Pictograms

Read a pictogram with its key, including half and quarter symbols, find what one symbol represents, draw a pictogram from a table, choose a sensible key, and see when a pictogram is a good choice.

Before you start

A fraction of a number: $\frac{1}{2}$ of 20 is 10, $\frac{1}{4}$ of 8 is 2, and $\frac{3}{4}$ of 8 is 6.
A mixed number times a whole number: $3\frac{1}{2} \times 14 = 3 \times 14 + \frac{1}{2} \times 14 = 42 + 7 = 49$ .
A part as a percentage of a total: 33 out of 150 is $\frac{33}{150} \times 100\% = 22\%$ .

What a Pictogram Shows

A pictogram shows data with small pictures called symbols. Each category of the data has a row of identical symbols, and a key states how many one whole symbol represents. The longer the row, the larger the number, so the categories can be compared at a glance.

Pictograms show counts in categories: the number of books of each kind sold at a book fair, the number of students in each CCA, the number of cups of bubble tea sold on each day of a week.

Pictogram

A pictogram uses rows (or columns) of identical symbols to show the frequency of each category.

The key states the number that one whole symbol represents.

Number for a row $=$ number of symbols in the row $\times$ the number one symbol represents.

A pictogram has a title, a label for each row, symbols of the same size and evenly spaced, every row starting from the same line, and a key.

Example 1: Reading a pictogram

The pictogram shows the numbers of books sold at a school book fair.

(a) How many comics were sold?

(b) How many more fiction books than biographies were sold?

(d) What fraction of the books sold were fiction books? Give the fraction in its simplest form.

Solution

The key says that each symbol represents 20 books.

(a) The row for comics has 5 symbols: $5 \times 20 = 100$ comics.

(b) Fiction: $4 \times 20 = 80$ books. Biography: $2 \times 20 = 40$ books. Difference: $80 - 40 = 40$ books.

A quicker way: the fiction row has $4 - 2 = 2$ more symbols, and $2 \times 20 = 40$ .

Check, row by row: $80 + 60 + 100 + 40 = 280$ .

(d) Fraction that were fiction $= \frac{80}{280} = \frac{2}{7}$ (dividing the numerator and the denominator by 40).

(a) 100 comics were sold. (b) 40 more fiction books than biographies were sold. (c) 280 books were sold altogether. (d) $\frac{2}{7}$ of the books sold were fiction books.

Part Symbols

A number that is not a whole number of symbols is shown with part of a symbol. Half a symbol represents half the number one whole symbol represents; a quarter of a symbol represents a quarter of that number. A part symbol is drawn by cutting a symbol from top to bottom and keeping the left part: its width shows the fraction.

In the drawing below each whole square represents 8. A square cut to three quarters of its width represents $\frac{3}{4} \times 8 = 6$ , a half square represents $\frac{1}{2} \times 8 = 4$ , and a quarter square represents $\frac{1}{4} \times 8 = 2$ .

Part symbols

If one whole symbol represents $k$ , then $\frac{1}{4}$ of a symbol represents $\frac{1}{4}k$ , $\frac{1}{2}$ of a symbol represents $\frac{1}{2}k$ , and $\frac{3}{4}$ of a symbol represents $\frac{3}{4}k$ .

Number for a row $=$ (number of whole symbols $+$ the fraction of the part symbol) $\times k$ .

Example 2: Half symbols

The pictogram shows the numbers of cups of bubble tea sold at a stall from Monday to Friday.

(a) How many cups were sold on Monday?

(b) On which day were the most cups sold, and how many were sold that day?

(d) On which days were fewer than 40 cups sold?

Solution

Each whole circle represents 10 cups, so half a circle represents $\frac{1}{2} \times 10 = 5$ cups.

(a) Monday has 3 whole circles and a half: $3 \times 10 + 5 = 35$ cups.

(b) Friday has the longest row, $6\frac{1}{2}$ circles: $6 \times 10 + 5 = 65$ cups.

(c) Number of circles altogether: $3\frac{1}{2} + 5 + 2\frac{1}{2} + 4 + 6\frac{1}{2} = 21\frac{1}{2}$ . Total $= 21\frac{1}{2} \times 10 = 215$ cups.

Check, row by row: $35 + 50 + 25 + 40 + 65 = 215$ .

(d) 40 cups is 4 circles. The rows shorter than 4 circles are Monday ( $3\frac{1}{2}$ circles, 35 cups) and Wednesday ( $2\frac{1}{2}$ circles, 25 cups). Thursday has exactly 40 cups, which is not fewer than 40.

(a) 35 cups were sold on Monday. (b) The most were sold on Friday: 65 cups. (c) 215 cups were sold altogether. (d) Fewer than 40 cups were sold on Monday and on Wednesday.

Reading a part symbol

Compare the part with a whole symbol in the row above or below it. A part that reaches a quarter of the way across a whole symbol is $\frac{1}{4}$ , halfway across is $\frac{1}{2}$ , and three quarters of the way across is $\frac{3}{4}$ . A square makes quarters easy to see, because a quarter of its width is also a quarter of its area.

Example 3: Quarter symbols

The pictogram shows the numbers of books four classes borrowed from the school library in one week.

(a) How many books did Class 1A borrow?

(b) How many books did Class 1B borrow?

(c) Which class borrowed the most books, and which the fewest? How many more books did the first class borrow than the second?

(d) Find the ratio of the number of books Class 1C borrowed to the number Class 1D borrowed. Give it in its simplest form.

Solution

Each whole square represents 8 books, so a quarter square represents $\frac{1}{4} \times 8 = 2$ books, a half square 4 books and three quarters of a square 6 books.

(a) Class 1A has 5 whole squares and a quarter: $5 \times 8 + 2 = 42$ books.

(b) Class 1B has 3 whole squares and three quarters: $3 \times 8 + 6 = 30$ books.

The most is Class 1D (48) and the fewest is Class 1B (30). Difference: $48 - 30 = 18$ books.

(d) Class 1C : Class 1D $= 36 : 48 = 3 : 4$ (dividing both by 12).

(a) Class 1A borrowed 42 books. (b) Class 1B borrowed 30 books. (c) Class 1D borrowed 18 more books than Class 1B. (d) The ratio is $3 : 4$ .

Finding What One Symbol Represents

Sometimes the key is missing, but the number for one row, or the total for the whole pictogram, is known. Count the symbols (with their parts), then divide.

Finding the key

Number one symbol represents $=$ number for a row $\div$ number of symbols in that row.

Number one symbol represents $=$ total $\div$ total number of symbols in the pictogram.

Example 4: The key from one row

Mei Ling drew a pictogram of the snacks chosen by students at a school canteen, but the key was torn off. The table gives the number of symbols in each row of her pictogram.

Snack	Number of symbols
Curry puff	$6$
Kaya toast	$4\frac{1}{2}$
Popiah	$3$
Sandwich	$2\frac{1}{2}$

She remembers that 48 students chose curry puffs.

(a) How many students does one symbol represent?

(b) How many students chose kaya toast?

Solution

(a) The curry puff row has 6 symbols for 48 students. One symbol represents $48 \div 6 = 8$ students.

(b) Kaya toast: $4\frac{1}{2} \times 8 = 4 \times 8 + \frac{1}{2} \times 8 = 32 + 4 = 36$ students.

Check, row by row: curry puff 48, kaya toast 36, popiah $3 \times 8 = 24$ , sandwich $2\frac{1}{2} \times 8 = 20$ ; $48 + 36 + 24 + 20 = 128$ .

Her pictogram with its key put back:

(a) One symbol represents 8 students. (b) 36 students chose kaya toast. (c) 128 students chose a snack altogether.

Example 5: The key from the total

Ravi stood by the main road outside his block one Saturday morning and counted the vehicles that passed in four 15-minute periods. He drew a pictogram, and the table gives the number of symbols in each of its rows.

Time	Number of symbols
7:00 to 7:15 am	$6\frac{1}{2}$
7:15 to 7:30 am	$5\frac{1}{2}$
7:30 to 7:45 am	$4$
7:45 to 8:00 am	$3$

Altogether 228 vehicles passed.

(a) Find the number of vehicles that one symbol represents.

(b) How many vehicles passed between 7:15 am and 7:30 am?

Solution

(a) Number of symbols altogether: $6\frac{1}{2} + 5\frac{1}{2} + 4 + 3 = 19$ .

One symbol represents $228 \div 19 = 12$ vehicles. (Check: $19 \times 12 = 228$ .)

Take care to divide by the number of symbols, 19, and not by the number of rows, 4.

(b) $5\frac{1}{2} \times 12 = 5 \times 12 + \frac{1}{2} \times 12 = 60 + 6 = 66$ vehicles.

The other rows: $6\frac{1}{2} \times 12 = 78$ , $4 \times 12 = 48$ and $3 \times 12 = 36$ . Check: $78 + 66 + 48 + 36 = 228$ .

(a) One symbol represents 12 vehicles. (b) 66 vehicles passed between 7:15 am and 7:30 am.

Drawing a Pictogram

To draw a pictogram from a table, first work out how many symbols each row needs, then draw them. Setting out the number of symbols as an extra column of the table keeps the working tidy and easy to check.

Drawing a pictogram

1. Write a title that says what the data are.

2. Write the categories down the left, one row for each.

3. For each category, number of symbols $=$ frequency $\div$ the number one symbol represents.

4. Draw the symbols the same size and evenly spaced, with every row starting from the same line. Draw a part symbol for a half or a quarter.

5. Write the key.

Example 6: Drawing a pictogram with half symbols

The table shows the numbers of plates of chicken rice sold at a hawker stall from Monday to Friday.

Day	Plates sold
Monday	$90$
Tuesday	$120$
Wednesday	$70$
Thursday	$100$
Friday	$150$

Draw a pictogram of the data, using a circle to represent 20 plates.

Solution

Number of circles $=$ plates sold $\div$ 20.

Day	Plates sold	Number of circles
Monday	$90$	$90 \div 20 = 4\frac{1}{2}$
Tuesday	$120$	$120 \div 20 = 6$
Wednesday	$70$	$70 \div 20 = 3\frac{1}{2}$
Thursday	$100$	$100 \div 20 = 5$
Friday	$150$	$150 \div 20 = 7\frac{1}{2}$
Total	$530$	$26\frac{1}{2}$

Check: $26\frac{1}{2} \times 20 = 530$ , which is the total number of plates.

A half circle represents 10 plates, so Monday, Wednesday and Friday each end in a half circle.

The pictogram has $4\frac{1}{2}$ , 6, $3\frac{1}{2}$ , 5 and $7\frac{1}{2}$ circles for Monday to Friday, a title, and the key "each circle represents 20 plates".

Example 7: Drawing a pictogram with quarter symbols

The 40 students of Class 1E were asked how they usually travel to school. The table shows the results.

Way of travelling	Number of students
MRT	$14$
Bus	$11$
Car	$6$
Walk	$9$

Draw a pictogram of the data, using a square to represent 4 students.

Solution

A quarter square represents $\frac{1}{4} \times 4 = 1$ student, a half square 2 students and three quarters of a square 3 students.

Way of travelling	Number of students	Number of squares
MRT	$14$	$14 \div 4 = 3\frac{1}{2}$
Bus	$11$	$11 \div 4 = 2\frac{3}{4}$
Car	$6$	$6 \div 4 = 1\frac{1}{2}$
Walk	$9$	$9 \div 4 = 2\frac{1}{4}$
Total	$40$	$10$

Check: $10 \times 4 = 40$ , the number of students in the class.

The pictogram has $3\frac{1}{2}$ squares for MRT, $2\frac{3}{4}$ for bus, $1\frac{1}{2}$ for car and $2\frac{1}{4}$ for walk, with the key "each square represents 4 students".

Choosing a Key

When the numbers are large, one symbol cannot stand for one thing: 2700 visitors would need 2700 symbols. A larger key is needed, and it has to be chosen with care.

If the key is too small, the rows are too long to draw and to count. If the key does not fit the numbers, the rows end in parts such as 0.4 or 0.7 of a symbol, which cannot be drawn or read accurately. A good place to start is a common factor of all the numbers, or a number whose half or quarter divides all of them.

A sensible key

Each number $\div$ the key is a whole number, or a whole number and a half (or a quarter, or three quarters).

The longest row has a sensible number of symbols, about 10 or fewer.

Example 8: Choosing a key for large numbers

A community library counted its visitors on four days.

Day	Thursday	Friday	Saturday	Sunday
Number of visitors	$1200$	$1500$	$2700$	$2400$

(a) Which of these keys is the most sensible for a pictogram of the data: each symbol represents 100, 300, 500 or 1000 visitors? Give a reason.

(b) Draw the pictogram with the key you chose.

Solution

(a) Work out the number of symbols each key would need.

Key	Thursday	Friday	Saturday	Sunday	Sensible?
$100$	$12$	$15$	$27$	$24$	No: rows of up to 27 symbols
$300$	$4$	$5$	$9$	$8$	Yes: whole symbols, at most 9
$500$	$2.4$	$3$	$5.4$	$4.8$	No: 0.4 and 0.8 of a symbol
$1000$	$1.2$	$1.5$	$2.7$	$2.4$	No: 0.2, 0.7 and 0.4 of a symbol

A key of 300 is the most sensible: 300 divides every number exactly ( $1200 \div 300 = 4$ , $1500 \div 300 = 5$ , $2700 \div 300 = 9$ , $2400 \div 300 = 8$ ), and the longest row has only 9 symbols.

A key of 600 would also work ( $2$ , $2\frac{1}{2}$ , $4\frac{1}{2}$ and $4$ symbols), but it was not one of the choices.

(b) With a key of 300 the rows have 4, 5, 9 and 8 symbols.

(a) Each symbol represents 300 visitors: every row is then a whole number of symbols, and no row is longer than 9 symbols. (b) The pictogram has 4, 5, 9 and 8 symbols for Thursday to Sunday.

Pictogram or Bar Graph?

A pictogram and a bar graph can show the same data. In a pictogram the length of a row is counted in symbols; in a bar graph the length of a bar is read against a scale.

Example 9: The same data drawn two ways

The pictogram of Example 2 and a bar graph both show the cups of bubble tea sold at the stall.

(a) Read the number of cups sold on Wednesday from each diagram.

(b) On Saturday the stall sold 37 cups. Can 37 cups be shown exactly on the pictogram, with the same key? Can it be shown on the bar graph?

Solution

(a) Pictogram: Wednesday has $2\frac{1}{2}$ circles, and $2\frac{1}{2} \times 10 = 25$ . Bar graph: the top of the Wednesday bar is halfway between 20 and 30, which is 25. Both give 25 cups.

(b) Pictogram: $37 \div 10 = 3.7$ circles. Seven tenths of a circle cannot be drawn or read accurately, so 37 cannot be shown exactly with this key.

Bar graph: a bar of height 37 can be drawn, seven tenths of the way from 30 to 40. The bar graph can show any value; how closely it can be read depends on its scale.

(c) The pictogram is attractive, and its value is found by counting symbols without a scale, so it is easy for anyone to understand. The bar graph can show any number, and the heights of its bars are read more precisely against the scale.

(a) 25 cups from each diagram. (b) Not exactly on the pictogram, since 3.7 symbols cannot be drawn; yes on the bar graph. (c) For example: the pictogram is easy to understand at a glance; the bar graph shows values more precisely.

Advantages and disadvantages of a pictogram
Advantages	Disadvantages
It is attractive and easy to understand, even for young children.	Part symbols are hard to draw and to read exactly.
The rows can be compared at a glance.	A number that is not a whole number of halves or quarters of the key cannot be shown exactly.
A value is found by counting symbols and using the key; no scale is needed.	With large numbers each symbol stands for many, so small differences are lost.
It suits a few categories with counts, such as items sold or students in groups.	It takes longer to draw than a bar graph, and it is not used for measurements such as heights or temperatures.

Common Mistakes

Counting the symbols and forgetting the key: 5 symbols is not 5 books when each symbol represents 20 books.
Reading a part symbol as a whole one, or as 1: half a symbol, when the key is 10, represents 5.
Mixing up a quarter and a half: compare the part with a whole symbol in the row above or below it.
Finding the key from a total by dividing by the number of rows instead of the number of symbols.
Drawing symbols of different sizes, uneven gaps, or rows that do not start from the same line: then a longer row need not mean a larger number.
Leaving out the key or the title: without the key nobody can tell what a symbol represents.
Choosing a key that leaves parts such as 0.4 or 0.7 of a symbol, which cannot be drawn or read accurately.

Practice Questions

Try each question first, then open the solution to check your working.

Practice 1: Reading a pictogram

The pictogram shows the numbers of ice-cream cones sold at a school canteen from Monday to Friday.

(a) How many cones were sold on Wednesday?

(b) On which day were the most cones sold?

ShowHide solution

Each circle represents 15 cones.

(a) Wednesday has 5 circles: $5 \times 15 = 75$ cones.

(b) Friday has the longest row, 6 circles, which is $6 \times 15 = 90$ cones.

Check: Friday $90$ , Thursday $2 \times 15 = 30$ , and $90 - 30 = 60$ .

(a) 75 cones (b) Friday (c) 60 more cones

Practice 2: Half symbols

The pictogram shows the numbers of Secondary 1 students in five CCAs.

(a) How many students are in Robotics?

(b) How many more students are in Football than in Art Club?

ShowHide solution

Each whole symbol represents 4 students, so half a symbol represents 2 students.

(a) Robotics: $3\frac{1}{2}$ symbols, $3 \times 4 + 2 = 14$ students.

(b) Football: $6 \times 4 = 24$ students. Art Club: $2\frac{1}{2}$ symbols, $2 \times 4 + 2 = 10$ students. Difference: $24 - 10 = 14$ students.

(c) Number of symbols: $4\frac{1}{2} + 6 + 3\frac{1}{2} + 2\frac{1}{2} + 5 = 21\frac{1}{2}$ . Total $= 21\frac{1}{2} \times 4 = 86$ students.

Check, row by row: Choir $4 \times 4 + 2 = 18$ , Scouts $5 \times 4 = 20$ ; $18 + 24 + 14 + 10 + 20 = 86$ .

(a) 14 students (b) 14 more students (c) 86 students

Practice 3: Quarter symbols

The pictogram shows the numbers of paper cranes four friends folded for a class project.

(a) Who folded the most cranes, and how many did he or she fold?

(b) How many cranes did Priya fold?

(d) How many cranes did the four friends fold altogether?

ShowHide solution

Each square represents 4 cranes, so a quarter square represents 1 crane, a half square 2 cranes and three quarters of a square 3 cranes.

(a) Siti has the longest row, 6 squares: $6 \times 4 = 24$ cranes.

(b) Priya: 3 squares and three quarters, $3 \times 4 + 3 = 15$ cranes.

(d) Ravi: $4\frac{1}{2}$ squares, $4 \times 4 + 2 = 18$ cranes. Total $= 21 + 15 + 18 + 24 = 78$ cranes.

Check with the symbols: $5\frac{1}{4} + 3\frac{3}{4} + 4\frac{1}{2} + 6 = 19\frac{1}{2}$ , and $19\frac{1}{2} \times 4 = 78$ .

(a) Siti, 24 cranes (b) 15 cranes (c) $7 : 5$ (d) 78 cranes

Practice 4: The key from one row

A pictogram shows the numbers of bowls of laksa sold at a hawker stall on four days. Its key has been smudged. The table gives the number of symbols in each row.

Day	Number of symbols
Thursday	$4\frac{1}{2}$
Friday	$6$
Saturday	$7\frac{1}{2}$
Sunday	$5$

On Thursday 54 bowls were sold.

(a) How many bowls does one symbol represent?

(b) How many bowls were sold on Saturday?

ShowHide solution

(a) Thursday: $4\frac{1}{2}$ symbols represent 54 bowls. $4\frac{1}{2}$ symbols are 9 half symbols, so one half symbol represents $54 \div 9 = 6$ bowls and one whole symbol represents $2 \times 6 = 12$ bowls.

Or: $54 \div 4\frac{1}{2} = 54 \div \frac{9}{2} = 54 \times \frac{2}{9} = 12$ .

(b) Saturday: $7\frac{1}{2} \times 12 = 84 + 6 = 90$ bowls.

(c) Sunday: $5 \times 12 = 60$ bowls. Difference: $90 - 60 = 30$ bowls. (Or: Saturday has $2\frac{1}{2}$ more symbols, and $2\frac{1}{2} \times 12 = 30$ .)

(a) 12 bowls (b) 90 bowls (c) 30 more bowls

Practice 5: The key from the total

At a Sports Day, the four houses won 90 medals altogether. A pictogram of the medals has lost its key; the table gives the number of symbols in each row.

House	Number of symbols
Red	$5$
Blue	$3\frac{1}{2}$
Green	$4$
Yellow	$2\frac{1}{2}$

(a) How many medals does one symbol represent?

(b) How many medals did Blue House win?

ShowHide solution

(a) Number of symbols altogether: $5 + 3\frac{1}{2} + 4 + 2\frac{1}{2} = 15$ . One symbol represents $90 \div 15 = 6$ medals.

(b) Blue: $3\frac{1}{2} \times 6 = 18 + 3 = 21$ medals.

Check: Green $4 \times 6 = 24$ , Yellow $2\frac{1}{2} \times 6 = 15$ ; $30 + 21 + 24 + 15 = 90$ .

(a) 6 medals (b) 21 medals (c) $\frac{1}{3}$

Practice 6: Drawing with quarter symbols

Hui Min watched the bird bath in a community garden near her block for half an hour on each of four mornings, and counted the birds that came to it. The table shows her results.

Day	Monday	Tuesday	Wednesday	Thursday
Number of birds	$14$	$9$	$20$	$17$

Draw a pictogram of the data, using a square to represent 4 birds.

ShowHide solution

A quarter square represents 1 bird, a half square 2 birds and three quarters of a square 3 birds.

Day	Number of birds	Number of squares
Monday	$14$	$14 \div 4 = 3\frac{1}{2}$
Tuesday	$9$	$9 \div 4 = 2\frac{1}{4}$
Wednesday	$20$	$20 \div 4 = 5$
Thursday	$17$	$17 \div 4 = 4\frac{1}{4}$
Total	$60$	$15$

Check: $15 \times 4 = 60$ .

The pictogram has $3\frac{1}{2}$ , $2\frac{1}{4}$ , 5 and $4\frac{1}{4}$ squares for Monday to Thursday, with the key "each square represents 4 birds".

Practice 7: From raw data to a pictogram

Twenty students in a CCA were asked for their favourite bubble tea flavour. Their answers were:

Taro, Brown sugar, Mango, Brown sugar, Original, Taro, Brown sugar, Mango, Original, Brown sugar, Taro, Brown sugar, Mango, Original, Taro, Brown sugar, Original, Mango, Brown sugar, Taro

(a) Make a tally and frequency table of the answers.

(b) Draw a pictogram of the data, using a circle to represent 2 students.

ShowHide solution

(a) Go through the list once, making a tally mark for each answer, then count the tallies.

Flavour	Tally	Frequency
Brown sugar		$7$
Taro		$5$
Mango		$4$
Original		$4$
Total		$20$

The total is 20, the number of students asked.

(b) Number of circles $=$ frequency $\div$ 2: brown sugar $7 \div 2 = 3\frac{1}{2}$ , taro $5 \div 2 = 2\frac{1}{2}$ , mango $4 \div 2 = 2$ , original $4 \div 2 = 2$ . Altogether $10$ circles, and $10 \times 2 = 20$ .

(a) Brown sugar 7, taro 5, mango 4, original 4 (total 20). (b) $3\frac{1}{2}$ , $2\frac{1}{2}$ , 2 and 2 circles. (c) 35% of the students chose brown sugar.

Practice 8: Choosing a key

The table shows the numbers of tickets sold for a school concert in each of four weeks.

Week	Week 1	Week 2	Week 3	Week 4
Tickets sold	$800$	$1800$	$600$	$1000$

(a) Siti suggests a key of 100 tickets and Ali suggests 300 tickets. Explain why neither key is sensible.

(b) Suggest a sensible key, and draw the pictogram.

ShowHide solution

(a) Key 100: the rows would have 8, 18, 6 and 10 symbols. A row of 18 symbols is far more than about 10: it is long to draw and to count.

Key 300: $800 \div 300 = 2\frac{2}{3}$ , and two thirds of a symbol cannot be drawn accurately. The same happens with $1000 \div 300 = 3\frac{1}{3}$ .

(b) Key 200: $800 \div 200 = 4$ , $1800 \div 200 = 9$ , $600 \div 200 = 3$ , $1000 \div 200 = 5$ . Every row is a whole number of symbols and the longest has 9.

A key of 400 also works: 2, $4\frac{1}{2}$ , $1\frac{1}{2}$ and $2\frac{1}{2}$ symbols.

(a) A key of 100 gives a row of 18 symbols, far too long; a key of 300 gives parts of $\frac{2}{3}$ and $\frac{1}{3}$ of a symbol, which cannot be drawn accurately. (b) For example, each symbol represents 200 tickets: 4, 9, 3 and 5 symbols.

Practice 9: A poor key

Four classes collected bags of rubbish at a beach clean-up: Class 1A 36 bags, Class 1B 45 bags, Class 1C 27 bags and Class 1D 54 bags. Ravi wants to draw a pictogram with each symbol representing 10 bags.

(a) Explain why 10 is a poor key for these data.

(b) Suggest a better key and give the number of symbols for each class.

ShowHide solution

(a) $36 \div 10 = 3.6$ , $45 \div 10 = 4.5$ , $27 \div 10 = 2.7$ and $54 \div 10 = 5.4$ . Only Class 1B gives a half; parts such as 0.6, 0.7 and 0.4 of a symbol cannot be drawn or read accurately.

(b) Look for a common factor: $36 = 4 \times 9$ , $45 = 5 \times 9$ , $27 = 3 \times 9$ , $54 = 6 \times 9$ . The highest common factor is 9.

With each symbol representing 9 bags, the rows have 4, 5, 3 and 6 symbols.

A key of 18 also works: 2, $2\frac{1}{2}$ , $1\frac{1}{2}$ and 3 symbols.

(a) Three of the numbers give parts of 0.6, 0.7 and 0.4 of a symbol (3.6, 2.7 and 5.4 symbols), which cannot be drawn or read accurately. (b) For example, a key of 9 bags: Class 1A 4, Class 1B 5, Class 1C 3 and Class 1D 6 symbols.

Practice 10: Pictogram and bar graph

The pictogram of Practice 2 and a bar graph both show the numbers of Secondary 1 students in five CCAs.

(a) Read the number of students in Choir from each diagram.

(b) Ali says, "The Football row is more than twice as long as the Art Club row, so more than twice as many students are in Football as in Art Club." Is he right? Show your working.

ShowHide solution

(a) Pictogram: Choir has $4\frac{1}{2}$ symbols, $4 \times 4 + 2 = 18$ . Bar graph: the top of the Choir bar is at 18. Both give 18 students.

(b) Football has 6 symbols and Art Club $2\frac{1}{2}$ , so the Football row is $6 \div 2\frac{1}{2} = 2.4$ times as long. The numbers are 24 and 10, and $24 \div 10 = 2.4$ . Ali is right: the symbols are the same size and evenly spaced, so the lengths of the rows are in the same ratio as the numbers.

(c) The pictogram is attractive and easy to understand without a scale. The bar graph can show any number of students, and each value is read directly from the scale.

(a) 18 students from each diagram. (b) Yes: $24 \div 10 = 2.4$ , more than twice as many. (c) For example: the pictogram is easy to understand at a glance; the bar graph shows each value more precisely against its scale.

Practice 11: Curry puffs at the canteen

The pictogram shows the numbers of curry puffs a school canteen sold from Monday to Friday.

(a) How many curry puffs were sold on Thursday?

(b) What percentage of the curry puffs sold during the week were sold on Wednesday?

ShowHide solution

Each square represents 8 curry puffs: a quarter square 2, a half square 4, three quarters of a square 6.

(a) Thursday: $6\frac{1}{4}$ squares, $6 \times 8 + 2 = 50$ curry puffs.

(b) Number of squares altogether: $5 + 3\frac{3}{4} + 4\frac{1}{2} + 6\frac{1}{4} + 5\frac{1}{2} = 25$ , so the week's total is $25 \times 8 = 200$ curry puffs.

Wednesday: $4\frac{1}{2}$ squares, $4 \times 8 + 4 = 36$ curry puffs. Percentage $= \frac{36}{200} \times 100\% = 18\%$ .

(c) Friday: $5\frac{1}{2}$ squares, $5 \times 8 + 4 = 44$ curry puffs. Money $= 44 \times 1.20 = 52.80$ , so the canteen received $52.80.

Check of the total, row by row: $40 + 30 + 36 + 50 + 44 = 200$ .

(a) 50 curry puffs (b) 18% (c) $52.80

Practice 12: A missing row

Mr Tan drew a pictogram of the bus services his students take to school, but he has not yet drawn the row for Service 52. Altogether 87 of his students take these four bus services.

(a) How many students take Service 52?

(b) How many symbols should Mr Tan draw in the row for Service 52?

ShowHide solution

Each symbol represents 6 students, so half a symbol represents 3.

Service 61: $3\frac{1}{2} \times 6 = 21$ . Service 154: $4 \times 6 = 24$ . Service 185: $2\frac{1}{2} \times 6 = 15$ .

(a) These three services: $21 + 24 + 15 = 60$ students. Service 52: $87 - 60 = 27$ students.

(b) $27 \div 6 = 4\frac{1}{2}$ symbols.

(a) 27 students (b) $4\frac{1}{2}$ symbols (c) $\frac{8}{29}$

Key Points

A pictogram shows the frequency of each category with a row of identical symbols; the key states what one whole symbol represents.
Number for a row $=$ number of symbols $\times$ the number one symbol represents.
A part symbol represents that fraction of the key: with a key of 8, a quarter symbol is 2, a half is 4 and three quarters is 6.
The key $=$ number for a row $\div$ its number of symbols, or total $\div$ total number of symbols.
To draw a pictogram, work out number of symbols $=$ frequency $\div$ key for each row; draw the symbols the same size, evenly spaced, from the same starting line, with a title and a key.
Choose a key that makes every row a whole number of symbols (or a half or quarter more), with no row too long.
A pictogram is attractive and quick to understand; a bar graph shows values more precisely and can show any number.