What is the total expenditure

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32 C H A P T E R 1 / I N T R O D U C T O R Y T O P I C S I : A L G E B R A
4. A 5-metre iron bar is to be produced. The bar may not deviate by more than 1 mm from its stated
length. Write a specification for the bar’s length x in metres: (a) by using a double inequality;
(b) with the aid of an absolute-value sign.
R E V I E W P R O B L E M S F O R C H A P T E R 1
1. (a) What is three times the difference between 50 and x?
(b) What is the quotient between x and the sum of y and 100?
(c) If the price of an item is a including 20% VAT (value added tax), what is the price before
VAT?
(d) A person buys x1, x2, and x3 units of three goods whose prices per unit are respectively p1,
p2, and p3. What is the total expenditure?
(e) A rental car costs F dollars per day in fixed charges and b dollars per kilometre. How much
must a customer pay to drive x kilometres in 1 day?
(f) A company has fixed costs of F dollars per year and variable costs of c dollars per unit
produced. Find an expression for the total cost per unit (total average cost) incurred by the
company if it produces x units in one year.
(g) A person has an annual salary of $L and then receives a raise of p% followed by a further
increase of q%. What is the person’s new yearly salary?
2. Express as single real numbers in decimal notation:
(a) 53 (b) 10-3 (c) 1
3-3 (d)
-1
10-3
(e) 3-233 (f) (3-2)-3 (g) – ‘5 3(0 (h) ‘-1 2(-3
3. Which of the following expressions are defined, and what are their values?
(a) (0 + 2)0 (b) 0-2 (c) (10)0
(0 + 1)0 (d)
(0 + 1)0
(0 + 2)0
4. Simplify:
(a) (232-5)3 (b) ’23(-1 – ’43(-1 (c) (3-2 – 5-1)-1 (d) (1.12)-3(1.12)3
⊂SM ⊃5. Simplify:
(a) (2x)4 (b) (2-1 – 4-1)-1 (c) 24x3y2z3
4x2yz2
(d) +-(-ab3)-3(a6b6)2,3 (e) a5 · a3 · a-2
a-3 · a6 (f) 7’x2 (3 · x8-2 8-3
6. Compute: (a) 12% of 300 (b) 5% of 2000 (c) 6.5% of 1500
R E V I E W P R O B L E M S F O R C H A P T E R 1 33
7. Give economic interpretations to each of the following expressions and then use a calculator to
find the approximate values:
(a) 100 · (1.01)8 (b) 50 000 · (1.15)10 (c) 6000 · (1.03)-8
8. (a) $100 000 is deposited into an account earning 8% interest per year. What is the amount
after 10 years?
(b) If the interest rate is 8% each year, how much money should you have deposited in a bank
6 years ago to have $25 000 today?
⊂SM ⊃9. Expand and simplify:
(a) a(a-1) (b) (x – 3)(x + 7) (c) -√3 %√3 – √6 & (d) %1-√2 &2
(e) (x – 1)3 (f) (1 – b2)(1 + b2) (g) (1 + x + x2 + x3)(1 – x) (h) (1 + x)4
10. Complete the following:
(a) x-1y-1 = 3 implies x3y3 = · · · (b) x7 = 2 implies (x-3)6(x2)2 = · · ·
(c) ‘ xy z (-2 = 3 implies ‘ xy z (6 = · · · (d) a-1b-1c-1 = 1/4 implies (abc)4 = · · ·
11. Factor the expressions
(a) 25x – 5 (b) 3×2 – x3y (c) 50 – x2 (d) a3 – 4a2b + 4ab2
⊂SM ⊃12. Factor the expressions
(a) 5(x + 2y) + a(x + 2y) (b) (a + b)c – d(a + b) (c) ax + ay + 2x + 2y
(d) 2×2 – 5yz + 10xz – xy (e) p2 – q2 + p – q (f) u3 + v3 – u2v – v2u
13. Compute the following without using a calculator:
(a) 161/4 (b) 243-1/5 (c) 51/7 · 56/7 (d) (48)-3/16
(e) 641/3 + √3 125 (f) (-8/27)2/3 (g) (-1/8)-2/3 + (1/27)-2/3 (h) 1000 √3 5–23/3
14. Solve the following equations for x:
(a) 22x = 8 (b) 33x+1 = 1/81 (c) 10×2-2x+2 = 100
15. Find the unknown x in each of the following equations:
(a) 255 · 25x = 253 (b) 3x – 3x-2 = 24 (c) 3x · 3x-1 = 81
(d) 35 + 35 + 35 = 3x (e) 4-6 + 4-6 + 4-6 + 4-6 = 4x (f) 226 – 223
226 + 223 =
x 9
⊂SM ⊃16. Simplify: (a) s
2s – 1

s
2s + 1
(b)
x
3 – x

1 – x
x + 3

24
x2 – 9 (c)
1
x2y –
1
xy2
1 2x

1 2y
34 C H A P T E R 1 / I N T R O D U C T O R Y T O P I C S I : A L G E B R A
⊂SM ⊃17. Reduce the following fractions:
(a)
25a3b2
125ab
(b)
x2 – y2
x + y
(c)
4a2 – 12ab + 9b2
4a2 – 9b2 (d)
4x – x3
4 – 4x + x2
18. Solve the following inequalities:
(a) 2(x – 4) < 5 (b)
1 3
(y – 3) + 4 ≥ 2 (c) 8 – 0.2x ≤
4 – 0.1x
0.5
(d)
x – 1
-3
>
-3x + 8
-5
(e) |5 – 3x| ≤ 8 (f) |x2 – 4| ≤ 2
19. Using a mobile phone costs $30 per month, and an additional $0.16 per minute of use.
(a) What is the cost for one month if the phone is used for a total of x minutes?
(b) What are the smallest and largest numbers of hours you can use the phone in a month if the
monthly telephone bill is to be between $102 and $126?
20. If a rope could be wrapped around the Earth’s surface at the equator, it would be approximately
circular and about 40 million metres long. Suppose we wanted to extend the rope to make it 1
metre above the equator at every point. How many more metres of rope would be needed? (The
circumference of a circle with radius r is 2πr.)
21. (a) Prove that a + a · p
100

)a + a100 · p * · p
100
= a71 – )100 p *28.
(b) An item initially costs $2000 and then its price is increased by 5%. Afterwards the price is
lowered by 5%. What is the final price?
(c) An item initially costs a dollars and then its price is increased by p%. Afterwards the (new)
price is lowered by p%. What is the final price of the item? (After considering this problem,
look at the expression in part (a).)
(d) What is the result if one first lowers a price by p% and then increases it by p%?
22. (a) If a > b, is it necessarily true that a2 > b2?
(b) Show that if a + b > 0, then a > b implies a2 > b2.
23. (a) If a > b, use numerical examples to check whether 1/a > 1/b, or 1/a < 1/b.
(b) Prove that if a > b and ab > 0, then 1/b > 1/a.
24. Prove that (i) |ab| = |a| · |b| and (ii) |a + b| ≤ |a| + |b|, for all real numbers a and b. (The
inequality in (ii) is called the triangle inequality.)
⊂SM ⊃25. Consider an equilateral triangle, and let P be an arbitrary point within the triangle. Let h1, h2,
and h3 be the shortest distances from P to each of the three sides. Show that the sum h1+h2+h3
is independent of where point P is placed in the triangle. (Hint: Compute the area of the triangle
as the sum of three triangles.)

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