Counting in Different Number Bases

A number base is a system of counting in which certain units make up a “bundle”. The "bundle" will form the next place value and the placing of the digits shows the value.

In number bases, counting is done in base ten. Base ten is sometimes called DECIMAL or DENARY.

Also it should be noted that base two is otherwise called BINARY.

Students should also note that when solving problems under number bases, the base in use should always be written in words and not figures.

For example; 233_eight, 87_six, ….

The placing of the digits shows the place value. For example; 6578_ten stands for 6 thousands, 5 hundreds, 7 tens, 8 units.

= 6 x 1000 + 5 x 100 + 7 x 10 + 8 x 1

= 6 x 10³ + 5 x 10² + 7 x 10¹ + 8 x 10⁰

Also 246_seven = 2 x forty nines, 4 sevens, 6 units.

= 2 x 49 + 4 x 7 + 6 x 1

= 2 x 7² + 4 x 7¹ + 6 x 7⁰

Please note that number base is calculated in its own power, e.g. base ten in powers of ten, base three in powers of three and so on.

Examples;

1. 476_eight

= 4 sixty four 7 eights, 6 units

= 4 x 64 + 7 x 8 + 6 x 1 
= 4 x 8² + 7 x 8¹ + 6 x 8⁰

2. 24013_five 

= 2 six hundred and twenty five, 4 one hundred and twenty five, 0 twenty five, 1 fives, 3 units

= 2 x 5⁴ + 4 x 5³ + 0 x 5² + 1 x 5¹ + 3 x 5⁰

Bearing and Distances

Bearing can be defined as the clockwise angular movement between two distant places.

Rules for Solving Bearing and Distances

1. Taking reading in bearing starts from the North Pole in clockwise direction and ends also at the North pole. I.e. North – North Pole reading.

2. All angles formed while taking reading in bearing is equal to 360 degrees.

3. All questions in bearing leads into the formation of a triangle.

Examples;

1. A boat sails 6km from a port X on a bearing of 065⁰ and thereafter 13km on a bearing of 136⁰. What is the distance and bearing of the boat from X?

Solution

Step 1

You will need to represent the question on a triangle.

You will actually need a ruler and pencil for drawing along with a good eraser in case you make mistake.

Take your pencil and draw a point X, take ruler and rule a straight line to another point Y and rule another straight line to point Z and finally close it up at point X again. See below;

Step 2

Use Cosine Rule to solve for ykm;

From Cosine Rule;

b² = a² + c² – 2ac Cos B

Step 3

Replace a, b, c and B with x, y, z and Y respectively;

y² = x² + z² – 2xz Cos Y

y² = 13² + 6² - 2 (13) (6) Cos 109⁰

y² = 169 + 36 – 156 (- 0.3256)

Note; ( - × - = + )

y² = 205 + 156 x 0.3256

y² = 205 + 50.7886

y² = 255.7886

Step4

Take square root of both sides;

Step5

To find the bearing of the boat from X, we will apply Sine Rule;

From Sine Rule;

Step 6

Cross Multiply;

Step 7

Divide both sides by 16km;

Step 8

The bearing of boat from the port will be 065⁰ + θ

Which is 065⁰ + 50⁰ = 115⁰ (answer).

2. An aircraft takes off from an airstrip at an average speed of 20km/hr on a bearing of 052⁰ for 3 hours. It then changes course and flies on a bearing of 028⁰ at an average speed of 3okm/hr for another 1¹/₂ hours. Find;

a. its distance from the starting point,

b. the bearing of the aircraft from the airstrip.

Solution

Step 1

Step 2

Use Cosine rule to solve for side b;

From Cosine Rule;

b² = a² + c² – 2ac Cos B

b² = 45² + 60² – 2 (45) (60) Cos 156⁰

b² = 2025 + 3600 – 5400 (- 0.9135)

b² = 5625 + 5400 x 0.9135

b² = 5625 + 4932.9

b² = 10557.9

b = 103km

Step 3

To find the bearing of the aircraft from the airstrip, you must first find θ;

Find θ using Sine Rule below;

From Sine Rule;

Step 4

Cross Multiply;

103km (Sin θ) = 45km x Sin 156⁰

Step 5

Divide both sides by 103km;

Note; (103km cancels itself and 45 x 0.4067 gives 18.3015)

Sin θ = 0.1776

θ = Sin^-1 0.1776 (Note; when taking Sin to the RHS, it becomes inverse)

θ = 10.2299⁰

To find the bearing will be 052⁰ – θ

Bearing = 052⁰ – 10.2299⁰

Bearing = 41.77⁰

Practice these questions below;

1. A ship leaves port and travels 21km on a bearing of 032⁰ and then 45km on a bearing of 287⁰.

(a) Calculate its distance from the port.

(b) Calculate the bearing of the port from the ship.

2. A surveyor leaves her base camp and drives 42km on a bearing of 032⁰, she then drives 28km on a bearing of 154⁰. How far is she then from her base camp and what is her bearing from it?

3. Two ships leave port at the same time and travels at 5km/hr on a bearing of 046⁰. The other travels at 9km/hr on a bearing of 127⁰. How far apart are the ships after 2 hours?

4. A triangular field has two sides 50m and 60m long, and the angle between them is 096⁰. How long is the third side?

Sine Rule

Only one of these formulas should be used at a time.

Note;

A, B and C represent angles while a, b and c represent sides. This indicates that the capital letters must be considered as angles while the small letters as sides.

Examples;

1. In a triangle ABC, A = 54⁰, B = 67⁰, a = 13.9m. Find b and c.

Solution

Step 1

From Sine Rule;

Substitution;

Step 2

Cross Multiply;

b X Sin 54⁰ = 13.9m X Sin 67⁰

b X 0.8090 = 13.9m X 0.9205

0.8090b = 12.79495

Step 3

Divide both sides by 0.8090;

Step 4

To find c, we will need to find angle C;

Angle C = 180⁰ – (54⁰ + 67⁰)

Angle C = 180⁰ - 121⁰

C = 59⁰.

Step 5

Substitution;

Step 6

Cross Multiply;

0.9205 X c = 0.8572 X 15.8m

0.9205c = 13.54376

Step 7

Divide both sides by 0.9205;

2. In a triangle PQR, R = 53⁰, q = 3.6m, r = 4.3m. Find Q.

Solution

Step 1

From Sine Rule;

Step 2

Cross Multiply;

4.3m X Sin Q = 3.6m X 0.7986

4.3mSinQ = 2.8751

Step 3

Divide both sides by 4.3m;

Sin Q = 0.6686

Practice these questions below;

1. In a triangle ABC, if A = 38⁰, B = 27⁰, b = 17m. Find a and c.

2. In a triangle XYZ, if X = 69⁰, y = 9cm, x = 19cm. Find Y and z.

3. In a triangle PQR, if Q = 36⁰, P = 88⁰, p = 9.5cm. Find q and r.

4. In the triangle EFG below, find G;

5. In a triangle ABC, if A = 39⁰, a = 8.2m, b = 5.6m. Find B and c.