Topic 2 - Number and Algebra
2.1
2.2
2.3
2.4
2.7
2.8
Organization of Numbers
Numbers in Calculations
Standard Form
International Units of Measure
Simultaneous Equations
Quadratic Equations
IB Math Studies
Review Sheet for Topic Two: Number and Algebra
You should be able to do the following things on the test:
Classify numbers as real, irrational, rational, integer, prime, and/or natural.
a
Express any decimal as a rational number in the form
, where a and b are integers
b
Arrange numbers in any form in increasing or decreasing order
Round any answer to a requested number of significant figures
Round any answer to a requested number of decimal places
Find the percent error between an exact answer and an approximate answer
Write any number in scientific notation
Perform operations with numbers in scientific notation
Perform operations involving metric units and conversions [i.e. m2  cm2]
Perform operations involving time units and conversions [i.e. hours  seconds]
Write a system of equations that represents a real-life situation
Solve a system of equations that represents a real-life situation
Identify the vertex, axis of symmetry, x- and y-intercepts of a quadratic function
Solve quadratic equations
Factor quadratic expressions
From previous knowledge, you should also know how to:
Solve any linear equation for x
Use geometric formulas for perimeter, area, and volume
Use the formula d=rt [distance = rate x time]
Section 2.1
Section 2.1
Section 2.1
Section 2.2
Section 2.2
Section 2.2
Section 2.3
Section 2.3
Section 2.4
Section 2.4
Section 2.6
Section 2.6
Section 2.7
Section 2.7
Section 2.7
IB Math Studies
Review Problems for Chapter Two: Number and Algebra
IB Math Studies
2.1 Organization of Numbers
5 , and -5. Complete the table below, showing
, these numbers belong to.
1.
Consider the numbers 5, 0.5,
which of the number sets,
2.
Given Z the set of integers, Q the set of rational numbers, and R the set of
real numbers:
a)
Write down an element that belongs to R ∩ Z.
b)
Write down an element that belongs to Q ∩ Z’.
c)
Write down an element that belongs to Q’.
d)
Use a Venn diagram to represent the sets Z, Q and R.
2.1 Practice
1. True or false? Explain your answer. If it is false, include a counter-example.
a. Some integers are whole numbers.
b. If a number is irrational, then it is a real number.
c. All rational numbers are integers.
2. Which of the numbers in this set  , 27, 2 , 100,  6, 2.57, 0.59, 20 , 0, 2  are also:
3
4


Remember, numbers may be used more than once!
a. Natural numbers
b. Whole numbers
c. Integers
d. Rational numbers
e. Irrational numbers
f. Real numbers
3. Show that 0.75 is a rational number
4. Explain why √7 is an irrational number
IB Math Studies
2.2 Numbers in Calculations
1.
2.
How many significant figures does each number have?
To what place value has the number been rounded?
a)
43.5
b)
5673.7
c)
1200
d)
4.001
e)
0.00452
f)
0.00340
g)
784000
h)
0.450
i)
4503450
Round the following values to the requested number of significant figures or
place value:
a)
2.526 [2 sf]
e)
0.4523 [2 sf]
b)
2.526 [hundredths]
f)
3.684 [tenths]
c)
24650 [1 sf]
g)
5.6720 [hundredths]
d)
45627 [3 sf]
h)
0.04537 [3 sf]
IB Math Studies
2.2 Numbers in Calculations
When we round our answers, we are changing an exact answer into an approximate
error. This introduces error into our calculation. IB expects you to find how much
error there is.
To find percent error, find the difference between the exact answer and the
approximate answer, and then divide by the exact answer. Multiply by 100 to
change to percent.
𝑃𝑒𝑟𝑐𝑒𝑛𝑡 𝑒𝑟𝑟𝑜𝑟 =
3.
4.
𝑣𝐴 − 𝑣𝐸
× 100
𝑣𝐸
Given the equation p  r 2  2qr ,
a)
Calculate the exact value of p then q = 3.6 and r = 24.
b)
Write your answer correct to two significant figures.
c)
Find the percentage error between a) and b).
For each figure below:
a)
b)
c)
Use your formula booklet to find the exact area.
Round your answer to the nearest tenth.
Calculate the percentage error between your two answers.
2.2 Practice
1. How many significant figures are in each of the following numbers?
a) 5.40
b) 210
c) 801.5
d) 1,000
e) 101.0100
f) 0.00120
g) 0.0102
h) 2,370.0
i) 890
j) 91010
k) 780.
l) 3400
2. Round each of the following to 3 significant figures
a) 5357
b) 64.845
c) 578900
d) 508.9
e) 790.1
f) 3.0063
g) 0.03407
h) 128.53
i) 435691
j) 707.5
k) 0.0003350
l) 2,300.2
3.
4.
Calculate 3.7 × 16.22 – 500, writing your answer
(a)
correct to two decimal places;
(b)
correct to three significant figures
(a)
Calculate exactly
(3  2.1) 3
.
7  1 .2
(b) Write the answer to part (a) correct to 2 significant figures.
(c) Calculate the percentage error when the answer to part (a) is
written correct to 2 significant figures.
IB Math Studies
2.3 Standard Form
1.
A box contains 1.81 x 1024 atoms. One third of them are carbon, the other
two thirds are oxygen. How many carbon atoms are in the box? How many
oxygen atoms?
2.
a)
Given x = 2.6 × 104 and y = 5.0 × 10−8, calculate the value of w = x × y.
Give your answer in the form 𝑎 × 10𝑘 where 1 ≤ 𝑎 < 10 and 𝑘 ∈ ℤ.
b)
Which two of the following statements about the nature of x, y, and w
are incorrect.
3.
4.
The total weight of 256 identical pencils is 4.24 kg.
Calculate the weight of one pencil, in kg.
a)
Give your answer exactly.
b)
Give your answer correct to three significant figures.
c)
Write your answer to part b) in the form a  10 k where 1  a  10 and
k  .
Let x  6.4  107 and y  1.6  108 .
Find
a)
x
y
b)
y  2x
giving your answers in the form a  10 k where 1  a  10 and k   .
2.3 Practice
1.
(a)
A girl’s height is 1.623 m. Write her height to the nearest cm.
(b)
The time taken to fill a tank was 2 hours 43 minutes. Write this
time to the nearest 5 minutes.
(c)
The attendance at a show was 2591 people. How many people, to the
nearest 100, were at the show?
(d)
The mean distance of the Moon from the Earth is approximately
384 403 km. Write this distance in the form a × 10k where 1 ≤ a < 10
and k ∈
2.
.
Let m = 6.0 ×103 and n = 2.4 ×10–5.
Express each of the following in the form a ×10k, where 1 ≤ a < 10 and k ∈
3.
5.
(a)
mn;
(b)
m
.
n
The speed of sound in air is given as 300 ms–l.
(a)
How many metres does sound travel in air in one hour?
(b)
Express your answer to part (a)
(i)
correct to two significant figures;
(ii)
in the form a × 10k, where 1 ≤ a < 10 and k ∈
.
A problem has an exact answer of x = 0.1265.
(a)
Write down the exact value of x in the form a×10k where k is an
integer and 1 ≤ a < 10.
(b)
State the value of x given correct to two significant figures.
(c)
Calculate the percentage error if x is given correct to two
significant figures.
.
IB Math Studies
2.4 International Units of Measure
1.
2.
A field is 91.4 m long and 68.5 m wide.
a)
Calculate the area of the field in m2.
b)
Calculate the area of the field in cm2.
c)
Express your answer to b) in the form
𝑎 × 10𝑘 , where 1 ≤ 𝑎 < 10 𝑎𝑛𝑑 𝑘 ∈ ℤ.
The speed of sound in air is given as 300 m s-1.
a)
How many metres does sound travel in air in one hour?
b)
Express your answer to part a) correct to two significant figures.
3.
a)
Convert 0.001673 litres to millilitres (ml). Give your answer to the nearest ml.
The SI unit for energy is Joules. An object with mass m travelling at speed v has energy
given by
b)
1 2
mv (Joules).
2
Calculate the energy of a comet of mass 351223 kg travelling at speed 176.334
m/sec. Give your answer correct to six significant figures.
In the SI system of units, distance is measured in metres (m), mass in kilograms (kg), and
time in seconds (s). The momentum of an object is given by the mass of the object
multiplied by its speed.
c)
Write down the correct combination of SI units (using m, kg, s) for momentum.
IB Math Studies
2.7 Simultaneous Equations
1.
2.
At Jumbo’s Burger Bar, Jumbo burgers cost ₤J each and regular cokes cost ₤C each. Two
Jumbo burgers and three regular cokes cost ₤5.95.
a)
Write an equation to show this.
b)
If one Jumbo Burger costs ₤2.15, what is the cost of one regular coke?
The cost c, in Australian dollars (AUD), of renting a bungalow for n weeks is given by the
linear relationship c = nr + s, where s is the security deposit and r is the amount of rent per
week.
Ana rented the bungalow for 12 weeks and paid a total of 2925 AUD.
Raquel rented the same bungalow for 20 weeks and paid a total of 4525 AUD.
a)
Write two equations to represent this information.
b)
Find the value of
i) r, the rent per week
ii) s, the security deposit
More 2.7 Practice
1.
2.
Jacques can buy six CDs and three video cassettes for $163.17
or he can buy nine CDs and two video cassettes for $200.53.
(a) Express the above information using two equations relating the
price of CDs and the price of video cassettes.
(b)
Find the price of one video cassette.
(c)
If Jacques has $180 to spend, find the exact amount of change he
will receive if he buys nine CDs.
A store sells bread and milk. On Tuesday, 8 loaves of bread and 5 litres
of milk were sold for $21.40. On Thursday, 6 loaves of bread and 9 litres
of milk were sold for $23.40.
If b = the price of a loaf of bread and m = the price of one litre of milk,
Tuesday’s sales can be written as 8b + 5m = 21.40.
(a) Using simplest terms, write an equation in b and m for Thursday’s
sales.
(b)
Find b and m.
(c)
Draw a sketch, in the space provided, to show how the prices can be
found graphically.
5
4
m
3
2
1
0
0
1
2
b
3
4
3. Mal is shopping for a school trip. He buys 50 tins of beans and 20 packets
of cereal. The total cost is 260 Australian dollars (AUD).
(a) Write down an equation showing this information, taking b to be the
cost of one tin of beans and c to be the cost of one packet of cereal
in AUD.
Stephen thinks that Mal has not bought enough so he buys 12 more tins
of beans and 6 more packets of cereal. He pays 66 AUD.
(b) Write down another equation to represent this information.
(ii)
(c)
Find the cost of one tin of beans.
(d)
(i)
Sketch the graphs of these two equations.
Write down the coordinates of the point of intersection of the two graphs
IB Math Studies
2.8 Solving Quadratic Functions
Sketch each of the following quadratic functions on the graph paper. Then find
the vertex, axis of symmetry, x-intercepts [solutions], and y-intercept.
Solutions
f ( x)  x  6 x  8
2
f ( x)  x 2  10 x  25
f ( x)  x 2  5 x  6
Factoring
IB Math Studies
2.8 Factoring Quadratic Equations
Show all your work in your notebooks.
Do NOT try to cram your answer into the space below…
IB Math Studies
2.8 Factoring Quadratic Equations
Special factoring pattern: The Difference of Two Squares
Multiply and simplify (𝑥 + 2)(𝑥 − 2)
Multiply and simplify (𝑥 − 7)(𝑥 + 7)
Multiply and simplify (3𝑥 − 2)(3𝑥 + 2)
Multiply and simplify (△ −∎)(△ +∎)
What pattern do you notice?
So, what is the factoring of…
𝑥 2 − 25 =
49𝑥 2 − 100 =
1.
2.
3.
a)
Factorize the expression 𝑥 2 − 25.
b)
Factorize the expression 𝑥 2 − 3𝑥 − 4.
c)
Using your answer to part b) or otherwise, solve the equation
𝑥 2 − 3𝑥 − 4 = 0.
a)
Find the solution of the equation 𝑥 2 − 5𝑥 − 24 = 0
b)
The equation 𝑎𝑥 2 − 9𝑥 − 30 = 0 has solutions 𝑥 = 5 and 𝑥 = −2.
Find the value of a.
a)
Factorize the expression 2𝑥 2 − 3𝑥 − 5.
b)
Hence, or otherwise, solve the equation 2𝑥 2 − 3𝑥 = 5.