Mathematics for Junior High School

By Marsigit

Proportion

Basic Competency :
To Use Proportion to solve the problems.
Have you ever seen a house, building, monument, or sport stadium?. Of course you have. How to build them? are they just built anyway?. Of course, they did not. Before to build them the worker develop the plan first. For example, the needed materials, the cost, and design a scale model. So what is the definition of a scale model? A scale model is a decried building with a certain proportion scale. By making the scale model, you can see the detailed of the buildings anyway.

What will you learn in this chapter?
• Scaled Picture
• Direct Proportion
• Inverse Proportions
A. Scaled Pictures
Scaled picture is a picture with a certain proportion or scale. Scaled picture
is made to represent the reality with a ratio (scale). Some example of scaled pictures are maps, design of a building, and cars model.

1. Definition of Scale
Scale is the ratio between the size of the picture and the actual size.

Scale is usually written as a ratio. For example, the map of Indonesia has a scale of 1 : 1.000.000. It means that 1 cm on the map represents 1.000.000 cm = 10 km on the actual size.

Example
The distance between Yogyakarta and Cilacap is 150 km. Find the distance between the cities on the map with the scale of 1 : 5.000.000.

Solution:
The real distance between Yogyakarta and Cilacap is 150 km = 15.000.000 cm.
The scale of the map is 1 : 5.000.000 =
As we know that,
Thus,


Distance on the map =
Hence, the distance between Yogyakarta and Cilacap on the map is 3 cm.


Exercise
1. Explain the meaning of the scale of 1 : 1.000.000 ?
2. Find the scale on a map if 5 cm on the map represent 30 km.
3. Let, the distance between Denpasar and Singaraja on the map is 3 cm. Find the real distance if the scale of the map is 1 : 6.000.000.
4. The length of a hedge is 16 m. Find the length of the hedge on the sketch with the scale of 1 : 400.
5. The distance between Makassar and Pare-pare is 155 km. If the distance between the cities on the map is 10 cm, find the scale of the map.

2. Scale Factor
Scale factor is the ratio between the size of the model and the actual size. Scale factor is used to determine the reduction or the enlargement of a scaled model or design.
If you enlarge a scaled model, you will get the scale factor k > 1. If you reduce a scaled model, you will get the scale factor 0 < k < 1.

Example
The length and the width of the handicraft model is 10 cm and 7,5 cm. The real length of the handicraft is 40 cm. Find the scale factor of the handicraft model, and then find the real width of the handicraft.

Solution:
The length of the handicraft model is 10 cm, and the width is 7.5 cm.
The real length is 40 cm.
• Scale factor,

Therefore, the scale factor of the handicraft model is .

•

Therefore, the real width of the handicraft is 30 cm.

Exercise
1. Complete the following ratio with the right number.
a. b.

2. The length and the width of the scaled picture of a football yard are 11 cm and 7 cm. Find the scale factor and the real width of the football yard if the real length is 110 m.
[Gambar lapangan sepak bola]

3. The size of photo is 4 x 6 cm. It will be enlarged by the scale factor of 1.5. Find the height of the photo after the enlargement.

[Gambar foto sawah]

4. The ratio of the length and the width of a photograph is 2 : 1. Find the length of the photograph if it’s width is reduced to be 4 cm.

5. The length and the width of a field are 100 m and 30 m. Rita wants to sketch the field. Determine the length of the sketch if the width of the sketch is 3 cm.
B. Direct Proportion

1. The Concept of Direct Proportion

The illustration below shows the example of a direct proportion. At the unit measurement of a gas station it shows the amount of the gas (liter) and the money should be paid by the buyer.

Amount of the gas (litre) Price (Rp.)
1 6.000
2 12.000
10 60.000
23 138.000

It shows that the more the gas, the more the price should be paid. This proportion is called a direct proportion. Direct proportion is indicated by a series of number. See the table below.

Number Direct Proportion
2, 7, 8, 28

18, 20, 27, 30

8, 4, 2, 1


Example
Determine the direct proportions of the following numbers.
1. 3, 7, 9, and 21 2. 52, 26, 12, and 6 3. 24, 16, 15, and 10

Solution:
1. 2. 3.

Exercise
Determine the direct proportions of the following numbers.
1. 450, 210, 60 and 28
2. 24, 42, 248 and 434
3. 320, 240, 48 and 36
4. 9, 26, 63, and 182
5. 6, 7, 24 and 28

2. Calculating Direct Proportion
There are two ways to calculate a direct proportion i.e. by unit method and by proportion.
a. Unit Method
To calculate a direct proportion by unit method, first the letter you choose must represent the unit of measure common to each part.

Example
Mom needs 10 eggs to make two cakes. How many eggs are needed for 7 cakes ?

Solution
First, find out how many eggs needed for one cake.
Two cakes need 10 eggs, therefore one cake needs,
eggs.
So, mom needs 7 x 5 eggs = 35 eggs to make 7 cakes.

Exercise
1. A motorcycle needs 4 liter of gas for 144 km. How much gas is needed for 54 km?
2. If the price of one dozen of spoons is Rp 18.000,00, how much it costs for 27 spoons?

b. Proportion

You can determine a direct proportion by proportion. See the example below.

Example
The ratio between the number of girls and boys in a class is 5 : 9. The total number of students in the class is 28 students. Determine the number of the girls and the boys in the class ?
Solution
The ratio between the girls and the boys in the class is 5 : 9.
The total number of students in the class is 28 students.
So the total boys in the class is,

students
By the same way you’ll find the total girls in the class, that is,
students
Check the answer by adding the total girls and the total boys in the class.
You find,
10 + 18 = 28 students.

Exercise
1. The perimeter of a garden is 30 meter. The ratio between the length and the width of the garden is 3 : 2. Determine the length and the width of the garden.
2. In the shelve there are Mathematic books, Indonesian Language books, and Physics books with the ratio of 4 : 2 : 3. Determine the numbers of Physics books and Mathematics books if the Indonesian Language books is 6 books.
3. The sum of two numbers is 72. The ratio between both two numbers is 4 : 5. Determine each of the numbers.
4. The total amount of Mr. Tanto’s money in two Banks are Rp 20.000.000,00. The ratio between the sum of money in each Bank is 3 : 2. How much Mr. Tanto’s has the money in each Bank?
5. Mother has spent Rp 60.000,00 to buy meat, rice, and vegetables by the ratio of 10 : 9 : 5. Determine the price of each of the goods.

2. The Graph of Direct Proportion

Consider the gasoline prices below. Can you draw a graph of the data from the table? To draw a graph, you need to make two axis, that is horizontal axis and vertical axis.
Indicate the horizontal line as the amount of gas and the vertical as the prices of the gasoline.


Amount of the gas (litre) Price (Rp.)
1 6.000
2 12.000
10 60.000
23 138.000

The graph is shows as follow.






Exercise
The table below shows the relation between the time and the distance of the journey by a car.

Time (hrs) 1 2 3 4 5
Distance (km) 60 120 … … …

1. Copy the table on your work-sheet and fill in the blank with appropriate numbers.
2. Sketch the graph representing the data of the table.
3. How long the distance traveling by the car for 11 hrs ?
4. How long time the car needs to travel for 780 km ?
5. How long time the car needs to travel for 20 km ?

C. Inverse Proportions

1. The Concept of Inverse Proportion

You can find some inverse proportions in daily life. For example, the ratio between the numbers of days needed to construct a house and the number of workers is the inverse proportion. The more the workers the sooner the house will be finished.

2. Calculating the Inverse Proportions
You can calculate the inverse proportion in two ways, based on the result of multiplication and proportion

a. Based on multiplication
An inverse proportion can be calculated in term of the result of multiplication. Look at the following example.

Example
A house can be built in two months by 10 workers. How long time it can be done by six workers?

Solution:
Make a table to help you to solve the problem.

Number of Workers
Time
10 2 months (60 days)
6

The multiplication results of each raw should be the similar, so you have an inverse proportion as follows.



It is concluded that if there are six workers, it can be done in 100 days.

Exercise
1. Complete the following proportion with the correct number!
a. b. c.
2. A bridge can be done by 18 workers in 20 days. If there are 12 workers, how long time it can be done?
b. Based on Proportion
An inverse proportion can be calculated by proportion method. Look at the
following example.
Example
Let, a box of candy has been distributed to 20 children, in such a way that
each child gets 12 candies. If there are 30 children, how many candies each child will
get?
Solution
Look at the following table!

Number of children
candies for each child
20 12
30
Based on the proportion, you can determine the amount of candies for each child.
. Hence, if there are 30 children. every child will get 8 candies.
Exercise
1. A basket of carrots can be finished by 10 rabbits in 6 days. How long time for 15 rabbits to eat them?
2. Father needs 20 minutes to go to the office by car in 60 km/hrs. Today he just takes 15 minutes to go to his office. Determine the speed of the car used to go to his office today!
3. The Graph of the Inverse Proportion

Do you know how to measure the speed? The speed can be measured by the following formula , with = velocity, distance, and = time. For example, you will go by bus to the city which the distance of 180 km from your house. The time you take for the trip depends on the speed of the bus you ride, as it shown in the following table.

[Tabel t –v]


Based on the table above, you can make a velocity graph towards time as follows.
[Grafik kecepatan-waktu]


Exercise
Anton is going to school by bike. He measures the speed of his bike and the time he needs to arrived at school on time. Look at the following result of Anton’s measurement!

(minute)
20 40 50
(minute)
15 7,5 6

1. Sketch a graph based on the table above!
2. How far the distance of Anton’s house to school? (use the formula )
3. If Anton want to arrive at school in 30 minutes, what does the speed of Anton’s bike?