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Reading Data and Graphs: Core practice

10 practice problems for this lesson. Work on paper, use hints when you need them, and check the answer or the full solution when you are ready.

Difficulty: Core (core-course level)

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Problem 1 of 10

Problem 1 The returned envelopes

Each letter identifies one collection box. A log records A with $0$ envelopes, B with $2$ , C with $1$ , D with $0$ , E with $2$ , F with $2$ , G with $3$ , H with $0$ , and C with $1$ again. Repeated records for the same box count just once. Complete the frequency table shown for the actual boxes, checking its total against the number of different boxes in the log.

A blank frequency table for the number of envelopes in each box.

Text description of this figure

A frequency table with two columns. The first column is headed Envelopes in a box and the second is headed Number of boxes. Below the headings are five rows, labeled 0, 1, 2, 3 and 4 in the first column, in that order. Every cell in the Number of boxes column is empty, and there is no total row.
1. Hint 1
  The table must count boxes, so repeated records of one box must not create extra tallies.
2. Hint 2
  Identify the label that occurs twice before counting the boxes in each row.
3. Hint 3
  Include a frequency of zero when no box has a listed envelope count, then add the frequencies to check the box count.
Answer

Frequencies: $0$ envelopes, $3$ boxes; $1$ , $1$ box; $2$ , $3$ boxes; $3$ , $1$ box; $4$ , $0$ boxes. Total $8$ boxes.

Full solution

Box C appears twice in the log, but it is one box with $1$ envelope.
Count it once.
The eight boxes A through H have amounts $0$ , $2$ , $1$ , $0$ , $2$ , $2$ , $3$ , $0$ .

Boxes A, D, and H contain $0$ envelopes, so that frequency is $3$ .
Box C gives a frequency of $1$ for $1$ envelope.
Boxes B, E, and F give a frequency of $3$ for $2$ envelopes.
Box G gives a frequency of $1$ for $3$ envelopes, and no box contains $4$ envelopes.

Fill the frequency column with $3$ , $1$ , $3$ , $1$ , $0$ in row order.
Check the total.
$3+1+3+1+0=8$
This matches the eight boxes, with each box counted once.

Answer

Frequencies: $0$ envelopes, $3$ boxes; $1$ , $1$ box; $2$ , $3$ boxes; $3$ , $1$ box; $4$ , $0$ boxes. Total $8$ boxes.

Key idea

A frequency table should count each actual observation once, even when a record has been copied.
Problem 1: Mark complete
Problem 2 The return stations

The pictograph shows returned envelopes at three stations. Aspen received $12$ envelopes. Complete the missing value in the key.

Returned envelopes at three stations, with the key's value missing.

Text description of this figure

A pictograph with three rows of circle symbols, all full circles the same size. The Aspen row shows one full circle followed by the left half of a circle. The Birch row shows two full circles. The Cedar row shows three full circles followed by the left half of a circle. Below the rows is a key: a full circle, then the words each full circle represents, a blank line, and envelopes. The value in the key is left blank.
1. Hint 1
  A pictograph key assigns the same value to every full symbol.
2. Hint 2
  Count the full and partial symbols in the Aspen row together.
3. Hint 3
  Divide the known Aspen total by its number of symbols.
Answer

Each full circle represents $8$ envelopes.

Full solution

The Aspen row has one full circle and one half circle, or $1.5$ circles altogether.
Its $12$ envelopes are shared over those symbols, so the value of a full circle is
$12\div1.5=8$
envelopes.

Check that the key gives a whole number of envelopes in each of the other rows.
Birch has $2$ circles, so it received
$2\times8=16$
envelopes.
Cedar has $3.5$ circles, so it received
$3.5\times8=28$
envelopes.
Both are whole numbers, as counts of envelopes must be, so the key fits every row.

Answer

Each full circle represents $8$ envelopes.

Key idea

A known row total can determine a pictograph key by division.
Problem 2: Mark complete
Problem 3 The storage containers

The bar graph shows water in three containers. Each container can hold $12$ liters. What fraction of Container C's capacity is filled? Give the fraction in lowest terms.

Water in three containers.

Text description of this figure

A vertical bar graph. The horizontal axis is labeled Container, with three bars for A, B and C. The vertical axis is labeled Water in liters and runs from 0 to 12, with a light gridline every 1 liter and number labels at 0, 2, 4, 6, 8, 10 and 12. Container A's bar reaches 4 liters, B's bar reaches 10 liters, and C's bar reaches 9 liters. No values are written on the bars.
1. Hint 1
  Read the amount in Container C from the vertical scale before comparing it with the capacity.
2. Hint 2
  The filled fraction is the amount of water divided by the full capacity.
Answer

$\frac{3}{4}$ .

Full solution

The top of Container C's bar is at $9$ liters.
Compare this with its $12$ liter capacity.
$\frac{9}{12}=\frac{3}{4}$
Thus three quarters of Container C is filled.

Check by taking that fraction of its capacity.
$\frac{3}{4}\times12=9$
This matches the graph.

Answer

$\frac{3}{4}$ .

Key idea

A bar gives an exact amount that can be compared with a stated whole to form a fraction.
Problem 3: Mark complete
Problem 4 The tank readings

The line graph shows the amount of water in a tank every $2$ minutes. List each interval between successive readings in which the amount changed by at least $3$ liters. For each interval, state whether the amount rose or fell and by how many liters.

Water in a tank, read every $2$ minutes.

Text description of this figure

A line graph. The horizontal axis is labeled Time in minutes and runs from 0 to 6, with ticks, labels and vertical gridlines at 0, 2, 4 and 6. The vertical axis is labeled Water in liters and runs from 0 to 16, with a light gridline every 1 liter and number labels at 0, 2, 4, 6, 8, 10, 12, 14 and 16. Four filled points are plotted: 14 liters at 0 minutes, 8 liters at 2 minutes, 8 liters at 4 minutes, and 11 liters at 6 minutes. Straight segments join neighboring points, and the line stops at the first and last points. No values are written beside the points.
1. Hint 1
  Compare the readings at the two ends of each interval between successive readings.
2. Hint 2
  A fall counts as a change too; compare the size of each change with $3$ liters.
3. Hint 3
  Include an interval whose change is exactly $3$ liters.
Answer

From $0$ to $2$ minutes: fell $6$ liters. From $4$ to $6$ minutes: rose $3$ liters.

Full solution

The readings at $0$ , $2$ , $4$ , and $6$ minutes are $14$ , $8$ , $8$ , and $11$ liters.
From $0$ to $2$ minutes the water falls by
$14-8=6$
liters, so that interval qualifies.

From $2$ to $4$ minutes the change is
$8-8=0$
liters, so that interval does not qualify.
From $4$ to $6$ minutes the water rises by
$11-8=3$
liters.
Exactly $3$ meets the requirement, so this interval qualifies too.

The listed intervals include both changes large enough to meet the condition and exclude the flat segment.

Answer

From $0$ to $2$ minutes: fell $6$ liters. From $4$ to $6$ minutes: rose $3$ liters.

Key idea

Reading successive values reveals both the direction and the size of each change over time.
Problem 4: Mark complete
Problem 5 The storage floor

The circle graph divides a $120$ square meter floor among three uses. Equipment and supplies will be combined into one category called Storage. The whole circle stands for one full turn, $360^\circ$ . Find the area assigned to Storage and the angle of its slice in the new circle graph.

How the floor is divided among three uses.

Text description of this figure

A circle graph of floor uses with three slices. Starting at the top of the circle and moving clockwise, the slices are Equipment, 45 percent, shaded with a blue tint; Supplies, 20 percent, shaded with diagonal stripes; and Walking space, 35 percent, shaded very lightly. Each slice is labeled outside the circle, joined to it by a short line, with its category name and percentage. No areas or angles are shown.
1. Hint 1
  The new category contains all of the two original categories.
2. Hint 2
  Add their shares before applying the result to the floor area.
3. Hint 3
  Take the same combined share of the full turn as you took of the floor area.
Answer

Area $78$ square meters; slice angle $234^\circ$ .

Full solution

Equipment is $45\%$ and supplies are $20\%$ of the floor.
Their combined share is
$45\%+20\%=65\%$

Apply the combined share to the full floor area.
$0.65\times120=78$
Storage therefore occupies $78$ square meters.

The Storage slice must occupy the same fraction of a full turn.
$0.65\times360^\circ=234^\circ$

Check with the remaining category, Walking space, which keeps the other $35\%$ .
Its area is $0.35\times120=42$ square meters, and
$120-78=42$
Its angle is $0.35\times360^\circ=126^\circ$ , and
$360^\circ-234^\circ=126^\circ$
Both pairs agree, so the Storage area and angle are consistent with the graph.

Answer

Area $78$ square meters; slice angle $234^\circ$ .

Key idea

Combining circle-graph categories adds their shares, which then apply to both the total amount and a full turn.
Problem 5: Mark complete
Problem 6 The bead packets

The double bar graph shows gold and blue beads in two crates. Combine the beads from both crates. Each packet needs $1$ gold bead and $2$ blue beads. What is the greatest number of complete packets you can make, and how many beads of each color remain?

Gold and blue beads in two crates.

Text description of this figure

A double bar graph. The horizontal axis is labeled Crate, with two groups, A and B. In each group the left bar is Gold, drawn with diagonal stripes, and the right bar is Blue, drawn with a solid light fill; a legend at the top shows both fills with their names. The vertical axis is labeled Number of beads and runs from 0 to 12, with a light gridline every 1 bead and number labels at 0, 2, 4, 6, 8, 10 and 12. In Crate A, the Gold bar reaches 6 and the Blue bar reaches 10. In Crate B, the Gold bar reaches 9 and the Blue bar reaches 8. No values are written on the bars.
1. Hint 1
  First find the available total of each color across both crates.
2. Hint 2
  Compare how many complete packets each color can supply.
3. Hint 3
  After choosing the greatest possible packet count, subtract the beads used from each available total.
Answer

$9$ packets; $6$ gold beads and $0$ blue beads remain.

Full solution

The gold bars show $6$ beads in Crate A and $9$ in Crate B; the blue bars show $10$ and $8$ .
Combine like colors.
$6+9=15$ $10+8=18$
There are $15$ gold beads and $18$ blue beads.

The gold beads could supply $15$ packets, but the blue beads supply only
$18\div2=9$
packets.
Both colors are needed, so the greatest possible count is $9$ packets.

Nine packets use $9$ gold beads and $18$ blue beads.
The leftovers are
$15-9=6$
gold beads and
$18-18=0$
blue beads.
A tenth packet would require more blue beads, so the result is as large as possible.

Answer

$9$ packets; $6$ gold beads and $0$ blue beads remain.

Key idea

Reading each bar of a double bar graph by its legend, then adding the bars for one legend entry across all the categories, gives the totals a problem needs.
Problem 6: Mark complete
Problem 7 The exhibition posters

The table shows all entries in an exhibition, with each entry counted once. One poster will compare the number of entries in the categories. A second will emphasize each category's share of the whole exhibition. Choose a bar graph, line graph, or circle graph for each purpose, and give the three percentages for the second poster.

Entries in the exhibition, by category.

Text description of this figure

A frequency table with two columns, headed Category and Number of entries. It has three rows, in this order: Drawing with 12 entries, Pottery with 9 entries, and Textiles with 3 entries. There is no total row and no percentages.
1. Hint 1
  Choose each display according to the question its poster must answer.
2. Hint 2
  Separate categories can be compared by amounts or by their shares of one total.
3. Hint 3
  Add the table frequencies, then divide each frequency by the total and convert to a percent.
Answer

First poster: bar graph. Second: circle graph. Drawing $50\%$ , pottery $37.5\%$ , textiles $12.5\%$ .

Full solution

A bar graph makes the entry counts easy to compare.
A circle graph emphasizes how the full exhibition splits into categories.
A line graph would not fit either purpose because these categories are not readings ordered in time.

The table contains $12$ drawings, $9$ pottery entries, and $3$ textile entries.
Find the total.
$12+9+3=24$

Divide each count by $24$ .
$\frac{12}{24}=0.5$ $\frac{9}{24}=0.375$ $\frac{3}{24}=0.125$
These are $50\%$ , $37.5\%$ , and $12.5\%$ .

Check that the shares account for the whole exhibition.
$50+37.5+12.5=100$
The percentages add to $100\%$ .

Answer

First poster: bar graph. Second: circle graph. Drawing $50\%$ , pottery $37.5\%$ , textiles $12.5\%$ .

Key idea

The same data can call for different displays when the question changes from comparing counts to comparing shares.
Problem 7: Mark complete
Problem 8 The printed reports

Report 1 and Report 2 each show a count for Room A and a count for Room B. In Report 2, Room B's bar is more than twice as long as Room A's. Tess says the percentage increase in count from Room A to Room B is the same in both reports. Is Tess right? Find the percentage increase from each report and justify your decision.

Counts for Rooms A and B in Report 1 and Report 2.

Text description of this figure

Two bar graphs stacked one above the other, with plotting areas of the same width and height. Each has a horizontal axis labeled Room, with bars for A and B, and a vertical axis labeled Count. The top graph, titled Report 1, has a vertical scale from 0 to 50 with a light gridline every 1 and number labels at 0, 5, 10, 15, 20, 25, 30, 35, 40, 45 and 50; its bars start at 0, with Room A's bar reaching 36 and Room B's bar reaching 45. The bottom graph, titled Report 2, has a vertical scale from 30 to 50 with a light gridline every 1 and number labels at 30, 35, 40, 45 and 50; its bars start at 30, and a zigzag break on the vertical axis below the 30 label shows that the scale does not start at 0; Room A's bar reaches 36 and Room B's bar reaches 45. No values are written on the bars.
1. Hint 1
  Read the value at each bar's top in both reports before using the lengths of the drawn bars.
2. Hint 2
  Compare the increase with the starting value for Room A.
3. Hint 3
  Consider which part of the drawing changes when the value axis begins at a different number.
Answer

Yes; the increase in count from Room A to Room B is $25\%$ in both reports.

Full solution

Read each bar's top against its own scale.
In Report 1, whose scale starts at $0$ , Room A reaches $36$ and Room B reaches $45$ .
In Report 2, whose scale starts at $30$ , Room A also reaches $36$ and Room B reaches $45$ .
So both reports show the same two counts.

The increase from Room A to Room B is
$45-36=9$
Compare the increase with Room A's count, the starting value.
$\frac{9}{36}=0.25$
The increase is $25\%$ in both reports, so Tess is right.

In Report 2 the drawn bars show only the parts above $30$ , so their lengths stand for $36-30=6$ and $45-30=15$ .
Room B's bar therefore looks two and a half times as long as Room A's, but those lengths are not the counts.
Comparing the lengths would suggest an increase of $150\%$ , but the labeled scale gives the counts $36$ and $45$ , and so the same $25\%$ .

Answer

Yes; the increase in count from Room A to Room B is $25\%$ in both reports.

Key idea

A cut-off value axis changes bar-length comparisons while still allowing calculations from correctly read scale values.
Problem 8: Mark complete
Problem 9 The weekend visits

The bar graph gives total visits to each park over Saturday and Sunday together. A student concludes that Park A had more visits than Park B on both days. Does that conclusion follow from the graph? Justify your decision.

Total visits to two parks over Saturday and Sunday.

Text description of this figure

A vertical bar graph titled Weekend visits. The horizontal axis is labeled Park, with bars for A and B. The vertical axis is labeled Number of visits and runs from 0 to 20, with labeled ticks and light gridlines every 2 visits. Park A's bar reaches 18 and Park B's bar reaches 14. The graph shows only these weekend totals; there are no separate Saturday or Sunday values and no numbers on the bars.
1. Hint 1
  A total combines days and does not show how the visits were divided between them.
2. Hint 2
  Try giving Park B more visits on Saturday while keeping both weekend totals correct.
3. Hint 3
  Find the Sunday counts by subtracting your Saturday counts from the graph totals.
Answer

No. One example: Saturday, A $4$ and B $8$ ; Sunday, A $14$ and B $6$ .

Full solution

The bars show $18$ visits to Park A and $14$ to Park B over the whole weekend.
They do not show either day separately.

For example, Park A could have $4$ visits on Saturday and $14$ on Sunday, while Park B has $8$ on Saturday and $6$ on Sunday.
These counts match both totals.
$4+14=18$ $8+6=14$

In this example, Park B has more visits on Saturday.
Thus the weekend graph does not establish that Park A had more on both days.
Other daily counts giving the same totals and a similar counterexample are also valid.

Answer

No. One example: Saturday, A $4$ and B $8$ ; Sunday, A $14$ and B $6$ .

Key idea

A larger combined total does not establish that each part of it was larger.
Problem 9: Mark complete
Problem 10 The archive donations

The circle graphs show all donations to an archive in April and May. Each donation belongs to exactly one of the two categories shown. Jo says fewer photographs were donated in May because their share of donations was smaller. Decide whether Jo is right, and compare the number of photograph donations in the two months.

Donations to the archive in April and May.

Text description of this figure

Two circle graphs stacked one above the other. The upper one is titled April, with the line Total: 50 donations beneath the title. Starting at the top of the circle and moving clockwise, its slices are Photographs, 60 percent, and Maps, 40 percent. The lower one is titled May, with the line Total: 80 donations. Starting at the top and moving clockwise, its slices are Photographs, 45 percent, and Maps, 55 percent. Photographs have the same blue tint in both circles and Maps the same very light tint. Each slice is labeled outside its circle, joined to it by a short line, with its category name and percentage. No counts are shown.
1. Hint 1
  A percentage becomes a count only after you apply it to the whole it describes.
2. Hint 2
  Read both the photograph share and the total for each month.
3. Hint 3
  Compute the photograph count for each month before comparing them.
Answer

Jo is wrong; May had $6$ more photograph donations, $36$ compared with $30$ in April.

Full solution

April has $50$ donations, and photographs are $60\%$ of them.
$0.60\times50=30$
There were $30$ photograph donations in April.

May has $80$ donations, and photographs are $45\%$ of them.
$0.45\times80=36$
There were $36$ photograph donations in May.

The count increased by
$36-30=6$
photograph donations.
Jo is wrong: the photograph share is smaller, but it is taken from a larger total.

Answer

Jo is wrong; May had $6$ more photograph donations, $36$ compared with $30$ in April.

Key idea

A smaller percentage can represent a larger count when the whole has grown.
Problem 10: Mark complete

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