Multiplication of Fractions – The teacher’s main goal in learning mathematics is to help students understand mathematics and encourage them to use mathematics to solve problems in everyday life, and enjoy learning mathematics. Learning mathematics is the process of providing learning experiences to students through a series of planned activities, so that students gain competency in the mathematics material being studied.
Learning multiplication of ordinary fractions is a subject that is tricky, because teachers usually teach it directly without the help of the media, and are introduced without knowing the basic concepts. If students do not know the basic concepts, children will forget the material they are learning more quickly.
Learning is the most important activity in the entire educational process in schools. This means that the success of achieving educational goals depends a lot on the learning process taking place effectively. A teacher’s understanding of the meaning of learning will greatly influence the way the teacher teaches.
Gagne and Riggs say that learning is a system that aims to help the student learning process, which contains a series of events that are designed, arranged in such a way as to influence and support the internal student learning process. So, the core of learning is all the efforts made by the teacher so that the learning process occurs in students.
Based on this opinion, it can be concluded that learning is a process of interaction of students with educators and learning resources that is done intentionally, so that it allows students to learn to do or demonstrate certain behaviors as well.
Mathematics is a study material that has an abstract object and is built through a process of deductive reasoning, namely previous truths, so that the linkages between concepts in mathematics are very strong and clear. Mathematics functions to develop reasoning abilities through investigative, exploratory and experimental activities as a means of solving problems through patterns of thinking and mathematical models, as well as a means of communication as symbols, tables, graphs, diagrams, in explaining ideas.
According to Karso, learning mathematics in elementary school (SD) has the following characteristics.
1. Learning Mathematics Using the Spiral Method
The spiral approach in learning mathematics is an approach to learning a concept or a mathematical topic that always relates or relates to the previous topic. The previous topic can be a prerequisite for understanding and studying a mathematical topic.
The new topics studied are the deepening and expansion of the previous topics. Giving concepts begins with concrete objects, then the concept is taught again in a more abstract form of understanding using notation that is more commonly used in mathematics.
2. Gradual Mathematics Learning
Mathematics subject matter is taught in stages, starting from simple concepts to more difficult concepts. In addition, learning mathematics starts from concrete to semi-concrete and finally to abstract concepts. To make it easier for students to understand mathematical objects, concrete objects are used at the concrete stage, then images at the semi-concrete stage and finally symbols at the abstract stage.
According to Karso, the purpose of learning mathematics is to train and develop ways of thinking systematically, logically, critically, creatively and consistently. In addition, this learning is also expected to develop a persistent and confident attitude in solving problems.
The learning theory that underlies the inquiry method, according to Sanjaya (2011: 196) is constructivism learning theory. Constructivism is a theory that explains that humans build or construct their own knowledge through their interactions with objects, phenomena, experiences, and their environment.
Knowledge cannot be simply transferred from one person to another, but knowledge is built by the learner (Suparno, 1997: 28-29 ) . So, the knowledge acquired by students does not come from teachers who impart their knowledge to students, but students themselves who build their understanding through interaction with their environment.
Fractions in Mathematics
sinaumedia friends, have you ever heard the word “fraction”? By the way, do you understand the difference between fractions and integers? Well, this article will discuss further about fractions, especially the multiplication of fractions.
Before we give the multiplication formula for fractions, you must first remember about fractions. In simple terms, fractions are numbers that are not integers and are in the form a/b, both a and b are integers and the value of b is not equal to 0 (zero). Fractions consist of two components, namely the numerator ( a ) and the denominator ( b ).
There are three types of fractions. The first type of fraction is a pure fraction, which is a fraction where the numerator is smaller than the denominator. The second type of fraction is an impure fraction, which is a fraction where the numerator is greater than the denominator. The average impure fraction is simplified to another fraction in arithmetic calculations. The last type of fraction is a mixed fraction, which is a combination of whole numbers and pure fractions.
Here’s a full explanation.
1. Definition of Fractions
2. Types of Fractions
Fractions can be divided into three, namely:
a. Decimal Numbers or Decimal Fractions
If you want to convert the fraction to a decimal, you must divide the quantifier and denominator to 1 : 2 = 0.5. The following table will provide some examples of how to read a decimal number.
| Number | How to read |
|---|---|
| 0.5 | zero point five |
| 0.75 | zero point seventy five |
| 0.025 | zero point zero twenty five |
b. Ordinary Fractional Numbers
| Number | How to read |
|---|---|
| half or one by two | |
| one third or one third | |
| quarter or one fourth | |
| one-fifth or one-fifth | |
| one-sixth or one-sixth | |
| one-seventh or one-seventh | |
| one-eighth or one-eighth | |
| ninth or one-ninth | |
| two-thirds | |
| three-quarter |
c. Mixed Fractions
The following table will provide examples of how to read mixed fractions.
| Number | How to read |
|---|---|
| one and a half | |
| two two thirds | |
| three three by four |
3. Counting Operations in Fractions
Operations arithmetic in fractions, namely addition and subtraction, multiplication, and division.
a. sum
The properties of addition in fractions, namely:
Examples of its application, namely:
- (The direct result is simplified by converting it to a mixed number).
b. Subtraction
The properties of reduction in fractions, namely:
Examples of its application, namely:
c. Multiplication
The properties of multiplication in fractions, namely:
Examples of its application, namely:
d. Distribution
The properties of division in fractions, namely:
Fraction Multiplication Formula
If you already understand the explanation of fractions above, now we will continue by explaining the three multiplication formulas for fractions.
1. The Ordinary Fractions Multiplication Formula
This formula is the most basic fraction multiplication formula, which is to calculate the multiplication between ordinary fractions. The formula is as follows:
Take a look at the formula above! You only need to multiply the quantifiers and the denominators in the basic fraction multiplication formula. An example of the question is as follows.
3/4 x 1/2
The steps that you have to take are to multiply the numbers 3 and 1 as co-quantifiers, and multiply the numbers 4 and 2 as common denominators. The solution is as follows.
3/4 x 1/2 = 3/8
Isn’t it very easy, sinaumedia friends? Well, now we will proceed to the second formula.
2. The Formula for Multiplication of Ordinary Fractions with Whole Numbers
So, now we will discuss the multiplication formula for fractions at the next level, namely the multiplication between fractions and whole numbers. Integers are numbers whose values are whole and not in the form of fractions, namely numbers that have been commonly encountered, for example 1, 2, 3, 4, 5, and so on.
You only need to multiply the quantifier in the multiplication between whole numbers and fractions. The formula is as follows.
By the way, do sinaumedia friends know the factors that cause us to only multiply the quantifier and the reason for the constant value of the denominator?
The answer is, if an integer is converted to a fraction, the denominator will be 1, for example you will change the number 2 to a fraction, the fraction will be 2/1.
Well, now you have to recall basic arithmetic logic. Any number multiplied by 1 remains the same. Therefore, we don’t need to convert integers to fractions and just multiply the quantifiers in this formula.
Are sinaumedia Friends still confused? In the following, we will provide an example question below so that you can understand it better.
5/4 x 3
If we convert the integer 3 into a fraction, the fractional form will be 3/1. Now, let’s arrange these numbers in the multiplication formula we learned earlier.
5/4 x 3/1 = 15/4
According to the basic fraction multiplication formula, you have to multiply the other numbers in the quantifier and denominator. Therefore, you have to multiply the numbers 5 and 3 to get the number 15. Meanwhile, if we multiply 4 and 1, the result is 4. As you explained before, any number multiplied by 1 will have the same value. .
So, to make calculations easier, if you meet multiplication between fractions and whole numbers, what’s being multiplied is only the numerator.
3. Mixed Fraction Multiplication Formula
The last fraction multiplication formula is used to multiply mixed fractions among other numbers. Remember, mixed fractions are fractions that consist of whole numbers and impure fractions. Multiplication of mixed fractions is actually easy and has the same concept as multiplication of basic fractions. However, a mixed number must be simplified and converted to an improper fraction first.
In the following, we will provide an example question below so that you can understand it better.
1 ½ x 2 ¼
The first step you have to do is change each mixed fraction into an improper fraction. The formula for converting a mixed fraction to an improper fraction is as follows.
To convert a mixed number into an improper fraction, all you have to do is multiply the whole number by the denominator, then add up the result. If applied to the problem, it can be explained as follows.
1 ½ x 2 ¼
= 3/2 x 9/4
Now, if you’ve converted it into an ordinary fraction, all you have to do is multiply the two fractions according to the basic fraction multiplication formula, namely multiplying the same quantifier and denominator. The solution is as follows.
3/2 x 9/4
= 27/8
So the combined product of 1 ½ and 2 ¼ is 27/8. Remember, you still have to simplify the results above because they are still impure fractions!
Examples of Multiplication Fractions Questions and Answers
1. First Exercise
The following section will explain the discussion and answers to the multiplication of two mixed fractions in a book entitled Mathematics Class 5 SD/MI which is the work of Purnomosidi, Wiyanto, Safiroh, and Ida Gannny.
1. 2 2/3 x 5 =
= 8/3 x 5
= (8 x 5)/3
= 40/3
= 13 1/3
2. 1 4/5 x 2 =
= 9/5 x 2
= (9 x 2)/5
= 18/5
= 3 3/5
3. 2 5/8 x 6
= 21/8 x 6
= (21 x 6)/8
= 126/8
= 15 6/8
= 15 3/4
4. 1 5/7 x 4 =
= 12/7 x 4
= (12 x 4)/7
= 48/7
= 6 6/7
5. 1 7/9 x 2 =
= 16/9 x 2
= (16 x 2)/9
= 32/9
= 3 5/9
6. 5 x 1 3/7 =
= 5 x 10/7
= 50/7
= 7 1/7
7. 6 x 1 9/10 =
= 6 x 19/10
= 114/10
= 11 4/10
8. 12 x 1 4/9 =
= 12 x 13/9
= 156/9
= 17 3/9
= 17 1/3
9. 100 x 1 2/3 =
= 100 x 5/3
= 500/3
= 166 2/3
10. 2 2/3 x 1/6 =
= 8/3 x 1/6
= (8 x 1)/(3 x 6)
= 8/18
= 4/9
11. 2 4/5 x 1/8 =
= 14/5 x 1/8
= (14 x 1)/(5 x 8)
= 14/40
= 7/20
12. 1 2/7 x 2/5 =
= 9/7 x 2/5
= (9 x 2)/(7 x 5)
= 18/35
13. 1 2/7 x 2/3 =
= 9/7 x 2/3
= (9 x 2)/(7 x 3)
= 18/21 = 6/7
14. 1 7/8 x 2/5 =
= 15/8 x 2/5
= (15 x 2)/(8 x 5)
= 30/40
= 3/4
15. 2/3 x 1 5/9 =
= 2/3 x 14/9
= (2 x 14)/(3 x 9)
= 28/27
= 1 1/27
16. 2/5 x 1 3/7 =
= 2/5 x 10/7
= (2 x 10)/(5 x 7)
= 20/35
= 4/7
17. 3/4 x 2 3/10 =
= 3/4 x 23/10
= (3 x 23)/(4 x 10)
= 69/40
= 1 29/40
18. 4/5 x 1 7/8 =
= 4/5 x 15/8
= (4 x 15)/(5×8)
= 60/40 = 3/2
= 1 1/2
19. 5/8 x 1 3/4 =
= 5/8 x 7/4
= (5 x 7)/(8 x 4)
= 35/32
= 1 3/32
2. Second Exercise
1. 2/3 x 2/5 = …
2. 3/4 x 5/6 = …
3. 3/5 x 1/2 = …
4. 5/8 x 2/7 = …
5. 5/7 x 7/8 = …
6. 4/9 x 2/3 = …
7. 3/2 x 2/3 = …
8. 4/6 x 5/8 = …
9. 6/7 x 3/6 = …
10 8/9 x 3/4 = …
11. 3/4 x 1/2 x 2/3 = …
12. 1/2 x 5/6 x 2/4 = …
13. 5/7 x 1/3 x 4/5 = …
14. 4/5 x 2/3 x 3/8 = …
15. 1/2 x 1/3 x 1/5 = …
16. 2 1/4 x 3 2/3 = …
17. 3 4/5 x 5 1/6 = …
18. 5 1/2 x 2 2/3 = …
19. 2 5/7 x 3 3/4 = …
20. 4 1/8 x 1 5/7 = …
Answer Key to Multiplication Fractions Questions
1. 2/3 x 1/4 = (2 x 1) / (3 x 4) = 2/12 = 1/6
2. 3/4 x 5/6 = (3 x 5) / (4 x 6) = 15/24 = 5/8
3. 3/5 x 1/2 = (3 x 1) / (5 x 2) = 3/10
4. 5/8 x 2/7 = (5 x 2) / ( 8 x 7) = 10/56 = 5/28
5. 5/7 x 7/8 = (5 x 7) / (7 x 8) = 35/56 = 5/8
6. 4/9 x 2/3 = (4 x 2) / (9 x 3) = 8/27
7. 3/2 x 2/3 = (3 x 2) / (2 x 3) = 5/6
8. 4/6 x 5/8 = (4 x 5) / (6 x 8) = 20/48 = 5/12
9. 6/7 x 3/6 = (6 x 3) / (7 x 6) = 18/42 = 3/7
10 8/9 x 3/4 = (8 x 3) / (9 x 4) = 24/36 = 2/3
11. 3/4 x 1/2 x 2/3 = (3 x 1 x 2) / (4 x 2 x 3) = 6/24 = 1/4
12. 1/2 x 5/6 x 2/4 = (1 x 5 x 2) / (2 x 6 x 4) = 10/48 = 5 /24
13. 5/7 x 1/3 x 4/5 = (5 x 1 x 4) / (7 x 3 x 5) = 20/105 = 4/21
14. 4/5 x 2/3 x 3/8 = (4 x 2 x 3) / (5 x 3 x 8) = 24/120 = 1/5
15. 1/2 x 1/3 x 1/5 = (1 x 1 x 1) / (2 x 3 x 5) = 1/30
16. 2 1/4 x 3 2/3 = 9/4 x 11/3 = (9 x 11) / (4 x 3 ) = 99/12 = 8 3/12 = 8 1/4
17. 3 4/5 x 5 1/6 = 19/5 x 31/6 = (19 x 31) / (5 x 6) = 589/30 = 19 19/30
18. 5 1/2 x 2 2/3 = 11/2 x 8/3 = (11 x 8) / (2 x 3) = 88/6 = 14 4/6 = 14 2/3
19. 2 5/7 x 3 3/4 = 19/7 x 15/4 = (19 x 15) / (7 x 4) = 285/28 = 10 5/28
20. 4 1/8 x 1 5/ 7 = 33/8 x 12/7 = (33 x 12) / (8 x 7) = 396/56 = 7 4/56 = 7 1/14
3. Third Exercise
Examples of Common Fraction Multiplication Problems
Problem 1. Multiply common fractions.
Calculate 1/3 x 1/7 = . . .?
Answer: 1/3 x 1/7 = 1×1 / 3×7 = 1/21
Problem 2. Multiplication of simplified fractions
Find 2/5 x 7/10 = . . .?
Answer: 2/5 x 7/10 = 2×7 / 5×10 = 14/50 = 7/25
Notice, the result is 14/50. The 14/50 value can be simplified by dividing the numerator and denominator by 2 or multiplied by 1/2. Therefore, 14 : 2 = 7, and 50 : 2 = 25, so we get 7/25.
Answer = 14/50 and 7/25 have the same value.
Problem 3. Multiplication of Three Common Fractions
Calculate the product of these 3 fractions 1/2 x 4/5 x 3/8 = . . .?
Answer: The question is a multiplication of three successive fractions.
1/2x 4/5 x 3/8 = 1x4x3 / 2x5x8 = 12/80 = 3/20
Note, the payoff is 12/80. The fractional value can still be simplified to 3/20.
Well, that’s a discussion about the meaning, the three formulas for multiplication of fractions, and their understanding that you can learn and use in your daily life. Hopefully useful and see you in the next discussion. Have a good study!