LESSON NOTES ON PROPORTIONS

   

         

SUBJECT: Number Theory.

TOPIC:  Proportions.

LESSON: Applying proportions to solve mathematical problems.

PREVIOUS KNOWLEDGE: Students can solve questions involving ratios.

LESSON OBJECTIVES: By the end of the lesson, students should be able to do the following: Define a proportion, differentiate between direct and indirect proportions, Solve exercises on direct and indirect proportions, and apply know of solving problems here to solve other problems in life.

TEACHING AIDS: Chalk, blackboard,

TEACHING METHODS:  Demonstration, questioning, group discussion, assignment.

CLASS LEVEL: Form Two

DURATION: 45 minutes

DATE:

INTRODUCTION

If m: 2=8:m. Calculate the positive value of m.

                       Solution

m: 2 = 8:m

m/2=8/m

m×m=8×2

m²=16(take the square root of both sides)

m=±4

m=4.

PRESENTATION

Definition: When two or more ratios are equal, the equation that results is called a proportion. For example if 2:3=4:9, the 2/3=4/9 is called a proportion. The majority of problems where proportion is involved are usually solved by finding the value of a unit quantity.

 Two types of proportions will be treated here namely direct and indirect proportions. In Direct proportion increase in one quantity leads to an increase in the other and vice versa. In indirect proportion, an increase in one quantity leads to a decrease in the other and vice versa.

EXAMPLE 1: A man is paid 25000FRS CFA for 10 days of work. Calculate his pay for 3days.

                                          Solution

                                 10days receives 25000 FRS

                                  3days receives   x

     Considering ratios; 3days/10days=x/25000FRS.

                                                  3/10=x/25000FRS (multiply through by 25000FRS)

                                                 (3/10)25000FRS=x

                                                      7500FRS=x

Therefore, the man’s pay in 5 days is 7500FRS.

EXAMPLE 2: Three students scored marks in a Geography test in the ratio 2:3:4. The least score by one of them is 8. Calculate the marks scored by the others.

                                      Solution

Ratio; 2:3:4

Sum of ratio: 2+3+4=9 ; sum of marks be x

Least ratio=2 ; least mark=8

                           9 ------x

                           2 ------8

Considering ratios

                         9:2=x:8

                         9/2=x/8

                         X/8=9/2

                           x=8(9/2)

                           x=36.

Total score is 36.

Marks scored by the other two are as follows:

>by second student; (3/9)36=12

>by third student; (4/9)36=16.

EXAMPLE 3: The three angles of a triangle are in the ratio 1:2:3. What are the angles of the triangle? Name the triangle.

                                            Solution

Let the angles of the triangle be <X, <Y, <Z.

Ratio, 1:2:3

Sum of ratio; 1+2+3=6. Sum of angles in any triangle, <X+<Y+<Z=180⁰

<X= (1/6)180⁰=30⁰

<Y= (2/6)180⁰=60⁰

<Z= (3/6)180⁰=90⁰

A right-angled triangle.

EXAMPLE 4: If x:3=12:x, calculate the positive value of x

                             SOLUTION

       x: 3=12: x (defines a proportion)

     x/3=12/x (multiply through by x)

     x²/3=12.   (Multiply through by 3)

       X²=36      (take the square root of both sides)

       x=6.

So far, we have been solving problems using direct proportions. The following example will be on indirect proportions.

EXAMPLE 5: Three men construct a fence round a building in 10 days.  How long will it take five men to do the same work?

                                       Solution

 3 men work in 10days

5 men work in x.

3 men---10days

5 men ----x

The proportion instead will be 3:5=x:10 not 3:5=10:x since it is indirect proportion.

3:5=x: 10

3/5=x/10

10days (3/5) =x

x=10days (3/5)

x=6 days

   

    EVALUATION

1)      A man earns $140 in 5 days. How much does he earn in 3 days?

2)      When $143 is divided among three children in the ratio of 2:4:5.What is the difference between the largest and the smallest share?

3)      A car uses 10 liters of petrol in 75 km. How far will it go on 8 liters of petrol?

4)      It takes 6 men to dig a hole 3 feet deep. How long will it take 10 men to dig that same hole?

     SUMMARY

>Equal ratios are called a proportion. Example 2/3=x/5

>Proportions are generally applied to solve mathematics, business, science, engineering, banking etc.

    CONCLUSION

1)      A man and his wife share a sum of money in the ratio 3:2.If the sum is doubled, in what ratio must they share so the man still receives the same amount?

2)      Nine milk bottles contain 4.5 liters of between them. How much will 5 liters contain?

3)      Find the cost of 1km of pipe at 7pence for every 4cm.

4)      A wheel turns through 90 revolutions per minute. How many degrees does it turn through in 1 second.

Every thing you need to know about math here. A great book for teachers and students