week 4 and from week 4 to week 5, 15 cm in each week.

(a) The height increased, slowly at first, fastest in weeks 4 and 5, then more slowly, and it stayed at 65 cm from week 7 to week 8. (b) From the end of week 3 to the end of week 5 (15 cm in each week). (c) 60 cm

An Axis That Starts Above 0

The temperature at a school field was read every 3 hours one day: 24 °C at 6 a.m., 28 °C at 9 a.m., 32 °C at 12 noon, 34 °C at 3 p.m. and 30 °C at 6 p.m. If the vertical axis starts at 0 and goes up in steps of 5, the line sits in the upper half of the graph, its rises and falls look small, and values such as 28 °C and 34 °C fall between gridlines and are hard to read.

A line graph shows each value by where its point sits against the scale, and its changes by the slope of the line, not by the length of a bar. So its vertical axis may start above 0, just below the least value. The part of the axis that is left out is shown by a zigzag break between 0 and the first number. The same temperatures are drawn below with the axis starting at 22 °C.

Starting the vertical axis above 0

The vertical axis of a line graph may start above 0 when all the values are far from 0. Mark the part left out with a zigzag break on the axis between 0 and the first number.

Read the scale with care: with a break, rises and falls look bigger than they are, and the height of a point is no longer in proportion to its value.

The value axis of a bar graph always starts at 0, because the length of a bar stands for its value.

Example 3: Reading a graph with a break

Use the temperature graph with the break, above.

(a) By how many degrees did the temperature rise from 6 a.m. to 3 p.m.?

(b) Express this rise as a percentage of the temperature at 6 a.m., correct to 1 decimal place.

(c) Ali looks at the graph and says, "The temperature at 3 p.m. was several times the temperature at 6 a.m." Is he right? Explain.

Solution

The axis has a break and then goes from 22 in steps of 2. The 6 a.m. point is on the gridline for 24 and the 3 p.m. point on the gridline for 34.

(a) Rise $= 34 - 24 = 10$ °C

(b) Percentage rise $= \frac{10}{24} \times 100\% = 41.66\ldots\% \approx 41.7\%$

(c) The 3 p.m. point is drawn far higher than the 6 a.m. point because the axis starts at 22, not at 0. The values themselves are 34 and 24, and $34 \div 24 \approx 1.4$ , so 34 °C is not even twice 24 °C.

(a) 10 °C (b) about 41.7% (c) No. The temperature rose only from 24 °C to 34 °C, about 1.4 times; the point looks much higher only because the axis starts at 22 °C.

Values Between the Readings

Some quantities change continuously: temperature, height, depth of water, distance travelled. Between two readings such a quantity takes every value in between. Joining the points by a straight line assumes that it changed steadily between the readings, so a point on the line between two readings gives an estimate of the value at that time.

Other data are counts or totals for each period: the sales of each day, the number of books borrowed each month, the rainfall of each month. There is nothing between Wednesday's sales and Thursday's sales, so a point on the line between them stands for no value at all. The line only shows the change from one reading to the next.

Reading between the points

For a quantity that changes continuously, a value between two readings can be estimated from the line. It is only an estimate, because the true change between the readings may not have been steady.

For counts or totals for each day, month or year, a point on the line between two readings has no meaning. The line shows only the rise or fall from one reading to the next.

Example 4: Estimating temperatures between the readings

Use the temperature graph with the break in the section above (24, 28, 32, 34 and 30 °C at 6 a.m., 9 a.m., 12 noon, 3 p.m. and 6 p.m.).

(a) Estimate the temperature at 7.30 a.m.

(b) Estimate the temperature at 4.30 p.m.

(d) Why are these answers only estimates?

Solution

Temperature changes continuously, so values between the readings may be estimated from the line.

(a) 7.30 a.m. is halfway between 6 a.m. and 9 a.m. Halfway along that piece of the line, the height is halfway between 24 and 28: $\frac{24 + 28}{2} = 26$ °C.

(b) 4.30 p.m. is halfway between 3 p.m. and 6 p.m.: $\frac{34 + 30}{2} = 32$ °C.

(c) Go across from 30 on the vertical axis. The line meets this level twice. On the way up, 30 is halfway between 28 (9 a.m.) and 32 (12 noon), so the time is halfway between 9 a.m. and 12 noon: 10.30 a.m. On the way down, the 6 p.m. reading is 30.

(d) The line assumes that the temperature changed steadily between readings. It may have changed faster or more slowly, for example if a cloud covered the sun.

(a) About 26 °C (b) About 32 °C (c) At about 10.30 a.m. and at 6 p.m. (d) The straight line assumes a steady change between readings, which may not be what happened.

Example 5: When a value in between means nothing

Ravi looks at the sales graph of the drinks stall. He finds the point on the line halfway between Wednesday and Thursday, reads $262.50, and says, "This is what the stall sold between Wednesday and Thursday." Explain why he is wrong, and say what that piece of the line does show.

Solution

Each point is the total sales for one whole day. There is no day between Wednesday and Thursday, so there are no sales to read there, and $262.50 is not the sales of any day.

The piece of the line from Wednesday ($275) to Thursday ($250) goes down, so it shows a fall of $275 - 250 = 25$ dollars from one day to the next.

Sales are a total for each whole day, so a point between two days stands for nothing. The line from Wednesday to Thursday shows only that the sales fell by $25.

Predicting Beyond the Data

A line graph is sometimes used to predict a value after the last reading, by supposing that the trend goes on. Such a prediction is only an estimate. Something may happen that changes the trend, and the further beyond the data we go, the less the prediction can be trusted.

Predicting with care

A prediction beyond the last reading assumes that the trend continues. Say that it is an estimate, and say what you have assumed.

Predict only a short way beyond the data. Far beyond it, a trend may stop or reverse, and the prediction can be badly wrong.

Example 6: Predicting the temperature

Use the temperature graph with the break (34 °C at 3 p.m. and 30 °C at 6 p.m.).

(a) Predict the temperature at 9 p.m. State what you assume.

(b) Mei Ling continues the line in the same way to 6 a.m. the next morning. What temperature does she predict? Is it a sensible prediction?

Solution

(a) From 3 p.m. to 6 p.m. the temperature fell by $34 - 30 = 4$ °C in 3 hours.

Assume it keeps falling at the same rate for the next 3 hours: $30 - 4 = 26$ °C at 9 p.m.

(b) From 6 p.m. to 6 a.m. is 12 hours, that is $12 \div 3 = 4$ steps of 3 hours.

A fall of 4 °C in each step gives $30 - 4 \times 4 = 30 - 16 = 14$ °C.

In Singapore the temperature at night stays in the mid-twenties, so 14 °C is far too low. The fall in the evening does not go on all night: the trend stops.

(a) About 26 °C, if the temperature keeps falling 4 °C every 3 hours; this is only an estimate. (b) 14 °C. It is not sensible: it is too far beyond the data, and the evening fall does not continue all night.

Two Lines on One Graph

Two sets of data taken at the same times can be drawn as two lines on one graph, on the same scale. A key (legend) says which line is which. The two lines can then be compared at each time: which is higher, by how much, where they meet, and which rises or falls faster.

A library counted the fiction and the non-fiction books borrowed each month from January to June.

Reading two lines

Use the key to tell the lines apart, and read both on the same scale.

Where the two points of a month are at the same height, the two values are equal. Where one line is above the other, that set has the greater value at that time.

Example 7: Fiction and non-fiction

Use the line graph of the books borrowed from the library above.

(a) In which month were equal numbers of fiction and non-fiction books borrowed?

(b) In which month were more non-fiction than fiction books borrowed? How many more?

(d) What percentage of the books borrowed in June were fiction?

Solution

Each step of the scale is 20.

Fiction: Jan 120, Feb 140, Mar 110, Apr 90, May 130, Jun 150.

Non-fiction: Jan 80, Feb 90, Mar 110, Apr 120, May 100, Jun 90.

(a) In March both points are halfway between 100 and 120, at 110.

(b) Only in April is the non-fiction line above the fiction line: $120 - 90 = 30$ more.

(c) Changes in fiction: $140 - 120 = 20$ , $110 - 140 = -30$ , $90 - 110 = -20$ , $130 - 90 = 40$ , $150 - 130 = 20$ . The greatest rise is 40, from April to May.

(d) Books borrowed in June $= 150 + 90 = 240$

Percentage that were fiction $= \frac{150}{240} \times 100\% = 62.5\%$

(a) March (110 of each) (b) April, 30 more (c) From April to May (a rise of 40) (d) 62.5%

Drawing a Line Graph

Steps for drawing a line graph

1. Draw the horizontal axis and mark the times on it in order, in equal steps. Label it.

2. Choose the scale of the vertical axis. Find the least and the greatest values. Take a step of 1, 2, 5, 10, 20, 25, 50, … so that the axis has about 5 to 10 steps and the values can be read. Start at 0, or, if all the values are far from 0, start just below the least value and draw a zigzag break. Label the axis, with its unit.

3. Plot a point for each reading, above its time and level with its value.

4. Join the points in order with straight lines, using a ruler.

5. Write a title.

Example 8: Drawing a line graph with a break

The table gives the number of runners who took part in a charity run each year from 2021 to 2026.

Number of runners in a charity run
Year	2021	2022	2023	2024	2025	2026
Number of runners	860	880	920	900	930	960

(a) Choose a suitable scale and draw a line graph of the data.

(b) Between which two years did the number of runners rise the most?

Solution

(a) Step 1. Mark the years 2021 to 2026 along the horizontal axis, equally spaced, and label it "Year".

Step 2. The least value is 860 and the greatest is 960. From 0, the axis would need steps of 100 (0 to 1000), the points would be crowded at the top, and differences of 20 could not be read. All the values are far from 0, so start at 840 with a zigzag break and go up in steps of 20 to 980. That is $\frac{980 - 840}{20} = 7$ steps. Label the axis "Number of runners".

Step 3. Plot the points: 860, 880, 920, 900 and 960 are on gridlines; 930 is halfway between 920 and 940.

Step 4. Join the points in order with straight lines.

Step 5. Title: "Number of runners in a charity run".

(b) The changes: $880 - 860 = 20$ , $920 - 880 = 40$ , $900 - 920 = -20$ , $930 - 900 = 30$ , $960 - 930 = 30$ .

The greatest rise is 40, from 2022 to 2023: the steepest piece of the line going up.

(a) The line graph above, with a break and a scale from 840 to 980 in steps of 20. (b) From 2022 to 2023, a rise of 40 runners.

When to Use a Line Graph

A line graph is the diagram for change over time: temperatures through a day, sales month by month, a child's height year by year. It is not for data whose categories have no order, such as favourite fruits (a bar graph or a pictogram suits them), or for showing how a whole is divided into parts (a pie chart suits that). Joining the points of such data with lines would suggest a trend that does not exist.

A line graph: advantages and disadvantages
Advantages	Disadvantages
Shows at a glance how a quantity changes over time: the trend, the rises and the falls.	Suits only data in time order; it cannot show categories with no order, or parts of a whole.
Two or more lines on one graph compare two sets of data over the same times.	Values cannot be read as exactly as from a table.
For a quantity that changes continuously, values between the readings can be estimated.	For counts or totals, the line between the points can be mistaken for values that do not exist.
The trend can suggest a careful prediction a short way ahead.	A broken axis, or the choice of scale, can make changes look bigger or smaller than they are.

Example 9: Is a line graph suitable?

For each set of data, say whether a line graph is suitable. Give a reason, and if it is not suitable, name a better diagram.

(a) The temperature of a patient, taken every 4 hours for two days.

(b) The favourite CCA of each of the 40 students in a class.

(d) How a household's monthly spending of $4000 is divided among food, rent, transport and savings.

Solution

(a) The data are readings taken over time, so a line graph shows how the temperature changed. Temperature changes continuously, so values between readings may even be estimated.

(b) The CCAs are categories with no order, so a line joining them would mean nothing. A bar graph (or a pictogram) suits them; since each student has one favourite, a pie chart could also show them as parts of the 40.

(c) The data are in time order, so a line graph shows the trend through the year. Each point is a total for a month, so the line between two months gives no value.

(d) The data are parts of one whole, so a pie chart suits them. A line graph shows change over time, and there is no time here.

(a) Suitable: change over time. (b) Not suitable: use a bar graph, a pictogram or a pie chart. (c) Suitable: change over the months. (d) Not suitable: use a pie chart.

Common mistakes

Misreading the scale. Find what one step stands for before reading a point, especially when the axis starts above 0. On an axis 22, 24, 26, …, a point halfway between 24 and 26 stands for 25.
Giving the highest value when the question asks for the greatest rise. The greatest rise is the greatest change between two readings next to each other, found by subtracting.
Reading values between the points of counts or totals. Sales each day or books each month have nothing between the readings; only continuous quantities such as temperature can be estimated there.
Judging a change by how high a point looks on a graph whose axis starts above 0. Read the values off the scale and compare those.
Leaving out the zigzag when the vertical axis starts above 0. Without the break, the reader may take the bottom of the axis to be 0.
Unequal time steps on the horizontal axis. Readings taken at 6 a.m., 9 a.m., 12 noon and 2 p.m. cannot be spaced equally: the gaps along the axis must match the gaps in time.
Treating a prediction as certain, or predicting far beyond the data.
Drawing a line graph for categories such as favourite fruits, where joining the points has no meaning.

Practice

The questions get harder as you go. Try each one before you open its solution.

Question 1

The line graph shows the number of students who visited the school library during recess each day of one week.

(a) How many students visited the library on Monday?

(b) On which day did the most students visit? How many?

ShowHide solution

Each step of the scale is 10.

(a) The Monday point is halfway between 30 and 40, so 35 students.

(b) The highest point is Thursday, on the gridline for 60.

Total number of visits $= 35 + 50 + 45 + 60 + 25 = 215$

This is the number of visits, not of different students: a student who went on two days is counted on both days.

(a) 35 students (b) Thursday, 60 students (c) 215 visits

Question 2

The line graph shows the number of cups of bubble tea sold by a stall each month from January to June.

(a) Between which two months was the greatest rise in sales? How many cups was it?

(b) Between which two months was the greatest fall? How many cups was it?

ShowHide solution

Each step of the scale is 1000. The values are 4000, 3500, 5000, 5500, 4500 and 7000 (3500, 5500 and 4500 are halfway between gridlines).

The changes: Jan to Feb $3500 - 4000 = -500$ ; Feb to Mar $5000 - 3500 = 1500$ ; Mar to Apr $5500 - 5000 = 500$ ; Apr to May $4500 - 5500 = -1000$ ; May to Jun $7000 - 4500 = 2500$ .

(a) The greatest rise is 2500, from May to June.

(b) The falls are 500 and 1000. The greatest fall is 1000, from April to May.

Percentage increase $= \frac{3000}{4000} \times 100\% = 75\%$

(a) From May to June, 2500 cups (b) From April to May, 1000 cups (c) 75%

Question 3

A tank was filled with water, and later some of the water was let out. The line graph shows the depth of the water every 5 minutes.

(a) Describe how the depth of the water changed over the 30 minutes.

(b) Was the depth changing faster while the tank was filling or while it was emptying? Explain.

ShowHide solution

Each step of the scale is 10. The depths are 0, 20, 40, 40, 40, 25 and 10 cm (25 is halfway between 20 and 30).

(a) From 0 to 10 minutes the depth rose steadily, 20 cm every 5 minutes, to 40 cm. From 10 to 20 minutes the line is flat: the depth stayed at 40 cm. From 20 to 30 minutes it fell steadily, $40 - 25 = 15$ cm and $25 - 10 = 15$ cm every 5 minutes, to 10 cm.

(b) Filling: 20 cm every 5 minutes. Emptying: 15 cm every 5 minutes. The rising part of the line is steeper than the falling part.

(c) Depth changes continuously, so it can be estimated from the line. 7.5 minutes is halfway between 5 and 10 minutes, so the depth is halfway between 20 and 40: $\frac{20 + 40}{2} = 30$ cm.

(a) It rose steadily to 40 cm in the first 10 minutes, stayed at 40 cm for 10 minutes, then fell steadily to 10 cm. (b) Faster while filling (20 cm against 15 cm every 5 minutes). (c) About 30 cm

Question 4

Siti poured hot water into a cup and measured its temperature every 4 minutes.

(a) Estimate the temperature of the water after 2 minutes.

(b) Estimate the temperature after 10 minutes.

(d) Describe the trend, and say why your answers to (a) to (c) are only estimates.

ShowHide solution

Each step of the scale is 10. The readings are 90, 70, 50, 40 and 35 °C. Temperature changes continuously, so values between readings can be estimated.

(a) 2 minutes is halfway between 0 and 4 minutes: $\frac{90 + 70}{2} = 80$ °C.

(b) 10 minutes is halfway between 8 and 12 minutes: $\frac{50 + 40}{2} = 45$ °C.

(c) 60 °C is halfway between 70 °C (4 minutes) and 50 °C (8 minutes), so the time is halfway between 4 and 8 minutes: 6 minutes.

(d) The temperature fell all the time: by 20 °C in each of the first two steps, then by 10 °C and by 5 °C. It cooled quickly at first and then more and more slowly. The answers are estimates because the straight lines assume a steady fall between readings.

(a) About 80 °C (b) About 45 °C (c) After about 6 minutes (d) The water cooled quickly at first, then more slowly; the line assumes a steady change between readings, so values read between them are estimates.

Question 5

Each set of data below is drawn as a line graph. For each, say whether a value read from the line between two readings has a meaning. Give a reason.

(a) The height of a tree, measured on 1 January each year.

(b) The number of students absent from a class each school day.

(d) The total money collected at a school fun fair in each hour it was open.

ShowHide solution

A value between readings has a meaning only for a quantity that changes continuously.

(a) Height grows continuously: the tree had a height at every moment between two measurements, so it can be estimated.

(b) Absences are a count for each school day. There is no school day between two days, so a point between them means nothing.

(c) The depth of water rises and falls continuously: it had a depth at every moment between two readings, so a value between them can be estimated.

(d) Each reading is a total for one whole hour. A point between two hours is not the takings of any hour.

(a) Yes, height changes continuously. (b) No, it is a count for each day. (c) Yes, the depth changes continuously. (d) No, it is a total for each hour.

Question 6

The line graph shows the rainfall each month from July to December at one place in Singapore, in two years.

(a) In which months was there more rain in 2025 than in 2024?

(b) How much more rain fell in December 2024 than in December 2025?

(d) Describe the trend in the rainfall in 2025.

ShowHide solution

Each step of the scale is 50.

2024: 150, 175, 125, 175, 250, 300.

2025: 125, 150, 175, 200, 275, 250.

(a) The 2025 line is above the 2024 line in September (175 against 125), October (200 against 175) and November (275 against 250).

(b) $300 - 250 = 50$ mm

2025: $125 + 150 + 175 + 200 + 275 + 250 = 1175$ mm

The totals are equal, so neither year was wetter over these six months.

(d) In 2025 the rainfall rose every month from July (125 mm) to November (275 mm), then fell to 250 mm in December.

(a) September, October and November (b) 50 mm (c) 1175 mm in each year; neither was wetter. (d) It increased every month from July to November, then fell a little in December.

Question 7

Hui Min wrote down the money in her savings account at the end of each month.

Hui Min's savings at the end of each month
Month	Jan	Feb	Mar	Apr	May	Jun
Savings ($)	40	60	90	80	110	150

(a) Choose a suitable scale and draw a line graph of her savings.

(b) In which month did her savings go down? What must have happened in that month? How do you see this on the graph?

ShowHide solution

(a) The least value is 40 and the greatest is 150. These are not far from 0, so the axis can start at 0 with no break. Steps of 20 from 0 to 160 give 8 steps; 90 and 110 are halfway between gridlines. (Steps of 10 would give 16 steps, too many; steps of 50 would leave 40, 60, 90 and 110 hard to read.)

Mark the months in equal steps, plot each point, join the points in order with a ruler, and write a title and the axis labels.

(b) Her savings went down from $90 at the end of March to $80 at the end of April. So in April she took out $10 more than she put in. (The graph shows only the balance at the end of each month, so it cannot tell whether she also took money out in other months while putting in more.) On the graph, the piece of line from March to April is the only one that goes down.

(c) The increases: $60 - 40 = 20$ , $90 - 60 = 30$ , $110 - 80 = 30$ , $150 - 110 = 40$ . The greatest is 40, from May to June.

(a) The line graph above, scale 0 to 160 in steps of 20. (b) In April: she took out $10 more than she put in; the line goes down from $90 to $80. (c) From May to June, an increase of $40.

Question 8

Ravi recorded the temperature in his classroom every hour from 8 a.m. to 1 p.m.

Temperature in a classroom
Time	8 a.m.	9 a.m.	10 a.m.	11 a.m.	12 noon	1 p.m.
Temperature (°C)	26	27	28	30	31	30.5

(a) Draw a line graph of the data. Explain your choice of scale.

(b) Estimate the temperature at 11.30 a.m.

ShowHide solution

(a) The values go from 26 to 31, all far from 0. From 0, the line would be squeezed into the top of the graph. So start the axis at 25 with a zigzag break and go up in steps of 1 to 32: 7 steps. 30.5 is halfway between 30 and 31.

(b) 11.30 a.m. is halfway between 11 a.m. (30 °C) and 12 noon (31 °C): $\frac{30 + 31}{2} = 30.5$ °C.

(a) The line graph above, with a break and a scale from 25 to 32 in steps of 1, because all the values are far from 0. (b) About 30.5 °C (c) At about 10.30 a.m.

Question 9

The line graph shows the number of members of a school's Robotics Club at the start of each year from 2022 to 2026.

(a) Describe the trend, and use it to predict the number of members at the start of 2027. State what you assume.

(b) Ali uses the same trend to predict the number of members at the start of 2040. What number does he get? Why should this prediction not be trusted?

ShowHide solution

Each step of the scale is 10. The values are 20, 25, 30, 35 and 40 (25 and 35 are halfway between gridlines).

(a) The number rose by 5 every year: a steady increase. If it goes on rising by 5 a year, the club will have $40 + 5 = 45$ members at the start of 2027.

(b) From 2026 to 2040 is $2040 - 2026 = 14$ years. Number $= 40 + 14 \times 5 = 40 + 70 = 110$ .

This is far beyond the data. A club cannot keep growing for ever: it is limited by the number of students, teachers, robots and the room it meets in, and interest may change. A trend over 5 years says little about 14 years later.

(a) A steady rise of 5 members a year; about 45 members in 2027, if the rise of 5 a year continues. (b) 110 members. It is too far beyond the data, and the trend is unlikely to continue for so long.

Question 10

The line graph shows the electricity bill of Mrs Goh's household each month from January to June.

(a) Between which two months did the bill rise the most?

(b) Find the total of the six bills.

(d) Mei Ling looks at the graph and says, "The bill in May was several times the bill in January." Explain why she is wrong.

ShowHide solution

The axis has a break and then goes from 70 in steps of 10. The bills are $80, $85, $95, $110, $120 and $105 (85, 95 and 105 are halfway between gridlines).

(a) The rises: $85 - 80 = 5$ , $95 - 85 = 10$ , $110 - 95 = 15$ , $120 - 110 = 10$ . The greatest is 15, from March to April.

(b) Total $= 80 + 85 + 95 + 110 + 120 + 105 = 595$ , so the total is $595.

Percentage $= \frac{40}{80} \times 100\% = 50\%$

(d) The axis starts at 70, not 0, so the January point is drawn near the bottom and the May point looks many times higher. The values are $80 and $120, and $120 \div 80 = 1.5$ : the May bill was one and a half times the January bill.

(a) From March to April ($15) (b) $595 (c) 50% (d) The axis starts at $70, so heights are not in proportion to the bills; $120 is only 1.5 times $80.

Question 11

Ali and Ravi cycled along a park connector from the same starting point at the same time. The line graph shows how far each had gone every 10 minutes.

(a) Who cycled faster in the first 10 minutes? How can you tell from the graph?

(b) After they set off, when had the two boys next cycled the same distance? What was that distance?

(d) Who was ahead after 50 minutes, and by how much?

ShowHide solution

Each step of the scale is 1 km.

Ali: 0, 2, 4, 6, 8, 10 km. Ravi: 0, 3, 5, 6, 7, 8 km.

(a) In the first 10 minutes Ravi went 3 km and Ali 2 km. Ravi's line is steeper there.

(b) The lines start together at 0 km. After that, Ravi's line is above Ali's until they meet again at 6 km at 30 minutes; after 30 minutes Ali's line is above.

(c) Distance changes continuously, so it can be estimated between the readings. 25 minutes is halfway between 20 and 30 minutes.

Ali: $\frac{4 + 6}{2} = 5$ km. Ravi: $\frac{5 + 6}{2} = 5.5$ km.

Ravi was ahead by $5.5 - 5 = 0.5$ km.

(d) After 50 minutes Ali had gone 10 km and Ravi 8 km: Ali was ahead by $10 - 8 = 2$ km.

(a) Ravi: he went 3 km to Ali's 2 km, and his line is steeper. (b) After 30 minutes, 6 km (c) About 0.5 km apart, Ravi ahead (d) Ali, by 2 km

Question 12

At the Secondary 1 camp, Mr Tan asked 40 students for their favourite camp activity. Siti drew a line graph of the results.

(a) Siti says, "The graph shows that the popularity rose from kayaking to climbing, fell at archery, rose to its highest at high ropes and then fell again." Explain why a line graph is not suitable for these data, and name a better diagram.

(b) The stall in Question 2 wants to show its sales of each month from January to June. Give one advantage of a line graph over a table for this, and one disadvantage.

ShowHide solution

(a) The activities are categories with no order: they could be listed in any order, and the line would go up and down differently each time. There is no "fall" or "rise" from one activity to the next, and the line between two activities means nothing. A bar graph (or a pictogram) shows these numbers well; since each student chose one activity, a pie chart could also show them as parts of the 40.

(b) Advantage: the line graph shows the trend at a glance: which months rose and fell, and that June was the best month, without comparing numbers one by one.

Disadvantage: the numbers cannot be read as exactly from a graph as from a table (for example, 3500 must be read halfway between two gridlines).

(a) The activities have no order, so joining them with a line shows a "trend" that does not exist; a bar graph (or a pictogram, or a pie chart) suits them. (b) Advantage: the rises, falls and trend are seen at a glance. Disadvantage: values cannot be read as exactly as from a table.

Key Points

A line graph shows how a quantity changes over time: the times along the horizontal axis in order and in equal steps, the values against a scale up the vertical axis, the points joined in order by straight lines.
To read a point, first find what one step of the scale stands for; a point halfway between two gridlines stands for the number halfway between them.
The change between two readings is the later value minus the earlier one; the steeper the piece of line, the greater the rise or fall.
Describe a trend by its general direction, where it changes and how fast, with values from the graph.
The vertical axis of a line graph may start above 0 if a zigzag break is marked; read the values from the scale, not from how high the points look.
Values between the readings can be estimated only for a quantity that changes continuously, such as temperature; for counts or totals of each day or month they have no meaning.
Two lines on one graph, with a key, compare two sets of data at the same times.
A prediction beyond the data assumes the trend continues: it is only an estimate, and it should not go far beyond the data.
Use a line graph for change over time; not for categories with no order, or for parts of a whole.