# Topic 2 - Number and Algebra

##### Topic 2 - Number and Algebra

2.1 Organization of Numbers

###### 2.2 Numbers in Calculations

2.3 Standard Form

2.4 International Units of Measure

2.7 Simultaneous Equations

2.8 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.Section 2.1

Express any decimal as a rational number in the form , where a and b are integersSection 2.1

Arrange numbers in any form in increasing or decreasing orderSection 2.1

Round any answer to a requested number of significant figuresSection 2.2

Round any answer to a requested number of decimal placesSection 2.2

Find the percent error between an exact answer and an approximate answerSection 2.2

Write any number in scientific notationSection 2.3

Perform operations with numbers in scientific notationSection 2.3

Perform operations involving metric units and conversions [i.e. m2 cm2]Section 2.4

Perform operations involving time units and conversions [i.e. hours seconds]Section 2.4

Write a system of equations that represents a real-life situationSection 2.6

Solve a system of equations that represents a real-life situationSection 2.6

Identify the vertex, axis of symmetry, x- and y-intercepts of a quadratic functionSection 2.7

Solve quadratic equationsSection 2.7

Factor quadratic expressionsSection 2.7

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]

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Review Problems for Chapter Two: Number and Algebra

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2.1 Organization of Numbers

1.Consider the numbers 5, 0.5, , 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.

- Some integers are whole numbers.
- If a number is irrational, then it is a real number.
- All rational numbers are integers.

2.Which of the numbers in this set are also:

*Remember, numbers may be used more than once!*

- Natural numbers

- Whole numbers

- Integers

- Rational numbers

- Irrational numbers

- Real numbers

3.Show that 0.75 is a rational number

4.Explain why is an irrational number

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2.2 Numbers in Calculations

1.How many significant figures does each number have?

To what place value has the number been rounded?

a)43.5b)5673.7c)1200

d)4.001e)0.00452f)0.00340

g)784000h)0.450i)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]

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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.

3.Given the equation ,

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 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) 0.00120g) 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) 3.0063g) 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

(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.

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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 and , calculate the value of . Give your answer in the form where and .

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 where and .

4.Let and.

Find

a)

b)

giving your answers in the form where and .

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).

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.

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2.4 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 .

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 (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.

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2.7 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 thelinear relationship c = nr + s, where s is the security deposit and r is the amount of rent perweek.

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 2.7Practice**

1.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.

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.

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

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2.8 SolvingQuadratic 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.

SolutionsFactoring

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2.8 FactoringQuadratic Equations

Show all your work in your notebooks.

Do NOT try to cram your answer into the space below…

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2.8 Factoring Quadratic Equations

Special factoring pattern: The Difference of Two Squares

Multiply and simplify

Multiply and simplify

Multiply and simplify

Multiply and simplify

What pattern do you notice?

So, what is the factoring of…

1.a)Factorize the expression .

b)Factorize the expression .

c)Using your answer to part b) or otherwise, solve the equation

.

2.a)Find the solution of the equation

b)The equation has solutions and .

Find the value of a.

3.a)Factorize the expression .

b)Hence, or otherwise, solve the equation .