Tables and line graphs · applications

Applications: Tables and Line Graphs

10 question types · Primary 4 · each worked step by step with a figure that follows the steps

PSLE · O-Level · SAT · GCSE Higher

01

Two-Way Table with One Missing Number

strategyRow Total Minus the Number You Know

A table shows the clubs that the children in one class chose. Of the boys, 12 chose Art and 15 chose Chess, which is 27 boys. Of the girls, 9 chose Chess, and there are 23 girls. The number of girls who chose Art is missing. In all, 26 children chose Art and 24 chose Chess, which is 50 children. (a) How many girls chose Art? (b) How many more girls than boys chose Art?

ArtChessTotalBoys121527Girls?923Total262450Girls: ? + 9 = 23
The missing number is in the row for the girls. That row ends with the total, 23.
Find the row for the girls. It has the missing number, the 9 girls who chose Chess, and the total of 23 girls.
step 1 of 4

The numbers in a row add up to the total at the end of the row. Use the row that holds the missing number, then check the answer with the column.

  1. Find the row for the girls. It has the missing number, the 9 girls who chose Chess, and the total of 23 girls.
  2. The two clubs in a row add up to the row total, so the missing number is 23 − 9 = 14.
  3. (a) 14 girls chose Art. Check with the Art column: 12 + 14 = 26, which is the total in the table.
  4. (b) Read down the Art column: 12 boys and 14 girls. 14 − 12 = 2, so 2 more girls than boys chose Art.

answer(a) 14 girls; (b) 2 more girls

techniqueReading Tables

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Taking 9 from the total of the whole table, 50 − 9 = 41. The 50 counts every child in the class. The girls' row adds up to 23, so the subtraction starts from 23.
  • Comparing the row totals for part (b), 27 − 23 = 4 more boys. Those totals count both clubs. The question is about Art only, so read the Art column.
02

Bus Timetable: Journey Time and the Latest Bus

strategyCount Up to the Hour, Then Read the Arrives Column

A bus timetable has three buses from Town to the Zoo. Bus A leaves Town at 7:10 and arrives at 7:45. Bus B leaves at 7:40 and arrives at 8:15. Bus C leaves at 8:05 and arrives at 8:40. (a) How many minutes does Bus B take? (b) Mei must be at the Zoo by 8:25. Which is the latest bus she can take, and how many minutes early will she be?

BusLeavesArrivesA7:107:45B7:408:15C8:058:40
Bus B leaves at 7:40 and arrives at 8:15.
Find the row for Bus B. It leaves at 7:40 and arrives at 8:15.
step 1 of 5

A journey time is counted from the leaving time up to the next hour, and then past it. To be in time, look at when each bus arrives, not when it leaves.

  1. Find the row for Bus B. It leaves at 7:40 and arrives at 8:15.
  2. Count up to the hour first. From 7:40 to 8:00 is 20 minutes. From 8:00 to 8:15 is 15 minutes.
  3. (a) 20 + 15 = 35, so Bus B takes 35 minutes.
  4. Read down the Arrives column: 7:45, 8:15 and 8:40. Mei must arrive by 8:25, so 8:40 is too late. The latest time that is not too late is 8:15, and that is Bus B.
  5. (b) Mei takes Bus B at 7:40. From 8:15 to 8:25 is 10 minutes, so she is 10 minutes early.

answer(a) 35 minutes; (b) Bus B at 7:40, and she is 10 minutes early

techniqueReading Tables · Time Durations

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Choosing Bus C because it leaves at 8:05, which is before 8:25. The time that matters is the arriving time. Bus C arrives at 8:40, which is after 8:25.
  • Subtracting the times like ordinary numbers, 815 − 740 = 75, and saying 75 minutes. An hour has 60 minutes, not 100, so count up to 8:00 and then past it.
03

Price Table Where Buying More Costs Less Each

strategyFind the Row for the Number Bought, Then Multiply

A shop sells notebooks, and the price of each notebook depends on how many are bought. For 1 to 5 notebooks, each costs $4. For 6 to 11 notebooks, each costs $3. For 12 or more notebooks, each costs $2. (a) Mrs Tan buys 7 notebooks. How much does she pay? (b) Mr Lee needs 11 notebooks. How much less does he pay if he buys 12 notebooks instead of 11?

Number boughtPrice of each1 to 5$46 to 11$312 or more$27 is between 6 and 11
7 notebooks are in the row for 6 to 11, so each costs $3.
Find the row for 7 notebooks. 7 is between 6 and 11, so each notebook costs $3.
step 1 of 5

The number bought decides which row of the table to use. Every notebook in the order then has the price in that row.

  1. Find the row for 7 notebooks. 7 is between 6 and 11, so each notebook costs $3.
  2. (a) 7 × 3 = 21, so Mrs Tan pays $21.
  3. 11 is in the same row, so 11 notebooks cost 11 × 3 = $33.
  4. 12 is in the row for 12 or more, so each notebook costs $2, and 12 notebooks cost 12 × 2 = $24.
  5. (b) 33 − 24 = 9, so Mr Lee pays $9 less when he buys 12 notebooks.

answer(a) $21; (b) $9 less

techniqueReading Tables · Two-Step Word Problems

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Using the first row for every order, 7 × 4 = $28. The price of $4 is only for 1 to 5 notebooks. Find the row that holds the number bought before multiplying.
  • Pricing the first 5 notebooks at $4 and the other 2 at $3. The table gives one price for every notebook in the order, so all 7 notebooks cost $3 each.
04

Temperature Graph: The Greatest Rise

strategySubtract Each Reading from the Next One

Ravi reads the temperature every two hours and draws a line graph. The readings are 24 degrees at 6 am, 25 at 8 am, 28 at 10 am, 32 at 12 noon, 33 at 2 pm, 30 at 4 pm and 27 at 6 pm. (a) Between which two readings next to each other does the temperature rise the most, and by how many degrees? (b) What is the difference between the highest and the lowest temperature?

2426283032346 am8 am10 amnoon2 pm4 pm6 pmtimetemperature, degrees
The line goes up from 6 am to 2 pm. Work out the rise for each part.
The line goes up from 6 am to 2 pm, so work out the rise for each part of the line there.
step 1 of 5

A rise is the later reading minus the earlier reading. The greatest rise is the steepest part of the line going up, which is not the same as the highest point.

  1. The line goes up from 6 am to 2 pm, so work out the rise for each part of the line there.
  2. From 6 am to 8 am: 25 − 24 = 1. From 8 am to 10 am: 28 − 25 = 3. From 10 am to 12 noon: 32 − 28 = 4. From 12 noon to 2 pm: 33 − 32 = 1.
  3. (a) The greatest rise is 4 degrees, between 10 am and 12 noon. This is the steepest part of the line going up.
  4. The highest point is 33 degrees at 2 pm. The lowest point is 24 degrees at 6 am.
  5. (b) 33 − 24 = 9, so the difference is 9 degrees.

answer(a) Between 10 am and 12 noon, by 4 degrees; (b) 9 degrees

techniqueLine Graphs

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Choosing 12 noon to 2 pm because the line is highest there. The highest point is not the greatest rise. The temperature rises only 1 degree in that part.
  • Taking the first reading from the last reading for part (b), 27 − 24 = 3. The highest temperature is 33 degrees at 2 pm, not the last reading of the day.
05

Plant Height Graph: A Value Between Two Points

strategyGo Halfway Along the Line Between Two Readings

Siti measures a bean plant every two weeks and draws a line graph. The plant is 2 cm tall at the start, 6 cm in week 2, 12 cm in week 4, 16 cm in week 6 and 18 cm in week 8. (a) Use the graph to estimate the height of the plant in week 3. (b) Why is this only an estimate, and between which two heights must the real height in week 3 be?

048121620012345678weekheight, cm
Week 3 has no dot. It is halfway between the dots for week 2 and week 4.
Siti did not measure in week 3. Week 3 is halfway between week 2 and week 4, so look at the line between those two dots.
step 1 of 5

Week 3 has no dot of its own. It is halfway between two dots, so read the height halfway along the line that joins them.

  1. Siti did not measure in week 3. Week 3 is halfway between week 2 and week 4, so look at the line between those two dots.
  2. The height is 6 cm in week 2 and 12 cm in week 4. The plant grew 12 − 6 = 6 cm in those two weeks.
  3. Halfway along the line is half of that growth: 6 ÷ 2 = 3 cm, and 6 + 3 = 9 cm.
  4. (a) The height in week 3 is about 9 cm.
  5. (b) Only the dots are measurements. The straight line joins two of them, and the plant may have grown more in one week than in the other. A plant does not get shorter, so the real height is between 6 cm and 12 cm.

answer(a) About 9 cm; (b) it was not measured in week 3, and the real height is between 6 cm and 12 cm

techniqueLine Graphs

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Reading the nearest dot and answering 6 cm or 12 cm. Those are the heights in week 2 and week 4. For week 3, go up from week 3 to the line and then across to the scale.
  • Giving 9 cm as a measured height. Siti measured only in weeks 0, 2, 4, 6 and 8. A height read from the line between two dots is an estimate.
06

Savings Graph: Passing a Target

strategyDraw the Target Line Across the Graph

Jun is saving for a skateboard that costs $60. His line graph shows his total savings at the end of each week. The totals for weeks 1 to 6 are $10, $25, $35, $50, $70 and $85. (a) At the end of which week does Jun first have more than $60? (b) How much more does he have at the end of week 6 than at the end of week 1?

020406080100123456weektotal savings, $$60
The line across the graph is at $60, the price of the skateboard.
Draw a line across the graph at $60, the price of the skateboard.
step 1 of 4

A line drawn across the graph at the target shows which points are below it and which are above it. The increase over the whole time is the last point minus the first point.

  1. Draw a line across the graph at $60, the price of the skateboard.
  2. Follow the graph from the left. The point for week 4 is $50, which is below the line. The point for week 5 is $70, which is above the line.
  3. (a) Week 5 is the first week in which Jun has more than $60.
  4. (b) Read the first point and the last point: $10 in week 1 and $85 in week 6. 85 − 10 = 75, so he has $75 more.

answer(a) Week 5; (b) $75 more

techniqueLine Graphs

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Choosing week 4 because $50 is the point nearest to $60. $50 is less than $60, so Jun cannot buy the skateboard yet.
  • Answering $85 for part (b). $85 is the total in week 6. The increase is the total in week 6 minus the total in week 1.
07

Table of Scores: A Tally as a Fraction of the Group

strategyTally the Scores That Pass, Then Write Part over Whole

A table shows the quiz scores of the 12 pupils in a group. The first row of the table is 6, 9, 7, 8. The second row is 5, 8, 10, 7. The third row is 6, 9, 7, 8. (a) How many pupils scored more than 6? (b) What fraction of the group scored more than 6? Give the fraction in its simplest form.

69785810769786, 5 and 6 are not more than 6
A score of 6 is not more than 6. Every other score gets a tally mark.
Go through the table one score at a time and make a tally mark for each score that is more than 6. A score of 6 is not more than 6.
step 1 of 5

Go through the table once and make a tally mark for each score that is more than 6. The tally is the numerator and the whole group is the denominator.

  1. Go through the table one score at a time and make a tally mark for each score that is more than 6. A score of 6 is not more than 6.
  2. The first row has 3 such scores: 9, 7 and 8. The second row has 3: 8, 10 and 7. The third row has 3: 9, 7 and 8.
  3. (a) 3 + 3 + 3 = 9, so 9 pupils scored more than 6.
  4. There are 12 pupils in the group, so the fraction is 912.
  5. (b) Divide the numerator and the denominator by 3: 912 = 34. Check: 3 pupils did not score more than 6, and 9 + 3 = 12.

answer(a) 9 pupils; (b) 34

techniqueReading Tables · A Fraction of a Group · Equivalent Fractions

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Counting the two scores of 6 as well, which gives 11 pupils. More than 6 means a score of 7 or higher.
  • Writing the fraction as 93, the pupils who scored more than 6 over the pupils who did not. The denominator is the whole group, which is 12 pupils.
08

Line Plot in Halves and Quarters: Most Common Length and Total

strategyRead the Tallest Stack, Then Pair Lengths into Whole Numbers

A class measures 10 small leaves to the nearest quarter of a centimeter and draws a line plot. It has 1 cross at 2 cm, 2 crosses at 214 cm, 4 crosses at 212 cm, 2 crosses at 234 cm and 1 cross at 3 cm. (a) Which length is the most common? (b) What is the total length of the 10 leaves?

22 1/42 1/22 3/43length, cmThe tallest stack has 4 crosses
Each cross is one leaf. The tallest stack is above 212 cm.
Each cross is one leaf. The tallest stack of crosses is above 212 cm, and it has 4 crosses.
step 1 of 5

Each cross is one leaf. The tallest stack shows the most common length. For the total, every cross is added, and lengths that make a whole number together are added first.

  1. Each cross is one leaf. The tallest stack of crosses is above 212 cm, and it has 4 crosses.
  2. (a) The most common length is 212 cm.
  3. For the total, pair lengths that make whole numbers. 2 + 3 = 5 cm. 214 + 234 = 5 cm, and there are two pairs like this, which make 10 cm.
  4. The four leaves of 212 cm make 4 × 212 = 10 cm.
  5. (b) 5 + 10 + 10 = 25, so the total length is 25 cm. Check: 1 + 2 + 4 + 2 + 1 = 10 leaves were added.

answer(a) 212 cm; (b) 25 cm

techniqueLine Plots in Halves and Fourths · Adding and Subtracting Mixed Numbers

examsPSLE

Common pitfalls

  • Answering 4 for part (a). 4 is the number of leaves with the most common length. The length is read from the scale under the stack.
  • Adding only the five numbers on the scale, 2 + 214 + 212 + 234 + 3. That adds each length once. Each cross is a leaf, so a length with 4 crosses is added 4 times.
09

Attendance Table: The Mean by Leveling

strategyMove the Extra from the Big Days to the Small Days

A table shows how many children came to a reading club in one week: 18 on Monday, 22 on Tuesday, 20 on Wednesday, 17 on Thursday and 23 on Friday. (a) Find the mean number of children for a day by leveling the five days. (b) How many children came in the whole week?

Mon18Tue22Wed20Thu17Fri23
Try to make every day 20, the number on Wednesday.
Leveling means moving children from the bigger days to the smaller days until every day is the same. Try 20, the number on Wednesday.
step 1 of 5

Leveling moves children from the days with more to the days with fewer until every day has the same number. That number is the mean, and the total does not change.

  1. Leveling means moving children from the bigger days to the smaller days until every day is the same. Try 20, the number on Wednesday.
  2. Tuesday has 2 more than 20 and Monday has 2 fewer. Move 2 from Tuesday to Monday, and both days are 20.
  3. Friday has 3 more than 20 and Thursday has 3 fewer. Move 3 from Friday to Thursday, and both days are 20.
  4. (a) Every day is now 20, so the mean is 20 children.
  5. (b) Leveling does not change the total, so 5 × 20 = 100 children came in the week. Check: 18 + 22 + 20 + 17 + 23 = 100.

answer(a) 20 children; (b) 100 children

techniqueThe Mean · Reading Tables

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Taking the number in the middle of the table as the mean without checking. Wednesday's 20 is the mean here only because the days above 20 have 5 extra children and the days below 20 are 5 short.
  • Adding children to a small day without taking them from a big day, so that Monday becomes 20 and Tuesday stays 22. Every child moved to one day must leave another day, or the total changes.
10

Two Lines on One Graph: Catching Up

strategyCompare the Two Lines Minute by Minute

Ali and Ben run for 5 minutes, and both start at 0 m. One graph shows how far each boy has run at the end of every minute. For Ali the distances are 200, 400, 600, 800 and 1000 m. For Ben they are 300, 500, 600, 700 and 750 m. (a) After how many minutes does Ali catch up with Ben? (b) How far ahead is Ali at the end of 5 minutes?

02004006008001000012345minutesdistance, mAliBenBen is ahead
After 1 minute and after 2 minutes, Ben's line is above Ali's line.
Compare the two lines minute by minute. After 1 minute Ben is ahead, with 300 m against 200 m. After 2 minutes Ben is still ahead, with 500 m against 400 m.
step 1 of 4

At each minute, the higher line belongs to the boy who has run further. Where the two lines cross, the boys have run the same distance.

  1. Compare the two lines minute by minute. After 1 minute Ben is ahead, with 300 m against 200 m. After 2 minutes Ben is still ahead, with 500 m against 400 m.
  2. After 3 minutes both boys have run 600 m. The two lines cross at this point.
  3. (a) Ali catches up with Ben after 3 minutes. After that, Ali's line is above Ben's line.
  4. (b) At 5 minutes Ali has run 1000 m and Ben has run 750 m. 1000 − 750 = 250, so Ali is 250 m ahead.

answer(a) After 3 minutes; (b) 250 m

techniqueLine Graphs

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Saying that Ben is ahead because his line is higher at the start. The lines cross at 3 minutes, and after that Ali's line is the higher one.
  • Reading the gap at the wrong minute, for example 800 − 700 = 100 m at 4 minutes. The question asks about the end of 5 minutes.
Mr. Chalk Read the guide