Topic 2 - Number and Algebra

2.1 Organization of Numbers2.2 Numbers in Calculations2.3 Standard Form2.4 International Units of Measure2.7 Simultaneous Equations2.8 Quadratic Equations

IB Math StudiesReview 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. Section 2.1

Express any decimal as a rational number in the form ab , where a and b are integersSection

2.1Arrange numbers in any form in increasing or decreasing order Section 2.1Round any answer to a requested number of significant figures Section 2.2Round any answer to a requested number of decimal places Section 2.2Find the percent error between an exact answer and an approximate answer Section 2.2Write any number in scientific notation Section 2.3Perform operations with numbers in scientific notation Section 2.3Perform operations involving metric units and conversions [i.e. m2 cm2] Section 2.4Perform operations involving time units and conversions [i.e. hours seconds] Section 2.4Write a system of equations that represents a real-life situation Section 2.6Solve a system of equations that represents a real-life situation Section 2.6Identify the vertex, axis of symmetry, x- and y-intercepts of a quadratic function Section 2.7Solve quadratic equations Section 2.7Factor quadratic expressions

Section 2.7

From previous knowledge, you should also know how to:

Solve any linear equation for xUse geometric formulas for perimeter, area, and volumeUse the formula d=rt [distance = rate x time]

IB Math StudiesReview Problems for Chapter Two: Number and Algebra

IB Math Studies2.1 Organization of Numbers

1. Consider the numbers 5, 0.5, √5 , and -5. Complete the table below, showing which of the number sets, , these numbers belong to.

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 are also: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 Studies2.2 Numbers in Calculations

1. 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

2. 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 Studies2.2 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=v A−vEvE

×100

3. Given the equation p=r2+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).

4. For each figure below:

a) Use your formula booklet to find the exact area.b) Round your answer to the nearest tenth.c) Calculate the percentage error between your two answers.

2.2 Practice1. 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 figuresa) 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. Calculate 3.7 × 16.22 – 500, writing your answer

(a) correct to two decimal places;

(b) correct to three significant figures

4. (a) Calculate exactly .

2.17)1.2(3 3

(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 Studies2.3 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. 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×10kwhere 1≤a<10 and k∈Ζ .

4. Let x=6 . 4×107 and y=1 . 6×108.

Find

a)xy

b) y−2x

giving your answers in the form a×10kwhere 1≤a<10 and k∈Ζ .

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 ∈ .

(a) mn;

(b) nm

.

3. 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 ∈ .

5. 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 Studies2.4 International Units of Measure

1. 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∧k∈Z.

2. 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 12mv2

(Joules).

b) 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 Studies2.7 Simultaneous Equations

1. 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?

2. 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. Jacques can buy six CDs and three video cassettes for $163.17or 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.

2. 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

3

2

1

00

1 2 3 4

m

b

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.

(c) Find the cost of one tin of beans.

(d) (i) Sketch the graphs of these two equations.

(ii) Write down the coordinates of the point of intersection of the two graphs

IB Math Studies2.8 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 Factoringf ( x )=x2+6 x+8

f ( x )=x2+10 x+25

f ( x )=x2+5 x−6

IB Math Studies2.8 Factoring Quadratic Equations

Show all your work in your notebooks. Do NOT try to cram your answer into the space below…

IB Math Studies2.8 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 (3 x−2 )(3 x+2)

Multiply and simplify (△−∎ )(△+∎)

What pattern do you notice?

So, what is the factoring of…

x2−25=¿

49 x2−100=¿

1. a) Factorize the expression x2−25.

b) Factorize the expression x2−3 x−4.

c) Using your answer to part b) or otherwise, solve the equation x2−3 x−4=0.

2. a) Find the solution of the equation x2−5 x−24=0

b) The equation a x2−9 x−30=0 has solutions x=5 and x=−2. Find the value of a.

3. a) Factorize the expression 2 x2−3x−5.

b) Hence, or otherwise, solve the equation 2 x2−3 x=5.