Scientific Methods Worksheet 2:

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Scientific Methods Worksheet 2:
Proportional Reasoning
Some problems adapted from Gibbs' Qualitative Problems for Introductory Physics
1. One hundred cm are equivalent to 1 m. How many cm are equivalent to 3 m? Briefly explain
how you could convert any number of meters into a number of centimeters.
xcm 100cm
3m = 300cm
One could set up a proportion as in
, and solve for x, but it

3m
1m
is simpler (and mathematically equivalent) to multiply the number of meters by the factor:
100cm
100cm
, as in 3m 
 300cm
1m
1m
2. Forty-five cm are equivalent to how many m? Briefly explain how you could convert any
number of cm into a number of m.
45cm 100cm
45 cm = 0.45 m Again, one could use a proportion
and solve for x, but

xm
1m
that is a 2-step process wheras one could simply make the conversion by multiplying the
1m
1m
number of centimeters by the factor
, as in 45cm 
 0.45m
100cm
100cm
3. One mole of water is equivalent to 18 grams of water. A glass of water has a mass of
200 g. How many moles of water is this? Briefly explain your reasoning.
1mole
 11.1moles
18g
as the new unit.
200g 
Multiplying by this factor cancels out the g, leaving moles
Use the metric prefixes table to answer the following questions:
4. The radius of the earth is 6378 km. What is the
diameter of the earth in meters?
6378km 
giga
mega
kilo
centi
milli
micro
nano
1000m
 6,378,000m
1km
6.378  106 m
5. In an experiment, you find the mass of a cart
to be 250 grams. What is the mass of the cart
in kilograms?
250g 
Metric prefixes:
= 1 000 000 000
= 1 000 000
= 1 000
= 1 / 100
= 1 / 1000
= 1 / 1 000 000
= 1 / 1 000 000 000
billion
million
thousand
hundredth
thousandth
millionth
billionth
1kg
 0.25kg
1000g
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6. How many megabytes of data can a 4.7 gigabyte DVD store?
Rather than try to figure out how many megabytes are in a gigabytes (or vice-versa), convert
to the base unit with one factor and to the desired unit with a second factor. (use B for byte)
109 B 1MB
4.7GB 
 6  4.7  103 MB
1GB 10 B
7. A mile is farther than a kilometer. Consider a fixed distance, like the diameter of the moon.
Would the number expressing this distance be larger in miles or in kilometers? Explain.
One would need more of a smaller unit than of a larger unit for a fixed distance. Consider an
easier example. A yard is 3 ft but 36 in.
8. One US dollar = 0.73 Euros (as of 12/13.) Which is worth more, one dollar or one Euro?
How many dollars is one Euro?
Since one dollar gets you less than one Euro, the Euro is worth more.
1USD
1Euro 
 $1.37
0.73Euro
9. In 2012, Germans paid 1.65 Euros per liter of gasoline. At the same time, American prices
were $3.90 per gallon.
a. How much would one gallon of European gas have cost in dollars?
b. How much would one liter of American gasoline have cost in Euros?
(One US dollar = 0.73 Euros, 1 gallon = 3.78 liters)
3.78L 1.65Euro
1USD
1gal 


 $8.54
1gal
1L
0.73Euro
1L 
1gal $3.90 0.73Euro


 0.75Euro
3.78L 1gal
1USD
10. A mile is equivalent to 1.6 km. When you are driving at 60 miles per hour, what is your
speed in meters per second? Clearly show how you used proportions to arrive at a solution.
60mi 1.6km 1000m
1hr
1min




 26.7 m This approach uses unitary factors.
s
1hr
1mi
1km
60min 60s
1mi 0.444m
, which is a useful factor to keep in mind for

hr
1s
converting speed in miles/hr to m/s.
These factors simplify to
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11. For each of the following mathematical relations, state what happens to the value of y when
the following changes are made. (k is a constant)
a. y = kx , x is tripled.
If k is a constant, then the relationship becomes y  x. Tripling x also triples y.
b. y = k/x, x is halved
If k is a constant, then the relationship becomes y  1/x. Halving x doubles y.
1
1
If y  , then ? y  1 . The (?) must be replaced by 2.
x
x
2

c. y = k/x2 , x is doubled
If k is a constant, then the relationship becomes y  1/x2. Doubling x decreases y by a
factor of 4.
1
1
1
If y  2 , then ? y 
 ? y  2 . The (?) must be replaced by 1/4.
2
x
4x
2x

 

d. y = kx2, x is tripled.
If k is a constant, then the relationship becomes y  x2. Tripling x increases y by a factor
of 9.
     ?y  9x
If y  x 2 , then ? y  3x
2
2
The (?) must be replaced by a 9.
12. When one variable is directly proportional to another, doubling one variable also doubles the
other. If y and x are the variables and a and b are constants, circle the following
relationships that are direct proportions. For those that are not direct proportions, explain
what kind of proportion does exist between x and y.
a. y = 3x
b. y = ax + b
c. y = x
d. y = ax2
e. y = a/x
f. y = ax
g. y = 1/x
h. y = a/x2
(a), (c), (f) are direct proportions. In (b), y and x are
linearly related, but not proportional.
In (d), y is proportional to the square of x.
In (e) and (g), y is inversely proportional to x.
In (h) y is inversely proportional to the square of x.
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13. The diagram shows a number of relationships between x and y.
a. Which relationships are linear? Explain.
(b), (d), (e) and (f) are linear relationships
The slope of these graphs are constant.
y
b. Which relationships are direct proportions?
Explain.
(b) and (e) - The lines pass through the origin,
so a change in x produces the same size change
in y.
a
b
c
d
e
c. Which relationships are inverse proportions?
Explain.
(a) As x increases, y decreases by a constant
factor.
f
x
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