for approximately 30 minutes would reasonably last a diver 30 minutes at a depth of 10 metres. The
formula you can use is
PiVi
Vi
P1 gh
t C
where
t time in minutes
Vi Volume of air in the tank in m3 (use 0.0150 m3)
Pi Pressure of air in the tank in Pa (use 2.02 107 Pa)
C rate of air consumption in m3/min
g 9.8 m/s2 (gravitational constant)
mass density of sea water in kg/m3
P1 air pressure at the surface of the water in Pa
h depth of the diver (given as 10 m)
Based on this formula, would a diver with a 30-minute tank be correct in expecting 30 minutes of air
during a 10 m dive?
Some variables are given; the rest you can determine through a little research.
STRUCTURAL ENGINEER—KAREN DECONTIE
CANADIAN BIOGRAPHY
Karen Decontie holds a master of engineering degree in civil engineering from the University of
Calgary. Her work with Public Works and Government Services Canada keeps the public safe
while enjoying the beauty of Canada’s national parks. Karen states that what she likes best
about her job is that it is “challenging, feeling that you can build something that is useful to
many people.” As a structural engineer, she is responsible for ensuring the safety of bridges
and structures. As a member of the Algonquin Nation, she believes that the more First Nations
engineers there are, “the more decisions we can make on our own.” In her own case, she says,
her engineering work enables her to contribute to society both for the present and for future
generations of visitors to the national parks.
Karen grew up in Kitigan Zibi in Quebec and went to McGill for her undergraduate degree
before moving to Calgary, where she has lived since. Leaving home was not easy, but the
experience gave her an opportunity to “learn different thinking,” and the process of becoming a
professional engineer “helped me to gain self-esteem and believe in myself as a person.”
Source: Native Access, Role Model: Karen Decontie,
http://www.nativeaccess.com/allabout/pop_rm/rm_decontie_karen.html
9–1
Simplification of Fractions
Parts of a Fraction
A fraction has a numerator, a denominator, and a fraction line.
fraction line
a
b
numerator
denominator
Quotient
A fraction is a way of indicating a quotient of two quantities. The fraction a/b can be read
“a divided by b.”
227
228
Chapter 9
◆
Fractions and Fractional Equations
Ratio
The quotient of two numbers or quantities is also spoken of as the ratio of those quantities. Thus
the ratio of x to y is x/y. The two forms ab and a/b are equally valid.
Division by Zero
Since division by zero is not permitted, it should be understood in our work with fractions that
the denominator cannot be zero.
◆◆◆
Example 1: What values of x are not permitted in the following fraction:
3x
x2 x 6
Solution: Factoring the denominator, we get
3x
x2 x 6
3x
(x 2)(x 3)
We see that an x equal to 2 or 3 will make (x 2) or (x 3) equal to zero. This will result in
◆◆◆
division by zero, so these values are not permitted.
Common Fractions
A fraction whose numerator and denominator are both integers is called a common fraction.
◆◆◆
Example 2: The following are common fractions:
2
3
,
9
5
,
124
125
and
18
11
◆◆◆
Algebraic Fractions
An algebraic fraction is one whose numerator and/or denominator contain literal quantities.
◆◆◆
Example 3: The following are algebraic fractions:
x
y
,
Recall that a polynomial is
an expression in which the
exponents are nonnegative
integers.
兹x 2
,
x
3
y
,
and
x2
x3
◆◆◆
Rational Algebraic Fractions
An algebraic fraction is called rational if the numerator and the denominator are both
polynomials.
◆◆◆
Example 4: The following are rational fractions:
x
y
,
3
w3
,
and
x2
x3
But
兹x 2
x
is not a rational fraction, since there is no perfect root for the numerator.
◆◆◆
Proper and Improper Fractions
A proper common fraction is one whose numerator is smaller than its denominator.
◆◆◆
9
Example 5: 35 , 13 , and are proper fractions, whereas 85 , 32 , and 74 are improper fractions.
11
◆◆◆
A proper algebraic fraction is a rational fraction whose numerator is of lower degree than
the denominator.
Section 9–1
◆◆◆
◆
229
Simplification of Fractions
Example 6: The following are proper fractions:
x
2
x2
x2 2x 3
x3 9
and
However,
x3 2
x3
x2
x2
y
and
◆◆◆
are improper fractions.
Mixed Form
A mixed number is the sum of an integer and a fraction.
◆◆◆
Example 7: The following are mixed numbers:
3
1
1
◆◆◆
2, 5, and 3
2
4
3
A mixed expression is the sum or difference of a polynomial and a rational algebraic fraction.
◆◆◆
Example 8: The following are mixed expressions:
1
3x 2 x
and
y
y y2 1
◆◆◆
Decimals and Fractions
To change a fraction to an equivalent decimal, simply divide the numerator by the denominator.
◆◆◆
9
Example 9: To write as a decimal, we divide 9 by 11.
11
9
11
We get a repeating decimal; the dots following the number indicate that the digits continue indefinitely.
◆◆◆
Repeating decimals are also commonly written with a bar over the repeating part: 0.81
0.818 181 81 . . . or 0.81
To change a decimal number to a fraction, write a fraction with the decimal number in the
numerator and 1 in the denominator. Multiply numerator and denominator by a multiple of 10
that will make the numerator a whole number. Finally, reduce to lowest terms.
◆◆◆
Example 10: Express 0.875 as a fraction.
Solution:
0.875
875
7
0.875 1
1000 8
◆◆◆
To express a repeating decimal as a fraction, follow the steps in the next example.
__
◆◆◆
Example 11: Change the repeating decimal 0.8181 to a fraction.
__
Solution: Let. x 0.81 81. Multiplying by 100, we have
__
100x 81.81
__
Subtracting the first equation, x 0.81 81 from the second gives us
99x 81
(exactly)
Dividing by 99 yields
9
81
x 99 11
(reduced)
◆◆◆
We’ll cover reducing fractions to
their lowest terms in more detail
later in this section.
230
Chapter 9
◆
Fractions and Fractional Equations
Simplifying a Fraction by Reducing to Lowest Terms
We reduce a fraction to lowest terms by dividing both numerator and denominator by any factor
that is contained in both.
Simplifying
Fractions
ad
bd
a
b
50
◆◆◆
Example 12: Reduce the following to lowest terms. Write the answer without negative
exponents.
3(3) 3
9
(a) 12 4(3) 4
3x2yz
3 x2 y z
x
(b) • • • 2
3
2
3
9 x y z
9xy z
3yz2
◆◆◆
When possible, factor the numerator and the denominator. Then divide both numerator and
denominator by any factors common to both.
◆◆◆
Example 13:
2x2 x (2x 1)x 2x 1
(a) 3x
3(x)
3
ab bc b(a c) a c
(b) bc bd b(c d) c d
(2x 1)(x 3)
2x2 5x 3
x3
(c) 2
(2x 1)(2x 1) 2x 1
4x 1
x(x a) 2b(x a)
x 2 ax 2bx 2ab
(d) (x a)(2x 3a)
2x2 ax 3a2
(x a)(x 2b)
x 2b
(x a)(2x 3a) 2x 3a
◆◆◆
The process of striking out the same factors from numerator and denominator is called
cancelling.
Common
Errors
Use caution when cancelling: If a factor is missing from even
one term in the numerator or denominator, that factor cannot
be cancelled.
xy z y z
wx
w
We may divide (or multiply) the numerator and denominator
by the same quantity (Eq. 50), but we may not add or subtract
the same quantity in the numerator and denominator, as this
will change the value of the fraction. For example,
3 31 4 2
5 51 6 3
Simplifying Fractions by Changing Signs
Recall from Chapter 2 that any two of the three signs of a fraction may be changed without
changing the value of a fraction.
Section 9–1
◆
231
Simplification of Fractions
a
b
a
b
a
b
a
b
a
b
Rules
of Signs
a
b
a
b
a
b
a
b
10
11
We can sometimes use this idea to simplify a fraction; that is, to reduce it to lowest terms.
◆◆◆
Example 14: Simplify the fraction
3x 2
2 3x
Solution: We can change the sign of the denominator and the sign of the entire fraction.
3x 2
3x 2
3x 2
3x 2
1
2 3x
(2 3x) 2 3x 3x 2
◆◆◆
changed
Exercise 1
◆
Simplification of Fractions
In each fraction, what values of x, if any, are not permitted?
x
12
1. 2. x
12
18
5x
3. 4. 2
x5
x 49
3x
7
5. 6. 2
2
x 3x 2
8x 14x 2
Change each fraction to a decimal. Work to four decimal places.
7
7. 12
125
10. 155
5
8. 9
11
11. 3
15
9. 16
25
12. 9
Change each decimal to a fraction.
13. 0.4375
14. 0.390 625
15. 0.6875
16. 0.281 25
17. 0.7777 . . .
18. 0.636 363 . . .
Simplify each fraction by manipulating the algebraic signs.
2x y
ab
19. 20. ba
y 2x
w(x y z)
(a b)(c d)
21. 22. yxz
ba
Reduce to lowest terms. Write your answers without negative exponents.
14
23. 21
36
26. 44
81
24. 18
2ab
27. 6b
75
25. 35
12m2n
28. 2
15mn
Hint: Factor the denominators in
problems 4, 5, and 6.
232
Chapter 9
◆
Fractions and Fractional Equations
21m2p2
29. 4
28mp
abx bx2
30. 2
acx cx
4a2 9b2
31. 4a2 6ab
3a2 6a
32. 2
a 4a 4
x2 5x
33. 2
x 4x 5
xy 3y2
34. x3 27y3
x2 4
35. x3 8
2a3 6a2 8a
36. 2a3 2a2 4a
2m3n 2m2n 24mn
37. 6m3 6m2 36m
9x3 30x2 25x
38. 3x4 11x3 10x2
2a2 2
39. a2 2a 1
3a2 4ab b2
40. a2 ab
x2 z2
41. x3 z3
2x2
42. 2
6x 4x
2a2 8
43. 2a2 2a 12
2a2 ab 3b2
44. a2 ab
x2 1
45. 2xy 2y
x3 a2x
46. x2 2ax a2
2x 4y 4 2
47. 3x 8y 8 3
(x y)2
49. x2 y2
18a2c 6bc
48. 42a2d 14bd
b2 a2 6b 9
51. 5b 15 5a
3w 3y 3
52. w2 y2 2wy 1
9–2
mw 3w mz 3z
50. m2 m 12
Multiplication and Division of Fractions
Multiplication
We multiply a fraction a/b by another fraction c/d as follows:
Multiplying
Fractions
a c
b d
ac
bd
• 51
The product of two or more fractions is a fraction whose numerator is the product of the
numerators of the original fractions and whose denominator is the product of the denominators
of the original fractions.
◆◆◆
Example 15:
2 5 2(5) 10
(a) • 3 7 3(7) 21
2 1 17 7 119
(c) 5 • 3 • 3 2
3 2
6
1 2 3 1(2)(3) 1
(b) • • 2 3 5 2(3)(5) 5
◆◆◆