Black – Divide by Decimals
1. How many quarters would it take to equal the volume of a piece of paper? (Round
your answer to the nearest quarter.)
A quarter has a diameter of .9375
inches and thickness of .0625 inches
The piece of paper is 10.5
inches by 8 inches and is
.0025 inches thick
2. A year is thought of as having 365 days. Actually the time it takes the earth to
revolve around the sun is 365.242 days. This is why we have a February 29th every 4
years. If we didn't have a leap year, eventually we would have the beginning of winter
on June 21st instead of December 21st. If the practice of having leap years was
discontinued, how many years would it take before winter started on June 21st?
3. If a kilometer is equal to .621 miles, how many kilometers is one mile equal to?
Limits of Accuracy
STARTER
A Lotto agency has some
advertising on their window. It
says: “If you are the only holder
of a winning ticket you should win
$1,300,000!”
Would you win exactly that
amount?
Explain how they might have
calculated that figure.
What is the least amount of
money you would expect to get if
their statement was correct to 2 sf?
What is the greatest amount of money you would expect to get if
their statement was correct to 2 sf?
Working backwards from a rounded number to a range of values that
it might have come from is called finding limits of accuracy.
To calculate limits of accuracy for a number x , go half-way up
or down to the closest numbers that have the same number of
significant figures as x .
Example
State the limits of accuracy for the measurement 188.
Answer
Consider
this
scale,
which shows 186
the number
188. 188 has 3 sf.
188 (3 sf)
187
188
187.5
189
190
191
188.5
Any number between 187.5 and 188.5, when rounded to 3 sf, would
give 188. Therefore the lower limit of accuracy is 187.5, and the
upper limit is 188.5.
Note that 187.5 is half-way between 187 and 188, and 188.5 is
half-way between 188 and 189.
One way of writing the answer is 187.5 < x < 188.5.
Example
A newspaper reports that there are 37,000 people at Ericsson
Stadium to watch an Auckland Warriors game. Give the limits of
accuracy for this estimate N.
Answer
37,000 has 2 significant figures. The closest numbers to 37,000 are
36,000 and 38,000. Going halfway in either direction from 37,000 to
these numbers we get 36,500 and 37,500.
Expressing as an inequality: 36,500 < N < 37,500.
A more advanced area of Mathematics involves studying what
happens when numbers that are inaccurate are used in calculations.
The effect of these errors can be studied to see how accurate the
final answer is.
Example
The measurements in this rectangle
are accurate to 2 sf.
Calculate the greatest and least
possible area of the rectangle.
27 m
64 m
Answer
The limits of accuracy for the base are 63.5 < b < 64.5.
The limits of accuracy for the height are 26.5 < h < 27.5.
Greatest possible area = 64.5 x 27.5 = 1773.75.
Least possible area = 63.5 x 26.5 = 1682.75.
Give the limits accuracy for these measurements.
4. 68 mm
5. 397 mm
6. 4 seconds
7. 50 g
8. 5890 kg
9. 820 cm
10. 92 kg
11. 89.1ºC
The following numbers are estimates. Give limits of accuracy for them.
12. Over five days a total of 60,000 people attended rugby games at Eden Park.
13. The total viewing audience each night for Shortland Street is 370,000.
14. 120 people attended a wedding.
15. The number of infant deaths in the Third World per year is 40,000,000.
16. A protest march in Ashburton against hospital cuts involved 5000 people.
17. Kaitoa throws a discus, and the distance reached is measured as 37 meters, 58 cm.
What is the lower limit of accuracy for this measurement?
18. The signpost shows the distance to Mathville.
a. What is the least possible distance the
cyclist will have to travel?
b. If the cyclist returns to the signpost after
reaching Mathville, what is the maximum
possible total distance the cyclist will
travel?
19. Calculate the greatest and least possible areas of these shapes.
a.
42.3 mm
15.9 mm
b. 5.2 m
8.6 m
20. Filling the Freezer
The Math Club meets one day a week after school to do projects and math
investigations. They enjoy having an ice cream snack at each meeting. Their project
this week is to buy the ice cream they'll be storing in their freezer for their
upcoming meetings.
Before they leave for the store they measure the freezer. The interior of the
freezer is 65 cm wide, 45 cm deep, and 35 cm tall.
At the store they measure an ice cream container. The dimensions of the top of the
box are 17.5 cm by 11.6 cm. The height is 12.2 cm.
One group, the Protractors, figures out how many boxes will fit in the freezer if
they place each box with the lid facing up and the 17.5 cm dimension along the front
of the freezer.
Another group, the Right Angles, agrees that they'll keep the lids facing upwards (to
avoid messes as some of the containers start to get eaten) and they figure out how
many boxes will fit if they turn the boxes and have the 11.6 cm dimension along the
front of the freezer.
Questions: Which group can fit more ice cream boxes in the freezer? How much
unused space would be left in the freezer if they buy that many boxes?
Extra: Describe a way to fit more whole containers of ice cream in the freezer. How
many can you fit?
Solutions
1. 5
The piece of paper has a volume of 10.5 x 8 x .0025 = .21 cubic inches
The radius of a quarter is .9375 ÷ 2 = .46875
The volume of a quarter is 3.14 x .46875 x .46875 x .0625 = .0431 cubic inches
.21 (volume of paper) ÷ .0431 (volume of quarter) = 4.872 quarters per piece of paper.
2. 754 years
Each year is .242 days "too long"
The problem is really asking how many .242 day pieces would it take to add up to half
of a year (365 ÷ 2 = 182.5 days)
182.5 ÷ .242 = 754 pieces or 754 years.
3. 1.61 kilometers
1 kilometer n kilometers
=
.621 miles
1 mile
Cross-multiplying: .621n = 1
Divide both sides by .621:n = 1.61
4. 67.5 ≤ X ≤ 68.5
5. 396.5 ≤ X ≤ 397.5
6. 3.5 ≤ X ≤ 4.5
7. 49.5 ≤ X ≤ 50.5
8. 5889.5 ≤ X ≤ 5890.5
9. 819.5 ≤ X ≤ 820.5
10. 91.5 ≤ X ≤ 92.5
11. 89.05 ≤ X ≤ 89.15
12. 59.500 ≤ N ≤ 60,500
13. 365,000 ≤ N ≤ 375,000
14. 115 ≤ N ≤ 125
15. 39,500,000 ≤ N ≤ 40,500,000
16. 4500 ≤ N ≤ 5500
17. 37.575
18. a. 7.75
b. 15.64
19. a. 42.25 ≤ b ≤ 42.35
15.85 ≤ h ≤ 15.95
Greatest 675.4825 mm2
Least
669.6625 mm2
b. 5.15 ≤ h ≤ 5.25
8.55 ≤ b ≤ 8.65
Greatest 22.70625 m2
Least
22.01625 m2
20. The Right Angles can fit more ice cream boxes in the freezer. If they buy 20 boxes
of ice cream, there will be 52,843 cubic centimeters of unused space in the freezer.
EXTRA: To fit more containers of ice cream in the freezer, they could put the
To find how many boxes the Protractors could fit in the freezer, I divided the width
of the freezer by the dimension of the ice cream box along the front of the freezer:
65 divided by 17.5=3.7 boxes across
I then divided the height of the freezer by the height of the box:
35 divided by 12.2=2.8 boxes high
I then divided the depth of the freezer by the depth of the box:
45 divided by 11.6=3.8 boxes deep
These numbers had to be rounded down to whole boxes of ice cream.
There would be 3 boxes across, 2 boxes high, and 3 boxes deep. I multiplied 3x2x3
to get a total of 18 boxes for the Protractors.
I used the same method to find how many boxes the Right Angles could fit in the
freezer.
65 divided by 11.6=5.6 boxes across
35 divided by 12.2=2.8 boxes high
45 divided by 17.5=2.5 boxes deep
Once again, I rounded down to get 5 boxes across, 2 boxes high, and 2 boxes deep. I
multiplied 5x2x2 to get a total of 20 boxes for the Right Angles.
To find how much unused space was left after putting 20 boxes in the freezer:
Volume of freezer=65cmx35cmx45cm=102,375 cubic cm
Volume of one box of ice cream=11.6cmx12.2cmx17.5cm=2,476.6 cubic cm
Volume of 20 boxes of ice cream=20x2,476.6 cubic cm=49,532 cubic cm
Volume of unused space=Volume of freezer-Volume of 20 boxes=
102,375 cubic cm-49,532 cubic cm=52,843 cubic cm
EXTRA: For this problem, I used the same calculations as above.
65 divided by 12.2=5.3 boxes across
35 divided by 17.5=2 boxes high
45 divided by 11.6=3.8 boxes deep
This would be 5 boxes across, 2 boxes high, and 3 boxes deep.
5x2x3=30 boxes of ice cream in the freezer
Bibliography Information
Teachers attempted to cite the sources for the problems included in this problem set. In some cases,
sources may not have been known.
Problems
Bibliography Information
1-3
Zaccaro, Edward. Challenge Math (Second
Edition): Hickory Grove Press, 2005.
Example and 4-19
Barton, David. Gamma Mathematics
Pearson Education. New Zealand, 2000
20
The Math Forum @ Drexel
(http://mathforum.org/)